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Номер 12, страница 69, 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. Задачи - номер 12, страница 69.
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" => 1119676 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/1-6zad-12" "field_display_title" => "12" "field_outside_task" => null "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: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #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_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 #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 [ …6] …17 } ] #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 {#1187 #items: array:4 [ 0 => App\Models\Branch {#1197 #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 {#1201 #items: array:1 [ 0 => App\Models\Term {#1198 #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 {#1230 #items: array:1 [ 0 => App\Models\Book {#1200 #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 {#1202 #items: array:1 [ 0 => App\Models\Term {#1199 #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 {#1203 #items: array:1 [ 0 => App\Models\Term {#1204 #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 {#1205 #items: array:1 [ 0 => App\Models\Term {#1206 #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 {#1207 #items: array:3 [ 0 => App\Models\Term {#1208 #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 {#1209 #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 {#1210 #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 {#1211 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1212 #items: array:1 [ 0 => App\Models\Term {#1213 #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 {#1214 #items: array:1 [ 0 => App\Models\Term {#1215 #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 {#1216 #items: array:1 [ 0 => App\Models\Term {#1217 #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 {#1218 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1219 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1220 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1221 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1222 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1223 #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 {#1224 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1225 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1226 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1227 #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 {#1202} "field_class" => Illuminate\Database\Eloquent\Collection {#1203} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1205} "field_author" => Illuminate\Database\Eloquent\Collection {#1207} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1211} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1212} "field_country" => Illuminate\Database\Eloquent\Collection {#1214} "field_city" => Illuminate\Database\Eloquent\Collection {#1216} "field_series" => Illuminate\Database\Eloquent\Collection {#1218} "field_umk" => Illuminate\Database\Eloquent\Collection {#1219} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1220} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1221} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1222} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1223} "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 {#1224} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1225} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1226} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1227} "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 {#1228 #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 {#1201} "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 {#1230} "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 => "*" ] } 1 => App\Models\Branch {#1229 #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 {#1234 #items: array:1 [ 0 => App\Models\Term {#1231 #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 {#1233 #items: array:1 [ 0 => App\Models\Book {#1200} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1232 #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 {#1234} "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 {#1233} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1232} ] #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 {#1235 #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 {#1239 #items: array:1 [ 0 => App\Models\Term {#1236 #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 {#1238 #items: array:1 [ 0 => App\Models\Book {#1200} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1279 #items: array:1 [ 0 => App\Models\Branch {#1237 #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 {#1241 #items: array:1 [ 0 => App\Models\Term {#1240 …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 {#1242 #items: array:1 [ 0 => App\Models\Book {#1243 …30} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1271 #items: array:1 [ 0 => App\Models\Branch {#1272 …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 {#1241} "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 {#1242} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1271} ] #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 {#1239} "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 {#1238} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1279} ] #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 {#1277 #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" => 1119357 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1278 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "67" "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 {#1280 #items: array:1 [ 0 => App\Models\Book {#1200} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1326 #items: array:1 [ 0 => App\Models\Branch {#1281 #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 {#1283 #items: array:1 [ 0 => App\Models\Term {#1282 …30} ] #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 {#1284 #items: array:1 [ 0 => App\Models\Book {#1285 …30} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1313 #items: array:1 [ 0 => App\Models\Branch {#1314 …30} ] #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 {#1283} "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 {#1284} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1313} ] #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" => 1119357 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1278} "field_page_start" => "67" "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 {#1280} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1326} ] #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 {#1142 #items: array:2 [ 0 => App\Models\Element {#1172 #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" => 1300361 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Edition {#1171 #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 {#1169 #items: array:1 [ 0 => App\Models\Term {#1170 …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 {#1168 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1167 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166 #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 {#1169} "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 {#1168} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1167} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166} "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" => "1119676" "type" => "task" ] "text" => "<p><strong>12. (1)</strong> Постройте графики функций на одной плоскости:</p><p><strong>а)</strong> $y=-x$;</p><p><strong>б)</strong> $y=-x-3$;</p><p><strong>в)</strong> $y=|-x-3|$;</p><p><strong>г)</strong> $y=|x+3|-5$;</p><p><strong>д)</strong> $y=||x+3|-5||$;</p><p><strong>е)</strong> $y=||-x+3|-5||$;</p><p><strong>ж)</strong> $y=||3-|x|-5||$. По графику ж) сделайте исследование функции.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "12-1.jpg" "alt" => null "width" => "1537" "height" => 419 "path" => "/media/algebra_10/baspasy-u19/0-1-6zad/12-1.webp?ts=1753534544" ] ] ] #original: array:7 [ "id" => 1300361 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1162} "task" => array:2 [ "refs" => "1119676" "type" => "task" ] "text" => "<p><strong>12. (1)</strong> Постройте графики функций на одной плоскости:</p><p><strong>а)</strong> $y=-x$;</p><p><strong>б)</strong> $y=-x-3$;</p><p><strong>в)</strong> $y=|-x-3|$;</p><p><strong>г)</strong> $y=|x+3|-5$;</p><p><strong>д)</strong> $y=||x+3|-5||$;</p><p><strong>е)</strong> $y=||-x+3|-5||$;</p><p><strong>ж)</strong> $y=||3-|x|-5||$. По графику ж) сделайте исследование функции.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "12-1.jpg" "alt" => null "width" => "1537" "height" => 419 "path" => "/media/algebra_10/baspasy-u19/0-1-6zad/12-1.webp?ts=1753534544" ] ] ] #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 {#1165 #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" => 1551397 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Edition {#1164 #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 {#1160 #items: array:1 [ 0 => App\Models\Term {#1163 …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 {#1161 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1158 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1159 #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 {#1160} "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 {#1161} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1158} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1159} "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" => "1119676" "type" => "task" ] "text" => "<p><strong>а)</strong> Функция $y=-x$. Это линейная функция, её график — прямая линия, которая проходит через начало координат (0,0) и точку (1,-1). Она является биссектрисой II и IV координатных четвертей.<br>Ответ: График функции $y=-x$ — это прямая, проходящая через точки (0,0) и (1,-1).</p><p><strong>б)</strong> Функция $y=-x-3$. График этой функции получается из графика функции $y=-x$ (из пункта а) путем параллельного переноса (сдвига) вдоль оси Oy на 3 единицы вниз. Прямая будет проходить через точки (0, -3) и (-3, 0).<br>Ответ: График функции $y=-x-3$ — это прямая, полученная сдвигом прямой $y=-x$ на 3 единицы вниз.</p><p><strong>в)</strong> Функция $y=|-x-3|$. Используя свойство модуля $|-a|=|a|$, можем переписать функцию как $y=|-(x+3)|=|x+3|$. График этой функции получается из графика базовой функции $y=|x|$ (график в виде "галочки" с вершиной в начале координат) путем сдвига на 3 единицы влево вдоль оси Ox. Вершина графика окажется в точке (-3, 0).<br>Альтернативно, график $y=|-x-3|$ можно получить из графика $y=-x-3$ (из пункта б), отразив ту часть графика, которая лежит ниже оси Ox, симметрично относительно оси Ox.<br>Ответ: График функции $y=|-x-3|$ эквивалентен графику $y=|x+3|$, который представляет собой график $y=|x|$, сдвинутый на 3 единицы влево.</p><p><strong>г)</strong> Функция $y=|x+3|-5$. График этой функции получается из графика функции $y=|x+3|$ (из пункта в) путем сдвига на 5 единиц вниз вдоль оси Oy. Вершина графика сместится из точки (-3, 0) в точку (-3, -5).<br>Ответ: График функции $y=|x+3|-5$ — это график $y=|x+3|$, сдвинутый на 5 единиц вниз.</p><p><strong>д)</strong> Функция $y=||x+3|-5|$. График этой функции получается из графика функции $y=|x+3|-5$ (из пункта г) путем симметричного отражения относительно оси Ox той части графика, которая лежит ниже оси Ox. Часть графика, которая находится выше или на оси Ox, остается без изменений. Вершина в точке (-3, -5) отразится в точку (-3, 5).<br>Ответ: График функции $y=||x+3|-5|$ получается из графика $y=|x+3|-5$ отражением его отрицательной части ($y<0$) относительно оси Ox.</p><p><strong>е)</strong> Функция $y=|-|x+3|-5|$. Проанализируем выражение под внешним модулем. Так как $|x+3| \ge 0$, то $-|x+3| \le 0$. Следовательно, выражение $-|x+3|-5$ всегда отрицательно (его значение не превышает -5). По определению модуля, $|a|=-a$ для $a \le 0$. Таким образом, $y = -(-|x+3|-5) = |x+3|+5$.<br>График этой функции получается из графика $y=|x+3|$ (из пункта в) сдвигом на 5 единиц вверх вдоль оси Oy. Вершина будет в точке (-3, 5).<br>Ответ: График функции $y=|-|x+3|-5|$ совпадает с графиком функции $y=|x+3|+5$, который получается сдвигом графика $y=|x+3|$ на 5 единиц вверх.</p><p><strong>ж)</strong> Построим график функции $y=||3-|x||-5|$ и проведем его исследование. Построение будем выполнять последовательно, используя преобразования графиков:<br>1. $y=|x|$ — базовый график, "галочка" с вершиной в (0,0).<br>2. $y=-|x|$ — график $y=|x|$, отраженный симметрично относительно оси Ox. "Галочка", перевернутая вниз.<br>3. $y=3-|x|$ — график $y=-|x|$, сдвинутый на 3 единицы вверх по оси Oy. Вершина в точке (0,3).<br>4. $y=|3-|x||$ — часть графика $y=3-|x|$, лежащая ниже оси Ox (при $|x|>3$), отражается симметрично относительно оси Ox. Получается график в форме буквы "W" с пиком в (0,3) и изломами в точках (-3,0) и (3,0).<br>5. $y=|3-|x||-5$ — график $y=|3-|x||$, сдвинутый на 5 единиц вниз по оси Oy. Пик "W" смещается в (0,-2), а точки излома в (-3,-5) и (3,-5).<br>6. $y=||3-|x||-5|$ — итоговый график. Часть графика $y=|3-|x||-5$, лежащая ниже оси Ox, отражается симметрично относительно этой оси. Точки пересечения с осью Ox ($x=\pm8$) остаются на месте. Точка (0,-2) переходит в (0,2). Точки (-3,-5) и (3,-5) переходят в пики (3,5) и (-3,5).<br><br><b>Исследование функции $y=||3-|x||-5|$ по графику:</b><br>1. <b>Область определения:</b> $D(y) = (-\infty, +\infty)$, так как выражение определено для любого действительного $x$.<br>2. <b>Область значений:</b> Функция является модулем, поэтому её значения неотрицательны. Минимальное значение равно 0. Максимального значения нет. $E(y) = [0, +\infty)$.<br>3. <b>Четность:</b> $y(-x) = ||3-|-x||-5| = ||3-|x||-5| = y(x)$. Функция является четной. Ее график симметричен относительно оси Oy.<br>4. <b>Нули функции:</b> $y=0$ при $||3-|x||-5|=0 \implies |3-|x||=5$. Это уравнение равносильно совокупности $3-|x|=5$ (что дает $|x|=-2$, нет решений) и $3-|x|=-5$ (что дает $|x|=8$). Нули функции: $x=-8$ и $x=8$.<br>5. <b>Промежутки знакопостоянства:</b> Так как $y \ge 0$ для всех $x$, то $y>0$ при $x \in (-\infty, -8) \cup (-8, 8) \cup (8, +\infty)$.<br>6. <b>Промежутки монотонности:</b><br> • Функция возрастает на промежутках $[-8, -3]$, $[0, 3]$ и $[8, +\infty)$.<br> • Функция убывает на промежутках $(-\infty, -8]$, $[-3, 0]$ и $[3, 8]$.<br>7. <b>Экстремумы:</b><br> • Точки локального минимума: $x_{min}=-8$, $x_{min}=0$, $x_{min}=8$.<br> Значения в точках минимума: $y(\pm8)=0$, $y(0)=2$.<br> • Точки локального максимума: $x_{max}=-3$, $x_{max}=3$.<br> Значения в точках максимума: $y(\pm3)=5$.<br> • Наименьшее значение функции равно 0 (достигается в точках $x=\pm8$).<br> • Наибольшего значения функция не имеет.<br><br>Ответ: График функции $y=||3-|x||-5|$ строится последовательными преобразованиями. Свойства функции: $D(y)=(-\infty, +\infty)$, $E(y)=[0, +\infty)$, функция четная, нули $x=\pm8$, возрастает на $[-8, -3]\cup[0, 3]\cup[8, \infty)$, убывает на $(-\infty, -8]\cup[-3, 0]\cup[3, 8]$, локальные минимумы в точках $x=0$ (значение 2) и $x=\pm8$ (значение 0), локальные максимумы в точках $x=\pm3$ (значение 5).</p>" ] #original: array:6 [ "id" => 1551397 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156} "task" => array:2 [ "refs" => "1119676" "type" => "task" ] "text" => "<p><strong>а)</strong> Функция $y=-x$. Это линейная функция, её график — прямая линия, которая проходит через начало координат (0,0) и точку (1,-1). Она является биссектрисой II и IV координатных четвертей.<br>Ответ: График функции $y=-x$ — это прямая, проходящая через точки (0,0) и (1,-1).</p><p><strong>б)</strong> Функция $y=-x-3$. График этой функции получается из графика функции $y=-x$ (из пункта а) путем параллельного переноса (сдвига) вдоль оси Oy на 3 единицы вниз. Прямая будет проходить через точки (0, -3) и (-3, 0).<br>Ответ: График функции $y=-x-3$ — это прямая, полученная сдвигом прямой $y=-x$ на 3 единицы вниз.</p><p><strong>в)</strong> Функция $y=|-x-3|$. Используя свойство модуля $|-a|=|a|$, можем переписать функцию как $y=|-(x+3)|=|x+3|$. График этой функции получается из графика базовой функции $y=|x|$ (график в виде "галочки" с вершиной в начале координат) путем сдвига на 3 единицы влево вдоль оси Ox. Вершина графика окажется в точке (-3, 0).<br>Альтернативно, график $y=|-x-3|$ можно получить из графика $y=-x-3$ (из пункта б), отразив ту часть графика, которая лежит ниже оси Ox, симметрично относительно оси Ox.<br>Ответ: График функции $y=|-x-3|$ эквивалентен графику $y=|x+3|$, который представляет собой график $y=|x|$, сдвинутый на 3 единицы влево.</p><p><strong>г)</strong> Функция $y=|x+3|-5$. График этой функции получается из графика функции $y=|x+3|$ (из пункта в) путем сдвига на 5 единиц вниз вдоль оси Oy. Вершина графика сместится из точки (-3, 0) в точку (-3, -5).<br>Ответ: График функции $y=|x+3|-5$ — это график $y=|x+3|$, сдвинутый на 5 единиц вниз.</p><p><strong>д)</strong> Функция $y=||x+3|-5|$. График этой функции получается из графика функции $y=|x+3|-5$ (из пункта г) путем симметричного отражения относительно оси Ox той части графика, которая лежит ниже оси Ox. Часть графика, которая находится выше или на оси Ox, остается без изменений. Вершина в точке (-3, -5) отразится в точку (-3, 5).<br>Ответ: График функции $y=||x+3|-5|$ получается из графика $y=|x+3|-5$ отражением его отрицательной части ($y<0$) относительно оси Ox.</p><p><strong>е)</strong> Функция $y=|-|x+3|-5|$. Проанализируем выражение под внешним модулем. Так как $|x+3| \ge 0$, то $-|x+3| \le 0$. Следовательно, выражение $-|x+3|-5$ всегда отрицательно (его значение не превышает -5). По определению модуля, $|a|=-a$ для $a \le 0$. Таким образом, $y = -(-|x+3|-5) = |x+3|+5$.<br>График этой функции получается из графика $y=|x+3|$ (из пункта в) сдвигом на 5 единиц вверх вдоль оси Oy. Вершина будет в точке (-3, 5).<br>Ответ: График функции $y=|-|x+3|-5|$ совпадает с графиком функции $y=|x+3|+5$, который получается сдвигом графика $y=|x+3|$ на 5 единиц вверх.</p><p><strong>ж)</strong> Построим график функции $y=||3-|x||-5|$ и проведем его исследование. Построение будем выполнять последовательно, используя преобразования графиков:<br>1. $y=|x|$ — базовый график, "галочка" с вершиной в (0,0).<br>2. $y=-|x|$ — график $y=|x|$, отраженный симметрично относительно оси Ox. "Галочка", перевернутая вниз.<br>3. $y=3-|x|$ — график $y=-|x|$, сдвинутый на 3 единицы вверх по оси Oy. Вершина в точке (0,3).<br>4. $y=|3-|x||$ — часть графика $y=3-|x|$, лежащая ниже оси Ox (при $|x|>3$), отражается симметрично относительно оси Ox. Получается график в форме буквы "W" с пиком в (0,3) и изломами в точках (-3,0) и (3,0).<br>5. $y=|3-|x||-5$ — график $y=|3-|x||$, сдвинутый на 5 единиц вниз по оси Oy. Пик "W" смещается в (0,-2), а точки излома в (-3,-5) и (3,-5).<br>6. $y=||3-|x||-5|$ — итоговый график. Часть графика $y=|3-|x||-5$, лежащая ниже оси Ox, отражается симметрично относительно этой оси. Точки пересечения с осью Ox ($x=\pm8$) остаются на месте. Точка (0,-2) переходит в (0,2). Точки (-3,-5) и (3,-5) переходят в пики (3,5) и (-3,5).<br><br><b>Исследование функции $y=||3-|x||-5|$ по графику:</b><br>1. <b>Область определения:</b> $D(y) = (-\infty, +\infty)$, так как выражение определено для любого действительного $x$.<br>2. <b>Область значений:</b> Функция является модулем, поэтому её значения неотрицательны. Минимальное значение равно 0. Максимального значения нет. $E(y) = [0, +\infty)$.<br>3. <b>Четность:</b> $y(-x) = ||3-|-x||-5| = ||3-|x||-5| = y(x)$. Функция является четной. Ее график симметричен относительно оси Oy.<br>4. <b>Нули функции:</b> $y=0$ при $||3-|x||-5|=0 \implies |3-|x||=5$. Это уравнение равносильно совокупности $3-|x|=5$ (что дает $|x|=-2$, нет решений) и $3-|x|=-5$ (что дает $|x|=8$). Нули функции: $x=-8$ и $x=8$.<br>5. <b>Промежутки знакопостоянства:</b> Так как $y \ge 0$ для всех $x$, то $y>0$ при $x \in (-\infty, -8) \cup (-8, 8) \cup (8, +\infty)$.<br>6. <b>Промежутки монотонности:</b><br> • Функция возрастает на промежутках $[-8, -3]$, $[0, 3]$ и $[8, +\infty)$.<br> • Функция убывает на промежутках $(-\infty, -8]$, $[-3, 0]$ и $[3, 8]$.<br>7. <b>Экстремумы:</b><br> • Точки локального минимума: $x_{min}=-8$, $x_{min}=0$, $x_{min}=8$.<br> Значения в точках минимума: $y(\pm8)=0$, $y(0)=2$.<br> • Точки локального максимума: $x_{max}=-3$, $x_{max}=3$.<br> Значения в точках максимума: $y(\pm3)=5$.<br> • Наименьшее значение функции равно 0 (достигается в точках $x=\pm8$).<br> • Наибольшего значения функция не имеет.<br><br>Ответ: График функции $y=||3-|x||-5|$ строится последовательными преобразованиями. Свойства функции: $D(y)=(-\infty, +\infty)$, $E(y)=[0, +\infty)$, функция четная, нули $x=\pm8$, возрастает на $[-8, -3]\cup[0, 3]\cup[8, \infty)$, убывает на $(-\infty, -8]\cup[-3, 0]\cup[3, 8]$, локальные минимумы в точках $x=0$ (значение 2) и $x=\pm8$ (значение 0), локальные максимумы в точках $x=\pm3$ (значение 5).</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" => "1119677" "type" => "task" ] "previous" => array:2 [ "refs" => "1119675" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Book {#1200} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => 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№12 (с. 69)
Условие. №12 (с. 69)
скриншот условия
12. (1) Постройте графики функций на одной плоскости:
а) $y=-x$;
б) $y=-x-3$;
в) $y=|-x-3|$;
г) $y=|x+3|-5$;
д) $y=||x+3|-5||$;
е) $y=||-x+3|-5||$;
ж) $y=||3-|x|-5||$. По графику ж) сделайте исследование функции.
Решение 2 (rus). №12 (с. 69)
а) Функция $y=-x$. Это линейная функция, её график — прямая линия, которая проходит через начало координат (0,0) и точку (1,-1). Она является биссектрисой II и IV координатных четвертей.
Ответ: График функции $y=-x$ — это прямая, проходящая через точки (0,0) и (1,-1).
б) Функция $y=-x-3$. График этой функции получается из графика функции $y=-x$ (из пункта а) путем параллельного переноса (сдвига) вдоль оси Oy на 3 единицы вниз. Прямая будет проходить через точки (0, -3) и (-3, 0).
Ответ: График функции $y=-x-3$ — это прямая, полученная сдвигом прямой $y=-x$ на 3 единицы вниз.
в) Функция $y=|-x-3|$. Используя свойство модуля $|-a|=|a|$, можем переписать функцию как $y=|-(x+3)|=|x+3|$. График этой функции получается из графика базовой функции $y=|x|$ (график в виде "галочки" с вершиной в начале координат) путем сдвига на 3 единицы влево вдоль оси Ox. Вершина графика окажется в точке (-3, 0).
Альтернативно, график $y=|-x-3|$ можно получить из графика $y=-x-3$ (из пункта б), отразив ту часть графика, которая лежит ниже оси Ox, симметрично относительно оси Ox.
Ответ: График функции $y=|-x-3|$ эквивалентен графику $y=|x+3|$, который представляет собой график $y=|x|$, сдвинутый на 3 единицы влево.
г) Функция $y=|x+3|-5$. График этой функции получается из графика функции $y=|x+3|$ (из пункта в) путем сдвига на 5 единиц вниз вдоль оси Oy. Вершина графика сместится из точки (-3, 0) в точку (-3, -5).
Ответ: График функции $y=|x+3|-5$ — это график $y=|x+3|$, сдвинутый на 5 единиц вниз.
д) Функция $y=||x+3|-5|$. График этой функции получается из графика функции $y=|x+3|-5$ (из пункта г) путем симметричного отражения относительно оси Ox той части графика, которая лежит ниже оси Ox. Часть графика, которая находится выше или на оси Ox, остается без изменений. Вершина в точке (-3, -5) отразится в точку (-3, 5).
Ответ: График функции $y=||x+3|-5|$ получается из графика $y=|x+3|-5$ отражением его отрицательной части ($y<0$) относительно оси Ox.
е) Функция $y=|-|x+3|-5|$. Проанализируем выражение под внешним модулем. Так как $|x+3| \ge 0$, то $-|x+3| \le 0$. Следовательно, выражение $-|x+3|-5$ всегда отрицательно (его значение не превышает -5). По определению модуля, $|a|=-a$ для $a \le 0$. Таким образом, $y = -(-|x+3|-5) = |x+3|+5$.
График этой функции получается из графика $y=|x+3|$ (из пункта в) сдвигом на 5 единиц вверх вдоль оси Oy. Вершина будет в точке (-3, 5).
Ответ: График функции $y=|-|x+3|-5|$ совпадает с графиком функции $y=|x+3|+5$, который получается сдвигом графика $y=|x+3|$ на 5 единиц вверх.
ж) Построим график функции $y=||3-|x||-5|$ и проведем его исследование. Построение будем выполнять последовательно, используя преобразования графиков:
1. $y=|x|$ — базовый график, "галочка" с вершиной в (0,0).
2. $y=-|x|$ — график $y=|x|$, отраженный симметрично относительно оси Ox. "Галочка", перевернутая вниз.
3. $y=3-|x|$ — график $y=-|x|$, сдвинутый на 3 единицы вверх по оси Oy. Вершина в точке (0,3).
4. $y=|3-|x||$ — часть графика $y=3-|x|$, лежащая ниже оси Ox (при $|x|>3$), отражается симметрично относительно оси Ox. Получается график в форме буквы "W" с пиком в (0,3) и изломами в точках (-3,0) и (3,0).
5. $y=|3-|x||-5$ — график $y=|3-|x||$, сдвинутый на 5 единиц вниз по оси Oy. Пик "W" смещается в (0,-2), а точки излома в (-3,-5) и (3,-5).
6. $y=||3-|x||-5|$ — итоговый график. Часть графика $y=|3-|x||-5$, лежащая ниже оси Ox, отражается симметрично относительно этой оси. Точки пересечения с осью Ox ($x=\pm8$) остаются на месте. Точка (0,-2) переходит в (0,2). Точки (-3,-5) и (3,-5) переходят в пики (3,5) и (-3,5).
Исследование функции $y=||3-|x||-5|$ по графику:
1. Область определения: $D(y) = (-\infty, +\infty)$, так как выражение определено для любого действительного $x$.
2. Область значений: Функция является модулем, поэтому её значения неотрицательны. Минимальное значение равно 0. Максимального значения нет. $E(y) = [0, +\infty)$.
3. Четность: $y(-x) = ||3-|-x||-5| = ||3-|x||-5| = y(x)$. Функция является четной. Ее график симметричен относительно оси Oy.
4. Нули функции: $y=0$ при $||3-|x||-5|=0 \implies |3-|x||=5$. Это уравнение равносильно совокупности $3-|x|=5$ (что дает $|x|=-2$, нет решений) и $3-|x|=-5$ (что дает $|x|=8$). Нули функции: $x=-8$ и $x=8$.
5. Промежутки знакопостоянства: Так как $y \ge 0$ для всех $x$, то $y>0$ при $x \in (-\infty, -8) \cup (-8, 8) \cup (8, +\infty)$.
6. Промежутки монотонности:
• Функция возрастает на промежутках $[-8, -3]$, $[0, 3]$ и $[8, +\infty)$.
• Функция убывает на промежутках $(-\infty, -8]$, $[-3, 0]$ и $[3, 8]$.
7. Экстремумы:
• Точки локального минимума: $x_{min}=-8$, $x_{min}=0$, $x_{min}=8$.
Значения в точках минимума: $y(\pm8)=0$, $y(0)=2$.
• Точки локального максимума: $x_{max}=-3$, $x_{max}=3$.
Значения в точках максимума: $y(\pm3)=5$.
• Наименьшее значение функции равно 0 (достигается в точках $x=\pm8$).
• Наибольшего значения функция не имеет.
Ответ: График функции $y=||3-|x||-5|$ строится последовательными преобразованиями. Свойства функции: $D(y)=(-\infty, +\infty)$, $E(y)=[0, +\infty)$, функция четная, нули $x=\pm8$, возрастает на $[-8, -3]\cup[0, 3]\cup[8, \infty)$, убывает на $(-\infty, -8]\cup[-3, 0]\cup[3, 8]$, локальные минимумы в точках $x=0$ (значение 2) и $x=\pm8$ (значение 0), локальные максимумы в точках $x=\pm3$ (значение 5).
Другие задания:
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