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Номер 5, страница 61, часть 1 - гдз по алгебре 10 класс учебник Пак, Ардакулы
Авторы: Пак О. В., Ардакулы Д., Ескендирова Е. В.
Тип: Учебник
Издательство: Алматыкітап баспасы
Год издания: 2019 - 2026
Часть: 1
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 1. Глава 1. Функции, её свойства и график. Параграф 6. Построение графиков и исследование функций. Упражнения - номер 5, страница 61.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "61" "field_page_end" => "64" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/1-6upr-5" "field_display_title" => "5" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\Term {#1043 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ 0 => App\Models\Term {#1045 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] …23 } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:3 [ 0 => App\Models\Term {#1049 …30} 1 => App\Models\Term {#1050 …30} 2 => App\Models\Term {#1051 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 0 => App\Models\Term {#1058 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042} "field_class" => Illuminate\Database\Eloquent\Collection {#1044} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046} "field_author" => Illuminate\Database\Eloquent\Collection {#1048} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053} "field_country" => Illuminate\Database\Eloquent\Collection {#1055} "field_city" => Illuminate\Database\Eloquent\Collection {#1057} "field_series" => Illuminate\Database\Eloquent\Collection {#1059} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:4 [ 0 => App\Models\Branch {#1154 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Term {#1155 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Book {#1157 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Term {#1156 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ 0 => App\Models\Term {#1161 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Term {#1163 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almatykitap baspasy" ] #original: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almatykitap baspasy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:3 [ 0 => App\Models\Term {#1165 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1166 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #original: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1167 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1168 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:1 [ 0 => App\Models\Term {#1170 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1171 #items: array:1 [ 0 => App\Models\Term {#1172 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:1 [ 0 => App\Models\Term {#1174 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1175 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1176 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1177 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1178 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1179 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1181 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1182 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1183 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1159} "field_class" => Illuminate\Database\Eloquent\Collection {#1160} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1162} "field_author" => Illuminate\Database\Eloquent\Collection {#1164} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1168} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1169} "field_country" => Illuminate\Database\Eloquent\Collection {#1171} "field_city" => Illuminate\Database\Eloquent\Collection {#1173} "field_series" => Illuminate\Database\Eloquent\Collection {#1175} "field_umk" => Illuminate\Database\Eloquent\Collection {#1176} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1177} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1178} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1179} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1180} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1181} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1182} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1183} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1184} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1158} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1187} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1186 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Функции, её свойства и график" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Term {#1188 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "10" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1190 #items: array:1 [ 0 => App\Models\Book {#1157} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Функции, её свойства и график" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1191} "field_page_start" => "10" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1190} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1189} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1192 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119355 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Построение графиков и исследование функций" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Term {#1193 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "60" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Book {#1157} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:1 [ 0 => App\Models\Branch {#1194 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Функции, её свойства и график" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Term {#1197 …30} ] #escapeWhenCastingToString: false } "field_page_start" => "10" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:1 [ 0 => App\Models\Book {#1200 …30} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:1 [ 0 => App\Models\Branch {#1229 …30} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Функции, её свойства и график" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1198} "field_page_start" => "10" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1199} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1228} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119355 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Построение графиков и исследование функций" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1196} "field_page_start" => "60" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1195} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1236} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1234 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119356 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1235 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "60" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Book {#1157} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1246 #items: array:1 [ 0 => App\Models\Branch {#1245 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119355 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Построение графиков и исследование функций" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:1 [ 0 => App\Models\Term {#1193} ] #escapeWhenCastingToString: false } "field_page_start" => "60" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1248 #items: array:1 [ 0 => App\Models\Book {#1157} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1249 #items: array:1 [ 0 => App\Models\Branch {#1194} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119355 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Построение графиков и исследование функций" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1247} "field_page_start" => "60" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1248} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1249} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119356 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1235} "field_page_start" => "60" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1237} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1246} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:2 [ 0 => App\Models\Element {#1132 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1300347 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Edition {#1131 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129 #items: array:1 [ 0 => App\Models\Term {#1130 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1128 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1127 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1126 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1128} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1127} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1126} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1119662" "type" => "task" ] "text" => "<p>Упражнение 5</p><p>Изобразите на координатной плоскости точку $A(4,2)$. Передвиньте точку на наименьшее расстояние так, чтобы она стала в 3 раза дальше от оси Ox, чем была. Как изменились ее координаты? Повторите все действия с точками $B(-4,2)$, $C(-1,3)$, $D(5,-1.5)$, $E(-3,0)$. Преобразование, описанное в упражнении 4, будем называть <strong>сжатием</strong> в 2 раза вдоль оси Ox относительно оси Oy. Преобразование, описанное в упражнении 5, будем называть <strong>растяжением</strong> в 3 раза вдоль оси Oy относительно оси Ox.</p><p>Основной вопрос, который мы будем рассматривать: как простейшие алгебраические преобразования аналитической формулы, которой задается функция, отражаются на преобразованиях графика? При этом будем считать, что график функции $y=f(x)$ уже построен.</p><p>В следующей таблице укажем алгебраические преобразования функции $y=f(x)$, соответствующие им геометрические преобразования графика и примеры. Во всех примерах черным цветом изображен график функции, красным – результат преобразования.</p><p><strong>Функция:</strong> $y=f(x)+b$</p><p><strong>Преобразование графика и обозначение:</strong> Параллельный перенос на $b$ единиц вдоль оси Oy (P1). Примечание. Если, например, $b=-2$, то фраза «перенос на -2 единицы вдоль оси Oy» означает сдвиг вниз.</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -200 -100 L -100 50 L 0 -100 L 100 -200 L 250 -50" stroke="red" stroke-width="2" fill="none"></path><!-- Label for b=-2 --><text x="180" y="-120" font-size="14" fill="black" text-anchor="middle" transform="scale(1, -1)">b=-2</text><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-100" y1="50" x2="-100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="50" x2="0" y2="50" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="-200" x2="100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="-200" x2="0" y2="-200" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation (downward shift) --><line x1="-100" y1="125" x2="-100" y2="75" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line><line x1="100" y1="-75" x2="100" y2="-125" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=f(x+a)$</p><p><strong>Преобразование графика и обозначение:</strong> Параллельный перенос на $a$ единиц вдоль оси Ox (P2)</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-300" y1="-3" x2="-300" y2="3" stroke="black" stroke-width="1"></line><text x="-300" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-6</text><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -300 0 L -200 150 L -100 0 L 0 -100 L 150 50" stroke="red" stroke-width="2" fill="none"></path><!-- Label for a=2 --><text x="180" y="-120" font-size="14" fill="black" text-anchor="middle" transform="scale(1, -1)">a=2</text><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-200" y1="150" x2="-200" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-200" y1="150" x2="0" y2="150" stroke="red" stroke-dasharray="2,2"></line><line x1="0" y1="-100" x2="0" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="0" y1="-100" x2="0" y2="-100" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation (leftward shift) --><line x1="-125" y1="150" x2="-175" y2="150" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line><line x1="75" y="-100" x2="25" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=f(kx), k>0$</p><p><strong>Преобразование графика и обозначение:</strong> Сжатие в $k$ раз вдоль оси Ox относительно оси Oy при $k>1$; растяжение в $k$ раз вдоль Ox относительно оси Oy при $0<k<1 (P3)$</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-150" y1="-3" x2="-150" y2="3" stroke="black" stroke-width="1"></line><text x="-150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-3</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="125" y1="-3" x2="125" y2="3" stroke="black" stroke-width="1"></line><text x="125" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2.5</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -100 0 L -50 150 L 0 0 L 50 -100 L 125 50" stroke="red" stroke-width="2" fill="none"></path><!-- Label for k=2 --><text x="150" y="120" font-size="14" fill="black" text-anchor="middle" transform="scale(1, -1)">k=2</text><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-50" y1="150" x2="-50" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-50" y1="150" x2="0" y2="150" stroke="red" stroke-dasharray="2,2"></line><line x1="50" y1="-100" x2="50" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="50" y1="-100" x2="0" y2="-100" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation (horizontal compress) --><line x1="-75" y1="150" x2="-25" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line><line x1="75" y="-100" x2="25" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=kf(x), k>0$</p><p><strong>Преобразование графика и обозначение:</strong> Растяжение в $k$ раз вдоль Oy относительно оси Ox при $k>1$; сжатие $\frac{1}{k}$ раз вдоль Oy относительно оси Ox при $O<k<1 (P4)$</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><line x1="-3" y1="-300" x2="3" y2="-300" stroke="black" stroke-width="1"></line><text x="10" y="-295" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">6</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -200 0 L -100 75 L 0 0 L 100 -50 L 250 25" stroke="red" stroke-width="2" fill="none"></path><!-- Label for k=1/2 --><text x="180" y="120" font-size="14" fill="black" text-anchor="middle" transform="scale(1, -1)">k=1/2</text><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-100" y1="75" x2="-100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="75" x2="0" y2="75" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="-50" x2="100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="-50" x2="0" y2="-50" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation (vertical compress) --><line x1="-100" y1="130" x2="-100" y2="90" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line><line x1="100" y="-70" x2="100" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=-f(x)$</p><p><strong>Преобразование графика и обозначение:</strong> Симметрия относительно оси Ox (P5)</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -200 0 L -100 -150 L 0 0 L 100 100 L 250 -50" stroke="red" stroke-width="2" fill="none"></path><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-100" y1="-150" x2="-100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="-150" x2="0" y2="-150" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="100" x2="100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="100" x2="0" y2="100" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation --><line x1="-100" y1="125" x2="-100" y2="-125" stroke="red" stroke-width="2" marker-start="url(#arrowhead-red)" marker-end="url(#arrowhead-red)"></line><line x1="100" y="-75" x2="100" stroke="red" stroke-width="2" marker-start="url(#arrowhead-red)" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=f(-x)$</p><p><strong>Преобразование графика и обозначение:</strong> Симметрия относительно оси Oy (P6)</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-250" y1="-3" x2="-250" y2="3" stroke="black" stroke-width="1"></line><text x="-250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-5</text><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M 200 0 L 100 150 L 0 0 L -100 -100 L -250 50" stroke="red" stroke-width="2" fill="none"></path><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="100" y1="150" x2="100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="150" x2="0" y2="150" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="-100" x2="-100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="-100" x2="0" y2="-100" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation --><line x1="-75" y1="150" x2="75" stroke="red" stroke-width="2" marker-start="url(#arrowhead-red)" marker-end="url(#arrowhead-red)"></line><line x1="75" y="-100" x2="-75" stroke="red" stroke-width="2" marker-start="url(#arrowhead-red)" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=|f(x)|$</p><p><strong>Преобразование графика и обозначение:</strong> Часть графика, лежащая в верхней полуплоскости относительно Ox, не изменяется. Часть графика, лежащая в нижней полуплоскости, отражается симметрично относительно Ox (P7)</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -200 0 L -100 150 L 0 0 L 100 100 L 250 50" stroke="red" stroke-width="2" fill="none"></path><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="100" y1="100" x2="100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="100" x2="0" y2="100" stroke="red" stroke-dasharray="2,2"></line></g></svg><p><strong>Функция:</strong> $y=f(|x|)$</p><p><strong>Преобразование графика и обозначение:</strong> Часть графика, лежащая в правой относительно Oy полуплоскости, остается без изменений; часть графика, лежащая в левой относительно Oy полуплоскости, заменяем на симметричное отражение правой относительно Oy (P8)</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-250" y1="-3" x2="-250" y2="3" stroke="black" stroke-width="1"></line><text x="-250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-5</text><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -250 50 L -100 -100 L 0 0 L 100 -100 L 250 50" stroke="red" stroke-width="2" fill="none"></path><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-100" y1="-100" x2="-100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="-100" x2="0" y2="-100" stroke="red" stroke-dasharray="2,2"></line></g></svg><p>Докажем, что описанные преобразования корректны.</p><p><strong>1) Преобразование P1: $y=f(x)+b$.</strong> Точка $(x_0, y_0)$ принадлежит графику $y=f(x)$, т.е. $y_0=f(x_0)$. В результате сдвига на $b$ единиц вдоль оси Oy точка $(x_0, y_0)$ переходит в точку $(x_0, y_0+b)$. Подставляя эти координаты в уравнение $y=f(x)+b$, получаем равенство $y_0+b=f(x_0)+b$, которое является верным в силу того, что равенство $y_0=f(x_0)$ верно.</p><p><strong>2) Преобразование P8: $y=f(|x|)$.</strong> Это значит, что на множестве $x \ge 0$, т.е. в правой относительно Oy полуплоскости, графики $y=f(|x|)$ и $y=f(x)$ совпадают. Заметим теперь, что $f(|-x|)=f(|x|)$, так как $|-x|=|x|$. Это значит, что функция $y=f(|x|)$ – четная и ось Oy является осью симметрии ее графика. Приходится левую часть графика $y=f(x)$ заменить на симметричное отражение правой.</p><p><strong>3) Преобразование </strong>P3: $y=f(kx)$, $0<k<1.$ Пусть $(x_0, y_0)$ – точка графика $y=f(x)$. Растяжение в $\frac{1}{k}$ раз вдоль оси Ox относительно оси Oy означает, что точка $(x_0, y_0)$ переходит в точку $(\frac{1}{k}x_0, y_0)$. Для доказательства того, что эта новая точка лежит на графике $y=f(kx)$, требуется показать, что ее координаты удовлетворяют уравнению $y=f(kx)$. Действительно, подставляем вместо $y$ число $y_0$, а вместо $x$ число $\frac{1}{k}x_0$: $y_0=f(k \cdot \frac{1}{k}x_0) \Leftrightarrow y_0=f(x_0)$. Последнее равенство является верным, так как, по предположению, $(x_0, y_0)$ – точка графика функции $y=f(x)$.</p><p>Корректность остальных преобразований доказывается аналогично.</p>" "img" => array:4 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "1686" "height" => 1626 "path" => "/media/algebra_10/baspasy-u19/0-1-6upr/5-1.webp?ts=1753534528" ] 1 => array:5 [ "name" => "5-2.jpg" "alt" => null "width" => "1686" "height" => 2424 "path" => "/media/algebra_10/baspasy-u19/0-1-6upr/5-2.webp?ts=1753534528" ] 2 => array:5 [ "name" => "5-3.jpg" "alt" => null "width" => "1757" "height" => 2465 "path" => "/media/algebra_10/baspasy-u19/0-1-6upr/5-3.webp?ts=1753534528" ] 3 => array:5 [ "name" => "5-4.jpg" "alt" => null "width" => "1777" "height" => 823 "path" => "/media/algebra_10/baspasy-u19/0-1-6upr/5-4.webp?ts=1753534528" ] ] ] #original: array:7 [ "id" => 1300347 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "1119662" "type" => "task" ] "text" => "<p>Упражнение 5</p><p>Изобразите на координатной плоскости точку $A(4,2)$. Передвиньте точку на наименьшее расстояние так, чтобы она стала в 3 раза дальше от оси Ox, чем была. Как изменились ее координаты? Повторите все действия с точками $B(-4,2)$, $C(-1,3)$, $D(5,-1.5)$, $E(-3,0)$. Преобразование, описанное в упражнении 4, будем называть <strong>сжатием</strong> в 2 раза вдоль оси Ox относительно оси Oy. Преобразование, описанное в упражнении 5, будем называть <strong>растяжением</strong> в 3 раза вдоль оси Oy относительно оси Ox.</p><p>Основной вопрос, который мы будем рассматривать: как простейшие алгебраические преобразования аналитической формулы, которой задается функция, отражаются на преобразованиях графика? При этом будем считать, что график функции $y=f(x)$ уже построен.</p><p>В следующей таблице укажем алгебраические преобразования функции $y=f(x)$, соответствующие им геометрические преобразования графика и примеры. Во всех примерах черным цветом изображен график функции, красным – результат преобразования.</p><p><strong>Функция:</strong> $y=f(x)+b$</p><p><strong>Преобразование графика и обозначение:</strong> Параллельный перенос на $b$ единиц вдоль оси Oy (P1). Примечание. Если, например, $b=-2$, то фраза «перенос на -2 единицы вдоль оси Oy» означает сдвиг вниз.</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -200 -100 L -100 50 L 0 -100 L 100 -200 L 250 -50" stroke="red" stroke-width="2" fill="none"></path><!-- Label for b=-2 --><text x="180" y="-120" font-size="14" fill="black" text-anchor="middle" transform="scale(1, -1)">b=-2</text><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-100" y1="50" x2="-100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="50" x2="0" y2="50" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="-200" x2="100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="-200" x2="0" y2="-200" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation (downward shift) --><line x1="-100" y1="125" x2="-100" y2="75" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line><line x1="100" y1="-75" x2="100" y2="-125" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=f(x+a)$</p><p><strong>Преобразование графика и обозначение:</strong> Параллельный перенос на $a$ единиц вдоль оси Ox (P2)</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-300" y1="-3" x2="-300" y2="3" stroke="black" stroke-width="1"></line><text x="-300" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-6</text><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -300 0 L -200 150 L -100 0 L 0 -100 L 150 50" stroke="red" stroke-width="2" fill="none"></path><!-- Label for a=2 --><text x="180" y="-120" font-size="14" fill="black" text-anchor="middle" transform="scale(1, -1)">a=2</text><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-200" y1="150" x2="-200" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-200" y1="150" x2="0" y2="150" stroke="red" stroke-dasharray="2,2"></line><line x1="0" y1="-100" x2="0" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="0" y1="-100" x2="0" y2="-100" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation (leftward shift) --><line x1="-125" y1="150" x2="-175" y2="150" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line><line x1="75" y="-100" x2="25" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=f(kx), k>0$</p><p><strong>Преобразование графика и обозначение:</strong> Сжатие в $k$ раз вдоль оси Ox относительно оси Oy при $k>1$; растяжение в $k$ раз вдоль Ox относительно оси Oy при $0<k<1 (P3)$</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-150" y1="-3" x2="-150" y2="3" stroke="black" stroke-width="1"></line><text x="-150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-3</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="125" y1="-3" x2="125" y2="3" stroke="black" stroke-width="1"></line><text x="125" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2.5</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -100 0 L -50 150 L 0 0 L 50 -100 L 125 50" stroke="red" stroke-width="2" fill="none"></path><!-- Label for k=2 --><text x="150" y="120" font-size="14" fill="black" text-anchor="middle" transform="scale(1, -1)">k=2</text><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-50" y1="150" x2="-50" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-50" y1="150" x2="0" y2="150" stroke="red" stroke-dasharray="2,2"></line><line x1="50" y1="-100" x2="50" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="50" y1="-100" x2="0" y2="-100" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation (horizontal compress) --><line x1="-75" y1="150" x2="-25" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line><line x1="75" y="-100" x2="25" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=kf(x), k>0$</p><p><strong>Преобразование графика и обозначение:</strong> Растяжение в $k$ раз вдоль Oy относительно оси Ox при $k>1$; сжатие $\frac{1}{k}$ раз вдоль Oy относительно оси Ox при $O<k<1 (P4)$</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><line x1="-3" y1="-300" x2="3" y2="-300" stroke="black" stroke-width="1"></line><text x="10" y="-295" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">6</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -200 0 L -100 75 L 0 0 L 100 -50 L 250 25" stroke="red" stroke-width="2" fill="none"></path><!-- Label for k=1/2 --><text x="180" y="120" font-size="14" fill="black" text-anchor="middle" transform="scale(1, -1)">k=1/2</text><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-100" y1="75" x2="-100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="75" x2="0" y2="75" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="-50" x2="100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="-50" x2="0" y2="-50" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation (vertical compress) --><line x1="-100" y1="130" x2="-100" y2="90" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line><line x1="100" y="-70" x2="100" stroke="red" stroke-width="2" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=-f(x)$</p><p><strong>Преобразование графика и обозначение:</strong> Симметрия относительно оси Ox (P5)</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -200 0 L -100 -150 L 0 0 L 100 100 L 250 -50" stroke="red" stroke-width="2" fill="none"></path><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-100" y1="-150" x2="-100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="-150" x2="0" y2="-150" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="100" x2="100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="100" x2="0" y2="100" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation --><line x1="-100" y1="125" x2="-100" y2="-125" stroke="red" stroke-width="2" marker-start="url(#arrowhead-red)" marker-end="url(#arrowhead-red)"></line><line x1="100" y="-75" x2="100" stroke="red" stroke-width="2" marker-start="url(#arrowhead-red)" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=f(-x)$</p><p><strong>Преобразование графика и обозначение:</strong> Симметрия относительно оси Oy (P6)</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-250" y1="-3" x2="-250" y2="3" stroke="black" stroke-width="1"></line><text x="-250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-5</text><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M 200 0 L 100 150 L 0 0 L -100 -100 L -250 50" stroke="red" stroke-width="2" fill="none"></path><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="100" y1="150" x2="100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="150" x2="0" y2="150" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="-100" x2="-100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="-100" x2="0" y2="-100" stroke="red" stroke-dasharray="2,2"></line><!-- Arrows showing transformation --><line x1="-75" y1="150" x2="75" stroke="red" stroke-width="2" marker-start="url(#arrowhead-red)" marker-end="url(#arrowhead-red)"></line><line x1="75" y="-100" x2="-75" stroke="red" stroke-width="2" marker-start="url(#arrowhead-red)" marker-end="url(#arrowhead-red)"></line></g></svg><p><strong>Функция:</strong> $y=|f(x)|$</p><p><strong>Преобразование графика и обозначение:</strong> Часть графика, лежащая в верхней полуплоскости относительно Ox, не изменяется. Часть графика, лежащая в нижней полуплоскости, отражается симметрично относительно Ox (P7)</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -200 0 L -100 150 L 0 0 L 100 100 L 250 50" stroke="red" stroke-width="2" fill="none"></path><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="100" y1="100" x2="100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="100" y1="100" x2="0" y2="100" stroke="red" stroke-dasharray="2,2"></line></g></svg><p><strong>Функция:</strong> $y=f(|x|)$</p><p><strong>Преобразование графика и обозначение:</strong> Часть графика, лежащая в правой относительно Oy полуплоскости, остается без изменений; часть графика, лежащая в левой относительно Oy полуплоскости, заменяем на симметричное отражение правой относительно Oy (P8)</p><p><strong>Пример:</strong></p><svg width="550" height="450" viewbox="0 0 550 450"><defs><marker id="arrowhead-red" markerwidth="10" markerheight="7" refx="0" refy="3.5" orient="auto"><polygon points="0 0, 10 3.5, 0 7" fill="red"></polygon></marker></defs><g transform="translate(300, 225) scale(1, -1)"><!-- Axes --><line x1="-300" y1="0" x2="250" y2="0" stroke="black" stroke-width="1"></line><line x1="0" y1="-200" x2="0" y2="200" stroke="black" stroke-width="1"></line><!-- Arrowheads for axes --><polygon points="250,0 245,5 245,-5" fill="black"></polygon><polygon points="0,200 5,195 -5,195" fill="black"></polygon><!-- Labels for axes --><text x="240" y="-10" font-size="12" fill="black" text-anchor="middle" transform="scale(1, -1)">x</text><text x="10" y="190" font-size="12" fill="black" text-anchor="start" transform="scale(1, -1)">y</text><!-- Origin label --><text x="-5" y="-5" font-size="10" fill="black" text-anchor="end" transform="scale(1,-1)">0</text><!-- Ticks and labels --><!-- x-axis ticks --><line x1="-250" y1="-3" x2="-250" y2="3" stroke="black" stroke-width="1"></line><text x="-250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-5</text><line x1="-200" y1="-3" x2="-200" y2="3" stroke="black" stroke-width="1"></line><text x="-200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-4</text><line x1="-100" y1="-3" x2="-100" y2="3" stroke="black" stroke-width="1"></line><text x="-100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">-2</text><line x1="50" y1="-3" x2="50" y2="3" stroke="black" stroke-width="1"></line><text x="50" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">1</text><line x1="100" y1="-3" x2="100" y2="3" stroke="black" stroke-width="1"></line><text x="100" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">2</text><line x1="150" y1="-3" x2="150" y2="3" stroke="black" stroke-width="1"></line><text x="150" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">3</text><line x1="200" y1="-3" x2="200" y2="3" stroke="black" stroke-width="1"></line><text x="200" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">4</text><line x1="250" y1="-3" x2="250" y2="3" stroke="black" stroke-width="1"></line><text x="250" y="-10" font-size="10" fill="black" text-anchor="middle" transform="scale(1, -1)">5</text><!-- y-axis ticks --><line x1="-3" y1="-150" x2="3" y2="-150" stroke="black" stroke-width="1"></line><text x="10" y="-145" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">3</text><line x1="-3" y1="-100" x2="3" y2="-100" stroke="black" stroke-width="1"></line><text x="10" y="-95" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">2</text><line x1="-3" y1="-50" x2="3" y2="-50" stroke="black" stroke-width="1"></line><text x="10" y="-45" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">1</text><line x1="-3" y1="50" x2="3" y2="50" stroke="black" stroke-width="1"></line><text x="10" y="55" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-1</text><line x1="-3" y1="100" x2="3" y2="100" stroke="black" stroke-width="1"></line><text x="10" y="105" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-2</text><line x1="-3" y1="150" x2="3" y2="150" stroke="black" stroke-width="1"></line><text x="10" y="155" font-size="10" fill="black" text-anchor="start" transform="scale(1, -1)">-3</text><!-- Original function (blue) --><path d="M -200 0 L -100 150 L 0 0 L 100 -100 L 250 50" stroke="blue" stroke-width="2" fill="none"></path><!-- Transformed function (red) --><path d="M -250 50 L -100 -100 L 0 0 L 100 -100 L 250 50" stroke="red" stroke-width="2" fill="none"></path><!-- Dashed lines --><!-- Original --><line x1="-100" y1="150" x2="-100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="-100" y1="150" x2="0" y2="150" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="100" y2="0" stroke="black" stroke-dasharray="2,2"></line><line x1="100" y1="-100" x2="0" y2="-100" stroke="black" stroke-dasharray="2,2"></line><!-- Transformed --><line x1="-100" y1="-100" x2="-100" y2="0" stroke="red" stroke-dasharray="2,2"></line><line x1="-100" y1="-100" x2="0" y2="-100" stroke="red" stroke-dasharray="2,2"></line></g></svg><p>Докажем, что описанные преобразования корректны.</p><p><strong>1) Преобразование P1: $y=f(x)+b$.</strong> Точка $(x_0, y_0)$ принадлежит графику $y=f(x)$, т.е. $y_0=f(x_0)$. В результате сдвига на $b$ единиц вдоль оси Oy точка $(x_0, y_0)$ переходит в точку $(x_0, y_0+b)$. Подставляя эти координаты в уравнение $y=f(x)+b$, получаем равенство $y_0+b=f(x_0)+b$, которое является верным в силу того, что равенство $y_0=f(x_0)$ верно.</p><p><strong>2) Преобразование P8: $y=f(|x|)$.</strong> Это значит, что на множестве $x \ge 0$, т.е. в правой относительно Oy полуплоскости, графики $y=f(|x|)$ и $y=f(x)$ совпадают. Заметим теперь, что $f(|-x|)=f(|x|)$, так как $|-x|=|x|$. Это значит, что функция $y=f(|x|)$ – четная и ось Oy является осью симметрии ее графика. Приходится левую часть графика $y=f(x)$ заменить на симметричное отражение правой.</p><p><strong>3) Преобразование </strong>P3: $y=f(kx)$, $0<k<1.$ Пусть $(x_0, y_0)$ – точка графика $y=f(x)$. Растяжение в $\frac{1}{k}$ раз вдоль оси Ox относительно оси Oy означает, что точка $(x_0, y_0)$ переходит в точку $(\frac{1}{k}x_0, y_0)$. Для доказательства того, что эта новая точка лежит на графике $y=f(kx)$, требуется показать, что ее координаты удовлетворяют уравнению $y=f(kx)$. Действительно, подставляем вместо $y$ число $y_0$, а вместо $x$ число $\frac{1}{k}x_0$: $y_0=f(k \cdot \frac{1}{k}x_0) \Leftrightarrow y_0=f(x_0)$. Последнее равенство является верным, так как, по предположению, $(x_0, y_0)$ – точка графика функции $y=f(x)$.</p><p>Корректность остальных преобразований доказывается аналогично.</p>" "img" => array:4 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "1686" "height" => 1626 "path" => "/media/algebra_10/baspasy-u19/0-1-6upr/5-1.webp?ts=1753534528" ] 1 => array:5 [ "name" => "5-2.jpg" "alt" => null "width" => "1686" "height" => 2424 "path" => "/media/algebra_10/baspasy-u19/0-1-6upr/5-2.webp?ts=1753534528" ] 2 => array:5 [ "name" => "5-3.jpg" "alt" => null "width" => "1757" "height" => 2465 "path" => "/media/algebra_10/baspasy-u19/0-1-6upr/5-3.webp?ts=1753534528" ] 3 => array:5 [ "name" => "5-4.jpg" "alt" => null "width" => "1777" "height" => 823 "path" => "/media/algebra_10/baspasy-u19/0-1-6upr/5-4.webp?ts=1753534528" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1124 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1551383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1125 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Term {#1123 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1121} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1115} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1119662" "type" => "task" ] "text" => "<p>В условии задачи "передвиньте точку на наименьшее расстояние так, чтобы она стала в 3 раза дальше от оси, чем была" есть неоднозначность, от какой именно оси нужно увеличить расстояние. Условие "на наименьшее расстояние" позволяет сделать вывод, что изменять нужно расстояние до ближайшей к точке оси. Если точка равноудалена от осей или лежит на одной из них, действовать нужно по ситуации.</p>\t<p>Общий алгоритм решения для точки с координатами $(x, y)$:</p>\t<p>1. Находим расстояния до осей координат: до оси $Ox$ расстояние равно $|y|$, до оси $Oy$ расстояние равно $|x|$.</p>\t<p>2. Сравниваем эти расстояния.</p>\t<p> - Если $|y| < |x|$, точка ближе к оси $Ox$. Чтобы перемещение было минимальным, двигаем точку перпендикулярно оси $Ox$ (т.е. вдоль оси $Oy$). Новая координата $y'$ становится $3y$. Новые координаты точки: $(x, 3y)$.</p>\t<p> - Если $|x| < |y|$, точка ближе к оси $Oy$. Двигаем точку перпендикулярно оси $Oy$ (т.е. вдоль оси $Ox$). Новая координата $x'$ становится $3x$. Новые координаты точки: $(3x, y)$.</p>\t<p> - Если точка лежит на одной из осей (например, $y=0$), ее расстояние до этой оси равно нулю. Увеличить его в 3 раза невозможно ($0 \times 3 = 0$). Поэтому нужно увеличить расстояние до другой оси. Для точки $(x, 0)$ новые координаты будут $(3x, 0)$.</p>\t<p>Применим этот алгоритм к каждой из заданных точек.</p>\t<p><strong>Точка A(4, 2):</strong></p>\t<p>\t\tНачальные координаты точки $A(4, 2)$. Расстояние от точки до оси $Ox$ равно $|2| = 2$. Расстояние до оси $Oy$ равно $|4| = 4$.\t\tПоскольку $2 < 4$, точка $A$ находится ближе к оси $Ox$.\t\tСледовательно, для перемещения на наименьшее расстояние мы должны увеличить ее расстояние от оси $Ox$ в 3 раза. Это достигается растяжением вдоль оси $Oy$ в 3 раза.\t\tКоордината $x$ остается неизменной, а координата $y$ умножается на 3: $y_{new} = 2 \times 3 = 6$.\t\tНовые координаты точки будут $A'(4, 6)$.\t\t<br>\t\tОтвет: Новые координаты точки A'(4, 6).\t</p>\t<p><strong>Точка B(-4, 2):</strong></p>\t<p>\t\tНачальные координаты точки $B(-4, 2)$. Расстояние от точки до оси $Ox$ равно $|2| = 2$. Расстояние до оси $Oy$ равно $|-4| = 4$.\t\tПоскольку $2 < 4$, точка $B$ находится ближе к оси $Ox$.\t\tУвеличиваем расстояние от оси $Ox$ в 3 раза, умножая координату $y$ на 3: $y_{new} = 2 \times 3 = 6$.\t\tНовые координаты точки будут $B'(-4, 6)$.\t\t<br>\t\tОтвет: Новые координаты точки B'(-4, 6).\t</p>\t<p><strong>Точка C(-1, 3):</strong></p>\t<p>\t\tНачальные координаты точки $C(-1, 3)$. Расстояние от точки до оси $Ox$ равно $|3| = 3$. Расстояние до оси $Oy$ равно $|-1| = 1$.\t\tПоскольку $1 < 3$, точка $C$ находится ближе к оси $Oy$.\t\tУвеличиваем расстояние от оси $Oy$ в 3 раза, умножая координату $x$ на 3: $x_{new} = -1 \times 3 = -3$.\t\tНовые координаты точки будут $C'(-3, 3)$.\t\t<br>\t\tОтвет: Новые координаты точки C'(-3, 3).\t</p>\t<p><strong>Точка D(5, -1,5):</strong></p>\t<p>\t\tНачальные координаты точки $D(5, -1.5)$. Расстояние от точки до оси $Ox$ равно $|-1.5| = 1.5$. Расстояние до оси $Oy$ равно $|5| = 5$.\t\tПоскольку $1.5 < 5$, точка $D$ находится ближе к оси $Ox$.\t\tУвеличиваем расстояние от оси $Ox$ в 3 раза, умножая координату $y$ на 3: $y_{new} = -1.5 \times 3 = -4.5$.\t\tНовые координаты точки будут $D'(5, -4.5)$.\t\t<br>\t\tОтвет: Новые координаты точки D'(5, -4.5).\t</p>\t<p><strong>Точка E(-3, 0):</strong></p>\t<p>\t\tНачальные координаты точки $E(-3, 0)$. Точка лежит на оси $Ox$. Ее расстояние до оси $Ox$ равно 0. Увеличить это расстояние в 3 раза нельзя (результат останется 0).\t\tПоэтому необходимо увеличить ее расстояние до другой оси — оси $Oy$. Расстояние до оси $Oy$ равно $|-3| = 3$.\t\tУвеличиваем это расстояние в 3 раза, умножая координату $x$ на 3: $x_{new} = -3 \times 3 = -9$.\t\tНовые координаты точки будут $E'(-9, 0)$.\t\t<br>\t\tОтвет: Новые координаты точки E'(-9, 0).\t</p>" ] #original: array:6 [ "id" => 1551383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1119662" "type" => "task" ] "text" => "<p>В условии задачи "передвиньте точку на наименьшее расстояние так, чтобы она стала в 3 раза дальше от оси, чем была" есть неоднозначность, от какой именно оси нужно увеличить расстояние. Условие "на наименьшее расстояние" позволяет сделать вывод, что изменять нужно расстояние до ближайшей к точке оси. Если точка равноудалена от осей или лежит на одной из них, действовать нужно по ситуации.</p>\t<p>Общий алгоритм решения для точки с координатами $(x, y)$:</p>\t<p>1. Находим расстояния до осей координат: до оси $Ox$ расстояние равно $|y|$, до оси $Oy$ расстояние равно $|x|$.</p>\t<p>2. Сравниваем эти расстояния.</p>\t<p> - Если $|y| < |x|$, точка ближе к оси $Ox$. Чтобы перемещение было минимальным, двигаем точку перпендикулярно оси $Ox$ (т.е. вдоль оси $Oy$). Новая координата $y'$ становится $3y$. Новые координаты точки: $(x, 3y)$.</p>\t<p> - Если $|x| < |y|$, точка ближе к оси $Oy$. Двигаем точку перпендикулярно оси $Oy$ (т.е. вдоль оси $Ox$). Новая координата $x'$ становится $3x$. Новые координаты точки: $(3x, y)$.</p>\t<p> - Если точка лежит на одной из осей (например, $y=0$), ее расстояние до этой оси равно нулю. Увеличить его в 3 раза невозможно ($0 \times 3 = 0$). Поэтому нужно увеличить расстояние до другой оси. Для точки $(x, 0)$ новые координаты будут $(3x, 0)$.</p>\t<p>Применим этот алгоритм к каждой из заданных точек.</p>\t<p><strong>Точка A(4, 2):</strong></p>\t<p>\t\tНачальные координаты точки $A(4, 2)$. Расстояние от точки до оси $Ox$ равно $|2| = 2$. Расстояние до оси $Oy$ равно $|4| = 4$.\t\tПоскольку $2 < 4$, точка $A$ находится ближе к оси $Ox$.\t\tСледовательно, для перемещения на наименьшее расстояние мы должны увеличить ее расстояние от оси $Ox$ в 3 раза. Это достигается растяжением вдоль оси $Oy$ в 3 раза.\t\tКоордината $x$ остается неизменной, а координата $y$ умножается на 3: $y_{new} = 2 \times 3 = 6$.\t\tНовые координаты точки будут $A'(4, 6)$.\t\t<br>\t\tОтвет: Новые координаты точки A'(4, 6).\t</p>\t<p><strong>Точка B(-4, 2):</strong></p>\t<p>\t\tНачальные координаты точки $B(-4, 2)$. Расстояние от точки до оси $Ox$ равно $|2| = 2$. Расстояние до оси $Oy$ равно $|-4| = 4$.\t\tПоскольку $2 < 4$, точка $B$ находится ближе к оси $Ox$.\t\tУвеличиваем расстояние от оси $Ox$ в 3 раза, умножая координату $y$ на 3: $y_{new} = 2 \times 3 = 6$.\t\tНовые координаты точки будут $B'(-4, 6)$.\t\t<br>\t\tОтвет: Новые координаты точки B'(-4, 6).\t</p>\t<p><strong>Точка C(-1, 3):</strong></p>\t<p>\t\tНачальные координаты точки $C(-1, 3)$. Расстояние от точки до оси $Ox$ равно $|3| = 3$. Расстояние до оси $Oy$ равно $|-1| = 1$.\t\tПоскольку $1 < 3$, точка $C$ находится ближе к оси $Oy$.\t\tУвеличиваем расстояние от оси $Oy$ в 3 раза, умножая координату $x$ на 3: $x_{new} = -1 \times 3 = -3$.\t\tНовые координаты точки будут $C'(-3, 3)$.\t\t<br>\t\tОтвет: Новые координаты точки C'(-3, 3).\t</p>\t<p><strong>Точка D(5, -1,5):</strong></p>\t<p>\t\tНачальные координаты точки $D(5, -1.5)$. Расстояние от точки до оси $Ox$ равно $|-1.5| = 1.5$. Расстояние до оси $Oy$ равно $|5| = 5$.\t\tПоскольку $1.5 < 5$, точка $D$ находится ближе к оси $Ox$.\t\tУвеличиваем расстояние от оси $Ox$ в 3 раза, умножая координату $y$ на 3: $y_{new} = -1.5 \times 3 = -4.5$.\t\tНовые координаты точки будут $D'(5, -4.5)$.\t\t<br>\t\tОтвет: Новые координаты точки D'(5, -4.5).\t</p>\t<p><strong>Точка E(-3, 0):</strong></p>\t<p>\t\tНачальные координаты точки $E(-3, 0)$. Точка лежит на оси $Ox$. Ее расстояние до оси $Ox$ равно 0. Увеличить это расстояние в 3 раза нельзя (результат останется 0).\t\tПоэтому необходимо увеличить ее расстояние до другой оси — оси $Oy$. Расстояние до оси $Oy$ равно $|-3| = 3$.\t\tУвеличиваем это расстояние в 3 раза, умножая координату $x$ на 3: $x_{new} = -3 \times 3 = -9$.\t\tНовые координаты точки будут $E'(-9, 0)$.\t\t<br>\t\tОтвет: Новые координаты точки E'(-9, 0).\t</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1119663" "type" => "task" ] "previous" => array:2 [ "refs" => "1119661" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Book {#1157} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1141 #items: array:1 [ 0 => App\Models\BookPage {#1108 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1099214 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "61" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-61" "field_display_title" => "61" "field_folder" => "2" "field_image_name" => "61" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Branch {#1118 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Term {#1117 …30} ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1114 #items: array:1 [ 0 => App\Models\Book {#1111 …30} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => 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№5 (с. 61)
Условие. №5 (с. 61)
скриншот условия
Упражнение 5
Изобразите на координатной плоскости точку $A(4,2)$. Передвиньте точку на наименьшее расстояние так, чтобы она стала в 3 раза дальше от оси Ox, чем была. Как изменились ее координаты? Повторите все действия с точками $B(-4,2)$, $C(-1,3)$, $D(5,-1.5)$, $E(-3,0)$. Преобразование, описанное в упражнении 4, будем называть сжатием в 2 раза вдоль оси Ox относительно оси Oy. Преобразование, описанное в упражнении 5, будем называть растяжением в 3 раза вдоль оси Oy относительно оси Ox.
Основной вопрос, который мы будем рассматривать: как простейшие алгебраические преобразования аналитической формулы, которой задается функция, отражаются на преобразованиях графика? При этом будем считать, что график функции $y=f(x)$ уже построен.
В следующей таблице укажем алгебраические преобразования функции $y=f(x)$, соответствующие им геометрические преобразования графика и примеры. Во всех примерах черным цветом изображен график функции, красным – результат преобразования.
Функция: $y=f(x)+b$
Преобразование графика и обозначение: Параллельный перенос на $b$ единиц вдоль оси Oy (P1). Примечание. Если, например, $b=-2$, то фраза «перенос на -2 единицы вдоль оси Oy» означает сдвиг вниз.
Пример:
Функция: $y=f(x+a)$
Преобразование графика и обозначение: Параллельный перенос на $a$ единиц вдоль оси Ox (P2)
Пример:
Функция: $y=f(kx), k>0$
Преобразование графика и обозначение: Сжатие в $k$ раз вдоль оси Ox относительно оси Oy при $k>1$; растяжение в $k$ раз вдоль Ox относительно оси Oy при $0<k<1 (P3)$
Пример:
Функция: $y=kf(x), k>0$
Преобразование графика и обозначение: Растяжение в $k$ раз вдоль Oy относительно оси Ox при $k>1$; сжатие $\frac{1}{k}$ раз вдоль Oy относительно оси Ox при $O<k<1 (P4)$
Пример:
Функция: $y=-f(x)$
Преобразование графика и обозначение: Симметрия относительно оси Ox (P5)
Пример:
Функция: $y=f(-x)$
Преобразование графика и обозначение: Симметрия относительно оси Oy (P6)
Пример:
Функция: $y=|f(x)|$
Преобразование графика и обозначение: Часть графика, лежащая в верхней полуплоскости относительно Ox, не изменяется. Часть графика, лежащая в нижней полуплоскости, отражается симметрично относительно Ox (P7)
Пример:
Функция: $y=f(|x|)$
Преобразование графика и обозначение: Часть графика, лежащая в правой относительно Oy полуплоскости, остается без изменений; часть графика, лежащая в левой относительно Oy полуплоскости, заменяем на симметричное отражение правой относительно Oy (P8)
Пример:
Докажем, что описанные преобразования корректны.
1) Преобразование P1: $y=f(x)+b$. Точка $(x_0, y_0)$ принадлежит графику $y=f(x)$, т.е. $y_0=f(x_0)$. В результате сдвига на $b$ единиц вдоль оси Oy точка $(x_0, y_0)$ переходит в точку $(x_0, y_0+b)$. Подставляя эти координаты в уравнение $y=f(x)+b$, получаем равенство $y_0+b=f(x_0)+b$, которое является верным в силу того, что равенство $y_0=f(x_0)$ верно.
2) Преобразование P8: $y=f(|x|)$. Это значит, что на множестве $x \ge 0$, т.е. в правой относительно Oy полуплоскости, графики $y=f(|x|)$ и $y=f(x)$ совпадают. Заметим теперь, что $f(|-x|)=f(|x|)$, так как $|-x|=|x|$. Это значит, что функция $y=f(|x|)$ – четная и ось Oy является осью симметрии ее графика. Приходится левую часть графика $y=f(x)$ заменить на симметричное отражение правой.
3) Преобразование P3: $y=f(kx)$, $0<k<1.$ Пусть $(x_0, y_0)$ – точка графика $y=f(x)$. Растяжение в $\frac{1}{k}$ раз вдоль оси Ox относительно оси Oy означает, что точка $(x_0, y_0)$ переходит в точку $(\frac{1}{k}x_0, y_0)$. Для доказательства того, что эта новая точка лежит на графике $y=f(kx)$, требуется показать, что ее координаты удовлетворяют уравнению $y=f(kx)$. Действительно, подставляем вместо $y$ число $y_0$, а вместо $x$ число $\frac{1}{k}x_0$: $y_0=f(k \cdot \frac{1}{k}x_0) \Leftrightarrow y_0=f(x_0)$. Последнее равенство является верным, так как, по предположению, $(x_0, y_0)$ – точка графика функции $y=f(x)$.
Корректность остальных преобразований доказывается аналогично.
Решение 2 (rus). №5 (с. 61)
В условии задачи "передвиньте точку на наименьшее расстояние так, чтобы она стала в 3 раза дальше от оси, чем была" есть неоднозначность, от какой именно оси нужно увеличить расстояние. Условие "на наименьшее расстояние" позволяет сделать вывод, что изменять нужно расстояние до ближайшей к точке оси. Если точка равноудалена от осей или лежит на одной из них, действовать нужно по ситуации.
Общий алгоритм решения для точки с координатами $(x, y)$:
1. Находим расстояния до осей координат: до оси $Ox$ расстояние равно $|y|$, до оси $Oy$ расстояние равно $|x|$.
2. Сравниваем эти расстояния.
- Если $|y| < |x|$, точка ближе к оси $Ox$. Чтобы перемещение было минимальным, двигаем точку перпендикулярно оси $Ox$ (т.е. вдоль оси $Oy$). Новая координата $y'$ становится $3y$. Новые координаты точки: $(x, 3y)$.
- Если $|x| < |y|$, точка ближе к оси $Oy$. Двигаем точку перпендикулярно оси $Oy$ (т.е. вдоль оси $Ox$). Новая координата $x'$ становится $3x$. Новые координаты точки: $(3x, y)$.
- Если точка лежит на одной из осей (например, $y=0$), ее расстояние до этой оси равно нулю. Увеличить его в 3 раза невозможно ($0 \times 3 = 0$). Поэтому нужно увеличить расстояние до другой оси. Для точки $(x, 0)$ новые координаты будут $(3x, 0)$.
Применим этот алгоритм к каждой из заданных точек.
Точка A(4, 2):
Начальные координаты точки $A(4, 2)$. Расстояние от точки до оси $Ox$ равно $|2| = 2$. Расстояние до оси $Oy$ равно $|4| = 4$. Поскольку $2 < 4$, точка $A$ находится ближе к оси $Ox$. Следовательно, для перемещения на наименьшее расстояние мы должны увеличить ее расстояние от оси $Ox$ в 3 раза. Это достигается растяжением вдоль оси $Oy$ в 3 раза. Координата $x$ остается неизменной, а координата $y$ умножается на 3: $y_{new} = 2 \times 3 = 6$. Новые координаты точки будут $A'(4, 6)$.
Ответ: Новые координаты точки A'(4, 6).
Точка B(-4, 2):
Начальные координаты точки $B(-4, 2)$. Расстояние от точки до оси $Ox$ равно $|2| = 2$. Расстояние до оси $Oy$ равно $|-4| = 4$. Поскольку $2 < 4$, точка $B$ находится ближе к оси $Ox$. Увеличиваем расстояние от оси $Ox$ в 3 раза, умножая координату $y$ на 3: $y_{new} = 2 \times 3 = 6$. Новые координаты точки будут $B'(-4, 6)$.
Ответ: Новые координаты точки B'(-4, 6).
Точка C(-1, 3):
Начальные координаты точки $C(-1, 3)$. Расстояние от точки до оси $Ox$ равно $|3| = 3$. Расстояние до оси $Oy$ равно $|-1| = 1$. Поскольку $1 < 3$, точка $C$ находится ближе к оси $Oy$. Увеличиваем расстояние от оси $Oy$ в 3 раза, умножая координату $x$ на 3: $x_{new} = -1 \times 3 = -3$. Новые координаты точки будут $C'(-3, 3)$.
Ответ: Новые координаты точки C'(-3, 3).
Точка D(5, -1,5):
Начальные координаты точки $D(5, -1.5)$. Расстояние от точки до оси $Ox$ равно $|-1.5| = 1.5$. Расстояние до оси $Oy$ равно $|5| = 5$. Поскольку $1.5 < 5$, точка $D$ находится ближе к оси $Ox$. Увеличиваем расстояние от оси $Ox$ в 3 раза, умножая координату $y$ на 3: $y_{new} = -1.5 \times 3 = -4.5$. Новые координаты точки будут $D'(5, -4.5)$.
Ответ: Новые координаты точки D'(5, -4.5).
Точка E(-3, 0):
Начальные координаты точки $E(-3, 0)$. Точка лежит на оси $Ox$. Ее расстояние до оси $Ox$ равно 0. Увеличить это расстояние в 3 раза нельзя (результат останется 0). Поэтому необходимо увеличить ее расстояние до другой оси — оси $Oy$. Расстояние до оси $Oy$ равно $|-3| = 3$. Увеличиваем это расстояние в 3 раза, умножая координату $x$ на 3: $x_{new} = -3 \times 3 = -9$. Новые координаты точки будут $E'(-9, 0)$.
Ответ: Новые координаты точки E'(-9, 0).
Другие задания:
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