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Номер 14, страница 87, 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 2. Тригонометрические функции. Параграф 2. Графики функций. 2.3. Примеры построения и исследования графиков. Задачи - номер 14, страница 87.
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" => 1119710 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "87" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/2-2-3zad-14" "field_display_title" => "14" "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 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ …1] #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 [ …4] ] "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 [ …4] ] "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 {#1186 #items: array:5 [ 0 => App\Models\Branch {#1198 #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 {#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: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 {#1231 #items: array:1 [ 0 => App\Models\Book {#1201 #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 {#1203 #items: array:1 [ 0 => App\Models\Term {#1200 #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 {#1204 #items: array:1 [ 0 => App\Models\Term {#1205 #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 {#1206 #items: array:1 [ 0 => App\Models\Term {#1207 #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 [ …6] "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 [ …6] "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 {#1208 #items: array:3 [ 0 => 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" => 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 {#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" => 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 {#1211 #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 {#1212 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Term {#1214 #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 [ …6] "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 [ …6] "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 {#1215 #items: array:1 [ 0 => App\Models\Term {#1216 #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 [ …6] "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 [ …6] "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 {#1217 #items: array:1 [ 0 => App\Models\Term {#1218 #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 [ …6] "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 [ …6] "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 {#1219 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1220 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1221 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1222 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1223 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1224 #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 {#1225 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1226 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1227 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1228 #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 {#1203} "field_class" => Illuminate\Database\Eloquent\Collection {#1204} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1206} "field_author" => Illuminate\Database\Eloquent\Collection {#1208} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1212} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1213} "field_country" => Illuminate\Database\Eloquent\Collection {#1215} "field_city" => Illuminate\Database\Eloquent\Collection {#1217} "field_series" => Illuminate\Database\Eloquent\Collection {#1219} "field_umk" => Illuminate\Database\Eloquent\Collection {#1220} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1221} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1222} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1223} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1224} "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 {#1225} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1226} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1227} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1228} "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 {#1229 #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 {#1202} "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 {#1231} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1229} ] #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 {#1230 #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" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ 0 => App\Models\Term {#1232 #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" => "71" "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 {#1234 #items: array:1 [ 0 => App\Models\Book {#1201} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1235} "field_page_start" => "71" "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 {#1234} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1233} ] #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 {#1236 #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" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1240 #items: array:1 [ 0 => App\Models\Term {#1237 #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" => "79" "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 {#1239 #items: array:1 [ 0 => App\Models\Book {#1201} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1280 #items: array:1 [ 0 => App\Models\Branch {#1238 #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" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1242 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_page_start" => "71" "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 {#1243 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1272 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1242} "field_page_start" => "71" "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 {#1243} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1272} ] #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" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1240} "field_page_start" => "79" "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 {#1239} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1280} ] #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 {#1278 #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" => 1119367 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.3. Примеры построения и исследования графиков" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1279 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "82" "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 {#1281 #items: array:1 [ 0 => App\Models\Book {#1201} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1327 #items: array:1 [ 0 => App\Models\Branch {#1282 #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" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1284 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_page_start" => "79" "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 {#1285 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1314 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1284} "field_page_start" => "79" "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 {#1285} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1314} ] #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" => 1119367 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.3. Примеры построения и исследования графиков" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1279} "field_page_start" => "82" "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 {#1281} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1327} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Branch {#1325 #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" => 1119368 "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 {#1326 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "86" "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 {#1328 #items: array:1 [ 0 => App\Models\Book {#1201} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1337 #items: array:1 [ 0 => App\Models\Branch {#1336 #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" => 1119367 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.3. Примеры построения и исследования графиков" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1338 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "82" "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 {#1339 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1340 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119367 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.3. Примеры построения и исследования графиков" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1338} "field_page_start" => "82" "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 {#1339} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1340} ] #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" => 1119368 "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 {#1326} "field_page_start" => "86" "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 {#1328} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1337} ] #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" => 1300394 "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 [ …1] #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" => "1119710" "type" => "task" ] "text" => "<p><strong>14. (3)</strong> Постройте графики функций и проведите исследование свойств функций по построенным графикам:</p><p><strong>а)</strong> $y=2\sin 2x+1$;</p><p><strong>б)</strong> $y=\sin \left(2x-\frac{\pi}{3}\right)$;</p><p><strong>в)</strong> $y=\left|\sin \left(x-\frac{\pi}{4}\right)\right|$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "14-1.jpg" "alt" => null "width" => "1668" "height" => 320 "path" => "/media/algebra_10/baspasy-u19/0-2-2-3zad/14-1.webp?ts=1753534556" ] ] ] #original: array:7 [ "id" => 1300394 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1162} "task" => array:2 [ "refs" => "1119710" "type" => "task" ] "text" => "<p><strong>14. (3)</strong> Постройте графики функций и проведите исследование свойств функций по построенным графикам:</p><p><strong>а)</strong> $y=2\sin 2x+1$;</p><p><strong>б)</strong> $y=\sin \left(2x-\frac{\pi}{3}\right)$;</p><p><strong>в)</strong> $y=\left|\sin \left(x-\frac{\pi}{4}\right)\right|$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "14-1.jpg" "alt" => null "width" => "1668" "height" => 320 "path" => "/media/algebra_10/baspasy-u19/0-2-2-3zad/14-1.webp?ts=1753534556" ] ] ] #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" => 1551430 "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 [ …1] #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" => "1119710" "type" => "task" ] "text" => "<p><strong>а) y = 2sin(2x) + 1</strong></p><p> <strong>Построение графика:</strong><br> График функции $y = 2\sin(2x) + 1$ получается из графика основной функции $y = \sin x$ с помощью следующих преобразований: 1. Сжатие графика вдоль оси Ох в 2 раза. Это преобразование функции $y = \sin x$ в $y = \sin(2x)$. Период функции уменьшается в 2 раза и становится равным $T = \frac{2\pi}{2} = \pi$. 2. Растяжение графика вдоль оси Оу в 2 раза. Это преобразование функции $y = \sin(2x)$ в $y = 2\sin(2x)$. Амплитуда колебаний увеличивается до 2, а область значений становится $[-2, 2]$. 3. Сдвиг графика вверх вдоль оси Оу на 1 единицу. Это преобразование функции $y = 2\sin(2x)$ в $y = 2\sin(2x) + 1$. Ось колебаний смещается на $y=1$, а область значений становится $[-1, 3]$.</p><p><strong>Исследование свойств функции:</strong></p><p>1. <strong>Область определения:</strong> $D(y) = (-\infty; +\infty)$, так как функция определена для любых действительных значений $x$.</p><p>2. <strong>Область значений:</strong> $E(y) = [-1; 3]$. Это следует из того, что $-1 \le \sin(2x) \le 1$, следовательно $-2 \le 2\sin(2x) \le 2$, и $-1 \le 2\sin(2x) + 1 \le 3$.</p><p>3. <strong>Четность, нечетность:</strong> Функция не является ни четной, ни нечетной (функция общего вида), так как $y(-x) = 2\sin(-2x) + 1 = -2\sin(2x) + 1 \neq y(x)$ и $y(-x) \neq -y(x)$.</p><p>4. <strong>Периодичность:</strong> Функция периодическая с наименьшим положительным периодом $T = \frac{2\pi}{2} = \pi$.</p><p>5. <strong>Нули функции:</strong> $y=0$ при $2\sin(2x) + 1 = 0$, то есть $\sin(2x) = -\frac{1}{2}$. Решения: $2x = (-1)^{k+1}\frac{\pi}{6} + \pi k$, откуда $x = (-1)^{k+1}\frac{\pi}{12} + \frac{\pi k}{2}$, где $k \in \mathbb{Z}$.</p><p>6. <strong>Промежутки знакопостоянства:</strong><br> $y > 0$ при $\sin(2x) > -\frac{1}{2}$, что соответствует $x \in (-\frac{\pi}{12} + \pi k; \frac{7\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.<br> $y < 0$ при $\sin(2x) < -\frac{1}{2}$, что соответствует $x \in (\frac{7\pi}{12} + \pi k; \frac{11\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.</p><p>7. <strong>Промежутки монотонности:</strong><br> Функция возрастает, когда ее производная $y' = 4\cos(2x)$ положительна, то есть на промежутках $(-\frac{\pi}{4} + \pi k; \frac{\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.<br> Функция убывает, когда ее производная отрицательна, то есть на промежутках $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.</p><p>8. <strong>Экстремумы:</strong><br> Точки максимума: $x_{max} = \frac{\pi}{4} + \pi k$, $k \in \mathbb{Z}$. Максимальное значение: $y_{max} = 3$.<br> Точки минимума: $x_{min} = -\frac{\pi}{4} + \pi k = \frac{3\pi}{4} + \pi n$ (при $n=k-1$), $k \in \mathbb{Z}$. Минимальное значение: $y_{min} = -1$.</p><p><strong>Ответ:</strong> Функция $y = 2\sin(2x) + 1$ — периодическая ($T=\pi$) функция общего вида, определенная на всей числовой оси, с областью значений $[-1, 3]$. Нули функции: $x = (-1)^{k+1}\frac{\pi}{12} + \frac{\pi k}{2}$. Возрастает на $(-\frac{\pi}{4} + \pi k; \frac{\pi}{4} + \pi k)$, убывает на $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$. Точки максимума $x = \frac{\pi}{4} + \pi k$ ($y_{max}=3$), точки минимума $x = \frac{3\pi}{4} + \pi k$ ($y_{min}=-1$), $k \in \mathbb{Z}$.</p><hr><p><strong>б) $y = \sin(2x - \frac{\pi}{3})$</strong></p><p> <strong>Построение графика:</strong><br> График функции $y = \sin(2x - \frac{\pi}{3})$ получается из графика $y = \sin x$. Удобно представить функцию в виде $y = \sin(2(x - \frac{\pi}{6}))$. 1. Сжатие графика $y = \sin x$ вдоль оси Ох в 2 раза, получаем $y = \sin(2x)$. Период становится $T = \pi$. 2. Сдвиг полученного графика $y = \sin(2x)$ вправо вдоль оси Ох на $\frac{\pi}{6}$. Это преобразование дает итоговый график $y = \sin(2(x - \frac{\pi}{6}))$.</p><p><strong>Исследование свойств функции:</strong></p><p>1. <strong>Область определения:</strong> $D(y) = (-\infty; +\infty)$.</p><p>2. <strong>Область значений:</strong> $E(y) = [-1; 1]$.</p><p>3. <strong>Четность, нечетность:</strong> Функция общего вида, так как $y(-x) = \sin(-2x - \frac{\pi}{3}) = -\sin(2x + \frac{\pi}{3}) \neq \pm y(x)$.</p><p>4. <strong>Периодичность:</strong> Периодическая с наименьшим положительным периодом $T = \frac{2\pi}{2} = \pi$.</p><p>5. <strong>Нули функции:</strong> $y=0$ при $\sin(2x - \frac{\pi}{3}) = 0$. Решения: $2x - \frac{\pi}{3} = \pi k$, откуда $x = \frac{\pi}{6} + \frac{\pi k}{2}$, где $k \in \mathbb{Z}$.</p><p>6. <strong>Промежутки знакопостоянства:</strong><br> $y > 0$ при $x \in (\frac{\pi}{6} + \pi k; \frac{2\pi}{3} + \pi k)$, $k \in \mathbb{Z}$.<br> $y < 0$ при $x \in (\frac{2\pi}{3} + \pi k; \frac{7\pi}{6} + \pi k)$, $k \in \mathbb{Z}$.</p><p>7. <strong>Промежутки монотонности:</strong><br> Производная $y' = 2\cos(2x - \frac{\pi}{3})$. Функция возрастает при $\cos(2x - \frac{\pi}{3}) > 0$, то есть на промежутках $(-\frac{\pi}{12} + \pi k; \frac{5\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.<br> Функция убывает при $\cos(2x - \frac{\pi}{3}) < 0$, то есть на промежутках $(\frac{5\pi}{12} + \pi k; \frac{11\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.</p><p>8. <strong>Экстремумы:</strong><br> Точки максимума (где $\sin(2x - \frac{\pi}{3}) = 1$): $x_{max} = \frac{5\pi}{12} + \pi k$, $k \in \mathbb{Z}$. Максимальное значение: $y_{max} = 1$.<br> Точки минимума (где $\sin(2x - \frac{\pi}{3}) = -1$): $x_{min} = -\frac{\pi}{12} + \pi k = \frac{11\pi}{12} + \pi n$, $k \in \mathbb{Z}$. Минимальное значение: $y_{min} = -1$.</p><p><strong>Ответ:</strong> Функция $y = \sin(2x - \frac{\pi}{3})$ — периодическая ($T=\pi$) функция общего вида, $D(y)=\mathbb{R}$, $E(y)=[-1, 1]$. Нули: $x = \frac{\pi}{6} + \frac{\pi k}{2}$. Возрастает на $(-\frac{\pi}{12} + \pi k; \frac{5\pi}{12} + \pi k)$, убывает на $(\frac{5\pi}{12} + \pi k; \frac{11\pi}{12} + \pi k)$. Точки максимума $x = \frac{5\pi}{12} + \pi k$ ($y_{max}=1$), точки минимума $x = \frac{11\pi}{12} + \pi k$ ($y_{min}=-1$), $k \in \mathbb{Z}$.</p><hr><p><strong>в) $y = |\sin(x - \frac{\pi}{4})|$</strong></p><p> <strong>Построение графика:</strong><br> График функции $y = |\sin(x - \frac{\pi}{4})|$ получается из графика $y = \sin x$ в два шага: 1. Сдвиг графика $y = \sin x$ вправо вдоль оси Ох на $\frac{\pi}{4}$. Получаем график функции $y = \sin(x - \frac{\pi}{4})$. 2. Применение операции взятия модуля: $y = |\sin(x - \frac{\pi}{4})|$. Часть графика, которая находится ниже оси Ох (где $y < 0$), симметрично отражается относительно оси Ох вверх.</p><p><strong>Исследование свойств функции:</strong></p><p>1. <strong>Область определения:</strong> $D(y) = (-\infty; +\infty)$.</p><p>2. <strong>Область значений:</strong> $E(y) = [0; 1]$, так как модуль — неотрицательная величина, а максимальное значение синуса по модулю равно 1.</p><p>3. <strong>Четность, нечетность:</strong> Функция общего вида, так как $y(-x) = |\sin(-x - \frac{\pi}{4})| = |\sin(x + \frac{\pi}{4})| \neq \pm y(x)$.</p><p>4. <strong>Периодичность:</strong> Функция периодическая. Период функции $y=\sin(x-\frac{\pi}{4})$ равен $2\pi$. При взятии модуля период функции $y=|\sin(ax+b)|$ становится в два раза меньше, поэтому наименьший положительный период $T = \frac{2\pi}{2} = \pi$.</p><p>5. <strong>Нули функции:</strong> $y=0$ при $\sin(x - \frac{\pi}{4}) = 0$. Решения: $x - \frac{\pi}{4} = \pi k$, откуда $x = \frac{\pi}{4} + \pi k$, где $k \in \mathbb{Z}$.</p><p>6. <strong>Промежутки знакопостоянства:</strong><br> $y > 0$ при всех $x$, кроме нулей функции, то есть при $x \neq \frac{\pi}{4} + \pi k$, $k \in \mathbb{Z}$.<br> Функция не принимает отрицательных значений ($y \ge 0$).</p><p>7. <strong>Промежутки монотонности:</strong><br> Функция возрастает на промежутках $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.<br> Функция убывает на промежутках $(\frac{3\pi}{4} + \pi k; \frac{5\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.</p><p>8. <strong>Экстремумы:</strong><br> Точки максимума (где $|\sin(x - \frac{\pi}{4})| = 1$): $x_{max} = \frac{3\pi}{4} + \pi k$, $k \in \mathbb{Z}$. Максимальное значение: $y_{max} = 1$.<br> Точки минимума (нули функции): $x_{min} = \frac{\pi}{4} + \pi k$, $k \in \mathbb{Z}$. Минимальное значение: $y_{min} = 0$.</p><p><strong>Ответ:</strong> Функция $y = |\sin(x - \frac{\pi}{4})|$ — периодическая ($T=\pi$) функция общего вида, $D(y)=\mathbb{R}$, $E(y)=[0, 1]$. Нули: $x = \frac{\pi}{4} + \pi k$. Функция неотрицательна. Возрастает на $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$, убывает на $(\frac{3\pi}{4} + \pi k; \frac{5\pi}{4} + \pi k)$. Точки максимума $x = \frac{3\pi}{4} + \pi k$ ($y_{max}=1$), точки минимума $x = \frac{\pi}{4} + \pi k$ ($y_{min}=0$), $k \in \mathbb{Z}$.</p>" ] #original: array:6 [ "id" => 1551430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156} "task" => array:2 [ "refs" => "1119710" "type" => "task" ] "text" => "<p><strong>а) y = 2sin(2x) + 1</strong></p><p> <strong>Построение графика:</strong><br> График функции $y = 2\sin(2x) + 1$ получается из графика основной функции $y = \sin x$ с помощью следующих преобразований: 1. Сжатие графика вдоль оси Ох в 2 раза. Это преобразование функции $y = \sin x$ в $y = \sin(2x)$. Период функции уменьшается в 2 раза и становится равным $T = \frac{2\pi}{2} = \pi$. 2. Растяжение графика вдоль оси Оу в 2 раза. Это преобразование функции $y = \sin(2x)$ в $y = 2\sin(2x)$. Амплитуда колебаний увеличивается до 2, а область значений становится $[-2, 2]$. 3. Сдвиг графика вверх вдоль оси Оу на 1 единицу. Это преобразование функции $y = 2\sin(2x)$ в $y = 2\sin(2x) + 1$. Ось колебаний смещается на $y=1$, а область значений становится $[-1, 3]$.</p><p><strong>Исследование свойств функции:</strong></p><p>1. <strong>Область определения:</strong> $D(y) = (-\infty; +\infty)$, так как функция определена для любых действительных значений $x$.</p><p>2. <strong>Область значений:</strong> $E(y) = [-1; 3]$. Это следует из того, что $-1 \le \sin(2x) \le 1$, следовательно $-2 \le 2\sin(2x) \le 2$, и $-1 \le 2\sin(2x) + 1 \le 3$.</p><p>3. <strong>Четность, нечетность:</strong> Функция не является ни четной, ни нечетной (функция общего вида), так как $y(-x) = 2\sin(-2x) + 1 = -2\sin(2x) + 1 \neq y(x)$ и $y(-x) \neq -y(x)$.</p><p>4. <strong>Периодичность:</strong> Функция периодическая с наименьшим положительным периодом $T = \frac{2\pi}{2} = \pi$.</p><p>5. <strong>Нули функции:</strong> $y=0$ при $2\sin(2x) + 1 = 0$, то есть $\sin(2x) = -\frac{1}{2}$. Решения: $2x = (-1)^{k+1}\frac{\pi}{6} + \pi k$, откуда $x = (-1)^{k+1}\frac{\pi}{12} + \frac{\pi k}{2}$, где $k \in \mathbb{Z}$.</p><p>6. <strong>Промежутки знакопостоянства:</strong><br> $y > 0$ при $\sin(2x) > -\frac{1}{2}$, что соответствует $x \in (-\frac{\pi}{12} + \pi k; \frac{7\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.<br> $y < 0$ при $\sin(2x) < -\frac{1}{2}$, что соответствует $x \in (\frac{7\pi}{12} + \pi k; \frac{11\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.</p><p>7. <strong>Промежутки монотонности:</strong><br> Функция возрастает, когда ее производная $y' = 4\cos(2x)$ положительна, то есть на промежутках $(-\frac{\pi}{4} + \pi k; \frac{\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.<br> Функция убывает, когда ее производная отрицательна, то есть на промежутках $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.</p><p>8. <strong>Экстремумы:</strong><br> Точки максимума: $x_{max} = \frac{\pi}{4} + \pi k$, $k \in \mathbb{Z}$. Максимальное значение: $y_{max} = 3$.<br> Точки минимума: $x_{min} = -\frac{\pi}{4} + \pi k = \frac{3\pi}{4} + \pi n$ (при $n=k-1$), $k \in \mathbb{Z}$. Минимальное значение: $y_{min} = -1$.</p><p><strong>Ответ:</strong> Функция $y = 2\sin(2x) + 1$ — периодическая ($T=\pi$) функция общего вида, определенная на всей числовой оси, с областью значений $[-1, 3]$. Нули функции: $x = (-1)^{k+1}\frac{\pi}{12} + \frac{\pi k}{2}$. Возрастает на $(-\frac{\pi}{4} + \pi k; \frac{\pi}{4} + \pi k)$, убывает на $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$. Точки максимума $x = \frac{\pi}{4} + \pi k$ ($y_{max}=3$), точки минимума $x = \frac{3\pi}{4} + \pi k$ ($y_{min}=-1$), $k \in \mathbb{Z}$.</p><hr><p><strong>б) $y = \sin(2x - \frac{\pi}{3})$</strong></p><p> <strong>Построение графика:</strong><br> График функции $y = \sin(2x - \frac{\pi}{3})$ получается из графика $y = \sin x$. Удобно представить функцию в виде $y = \sin(2(x - \frac{\pi}{6}))$. 1. Сжатие графика $y = \sin x$ вдоль оси Ох в 2 раза, получаем $y = \sin(2x)$. Период становится $T = \pi$. 2. Сдвиг полученного графика $y = \sin(2x)$ вправо вдоль оси Ох на $\frac{\pi}{6}$. Это преобразование дает итоговый график $y = \sin(2(x - \frac{\pi}{6}))$.</p><p><strong>Исследование свойств функции:</strong></p><p>1. <strong>Область определения:</strong> $D(y) = (-\infty; +\infty)$.</p><p>2. <strong>Область значений:</strong> $E(y) = [-1; 1]$.</p><p>3. <strong>Четность, нечетность:</strong> Функция общего вида, так как $y(-x) = \sin(-2x - \frac{\pi}{3}) = -\sin(2x + \frac{\pi}{3}) \neq \pm y(x)$.</p><p>4. <strong>Периодичность:</strong> Периодическая с наименьшим положительным периодом $T = \frac{2\pi}{2} = \pi$.</p><p>5. <strong>Нули функции:</strong> $y=0$ при $\sin(2x - \frac{\pi}{3}) = 0$. Решения: $2x - \frac{\pi}{3} = \pi k$, откуда $x = \frac{\pi}{6} + \frac{\pi k}{2}$, где $k \in \mathbb{Z}$.</p><p>6. <strong>Промежутки знакопостоянства:</strong><br> $y > 0$ при $x \in (\frac{\pi}{6} + \pi k; \frac{2\pi}{3} + \pi k)$, $k \in \mathbb{Z}$.<br> $y < 0$ при $x \in (\frac{2\pi}{3} + \pi k; \frac{7\pi}{6} + \pi k)$, $k \in \mathbb{Z}$.</p><p>7. <strong>Промежутки монотонности:</strong><br> Производная $y' = 2\cos(2x - \frac{\pi}{3})$. Функция возрастает при $\cos(2x - \frac{\pi}{3}) > 0$, то есть на промежутках $(-\frac{\pi}{12} + \pi k; \frac{5\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.<br> Функция убывает при $\cos(2x - \frac{\pi}{3}) < 0$, то есть на промежутках $(\frac{5\pi}{12} + \pi k; \frac{11\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.</p><p>8. <strong>Экстремумы:</strong><br> Точки максимума (где $\sin(2x - \frac{\pi}{3}) = 1$): $x_{max} = \frac{5\pi}{12} + \pi k$, $k \in \mathbb{Z}$. Максимальное значение: $y_{max} = 1$.<br> Точки минимума (где $\sin(2x - \frac{\pi}{3}) = -1$): $x_{min} = -\frac{\pi}{12} + \pi k = \frac{11\pi}{12} + \pi n$, $k \in \mathbb{Z}$. Минимальное значение: $y_{min} = -1$.</p><p><strong>Ответ:</strong> Функция $y = \sin(2x - \frac{\pi}{3})$ — периодическая ($T=\pi$) функция общего вида, $D(y)=\mathbb{R}$, $E(y)=[-1, 1]$. Нули: $x = \frac{\pi}{6} + \frac{\pi k}{2}$. Возрастает на $(-\frac{\pi}{12} + \pi k; \frac{5\pi}{12} + \pi k)$, убывает на $(\frac{5\pi}{12} + \pi k; \frac{11\pi}{12} + \pi k)$. Точки максимума $x = \frac{5\pi}{12} + \pi k$ ($y_{max}=1$), точки минимума $x = \frac{11\pi}{12} + \pi k$ ($y_{min}=-1$), $k \in \mathbb{Z}$.</p><hr><p><strong>в) $y = |\sin(x - \frac{\pi}{4})|$</strong></p><p> <strong>Построение графика:</strong><br> График функции $y = |\sin(x - \frac{\pi}{4})|$ получается из графика $y = \sin x$ в два шага: 1. Сдвиг графика $y = \sin x$ вправо вдоль оси Ох на $\frac{\pi}{4}$. Получаем график функции $y = \sin(x - \frac{\pi}{4})$. 2. Применение операции взятия модуля: $y = |\sin(x - \frac{\pi}{4})|$. Часть графика, которая находится ниже оси Ох (где $y < 0$), симметрично отражается относительно оси Ох вверх.</p><p><strong>Исследование свойств функции:</strong></p><p>1. <strong>Область определения:</strong> $D(y) = (-\infty; +\infty)$.</p><p>2. <strong>Область значений:</strong> $E(y) = [0; 1]$, так как модуль — неотрицательная величина, а максимальное значение синуса по модулю равно 1.</p><p>3. <strong>Четность, нечетность:</strong> Функция общего вида, так как $y(-x) = |\sin(-x - \frac{\pi}{4})| = |\sin(x + \frac{\pi}{4})| \neq \pm y(x)$.</p><p>4. <strong>Периодичность:</strong> Функция периодическая. Период функции $y=\sin(x-\frac{\pi}{4})$ равен $2\pi$. При взятии модуля период функции $y=|\sin(ax+b)|$ становится в два раза меньше, поэтому наименьший положительный период $T = \frac{2\pi}{2} = \pi$.</p><p>5. <strong>Нули функции:</strong> $y=0$ при $\sin(x - \frac{\pi}{4}) = 0$. Решения: $x - \frac{\pi}{4} = \pi k$, откуда $x = \frac{\pi}{4} + \pi k$, где $k \in \mathbb{Z}$.</p><p>6. <strong>Промежутки знакопостоянства:</strong><br> $y > 0$ при всех $x$, кроме нулей функции, то есть при $x \neq \frac{\pi}{4} + \pi k$, $k \in \mathbb{Z}$.<br> Функция не принимает отрицательных значений ($y \ge 0$).</p><p>7. <strong>Промежутки монотонности:</strong><br> Функция возрастает на промежутках $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.<br> Функция убывает на промежутках $(\frac{3\pi}{4} + \pi k; \frac{5\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.</p><p>8. <strong>Экстремумы:</strong><br> Точки максимума (где $|\sin(x - \frac{\pi}{4})| = 1$): $x_{max} = \frac{3\pi}{4} + \pi k$, $k \in \mathbb{Z}$. Максимальное значение: $y_{max} = 1$.<br> Точки минимума (нули функции): $x_{min} = \frac{\pi}{4} + \pi k$, $k \in \mathbb{Z}$. Минимальное значение: $y_{min} = 0$.</p><p><strong>Ответ:</strong> Функция $y = |\sin(x - \frac{\pi}{4})|$ — периодическая ($T=\pi$) функция общего вида, $D(y)=\mathbb{R}$, $E(y)=[0, 1]$. Нули: $x = \frac{\pi}{4} + \pi k$. Функция неотрицательна. Возрастает на $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$, убывает на $(\frac{3\pi}{4} + \pi k; \frac{5\pi}{4} + \pi k)$. 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№14 (с. 87)
Условие. №14 (с. 87)
скриншот условия
14. (3) Постройте графики функций и проведите исследование свойств функций по построенным графикам:
а) $y=2\sin 2x+1$;
б) $y=\sin \left(2x-\frac{\pi}{3}\right)$;
в) $y=\left|\sin \left(x-\frac{\pi}{4}\right)\right|$.
Решение 2 (rus). №14 (с. 87)
а) y = 2sin(2x) + 1
Построение графика:
График функции $y = 2\sin(2x) + 1$ получается из графика основной функции $y = \sin x$ с помощью следующих преобразований: 1. Сжатие графика вдоль оси Ох в 2 раза. Это преобразование функции $y = \sin x$ в $y = \sin(2x)$. Период функции уменьшается в 2 раза и становится равным $T = \frac{2\pi}{2} = \pi$. 2. Растяжение графика вдоль оси Оу в 2 раза. Это преобразование функции $y = \sin(2x)$ в $y = 2\sin(2x)$. Амплитуда колебаний увеличивается до 2, а область значений становится $[-2, 2]$. 3. Сдвиг графика вверх вдоль оси Оу на 1 единицу. Это преобразование функции $y = 2\sin(2x)$ в $y = 2\sin(2x) + 1$. Ось колебаний смещается на $y=1$, а область значений становится $[-1, 3]$.
Исследование свойств функции:
1. Область определения: $D(y) = (-\infty; +\infty)$, так как функция определена для любых действительных значений $x$.
2. Область значений: $E(y) = [-1; 3]$. Это следует из того, что $-1 \le \sin(2x) \le 1$, следовательно $-2 \le 2\sin(2x) \le 2$, и $-1 \le 2\sin(2x) + 1 \le 3$.
3. Четность, нечетность: Функция не является ни четной, ни нечетной (функция общего вида), так как $y(-x) = 2\sin(-2x) + 1 = -2\sin(2x) + 1 \neq y(x)$ и $y(-x) \neq -y(x)$.
4. Периодичность: Функция периодическая с наименьшим положительным периодом $T = \frac{2\pi}{2} = \pi$.
5. Нули функции: $y=0$ при $2\sin(2x) + 1 = 0$, то есть $\sin(2x) = -\frac{1}{2}$. Решения: $2x = (-1)^{k+1}\frac{\pi}{6} + \pi k$, откуда $x = (-1)^{k+1}\frac{\pi}{12} + \frac{\pi k}{2}$, где $k \in \mathbb{Z}$.
6. Промежутки знакопостоянства:
$y > 0$ при $\sin(2x) > -\frac{1}{2}$, что соответствует $x \in (-\frac{\pi}{12} + \pi k; \frac{7\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.
$y < 0$ при $\sin(2x) < -\frac{1}{2}$, что соответствует $x \in (\frac{7\pi}{12} + \pi k; \frac{11\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.
7. Промежутки монотонности:
Функция возрастает, когда ее производная $y' = 4\cos(2x)$ положительна, то есть на промежутках $(-\frac{\pi}{4} + \pi k; \frac{\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.
Функция убывает, когда ее производная отрицательна, то есть на промежутках $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.
8. Экстремумы:
Точки максимума: $x_{max} = \frac{\pi}{4} + \pi k$, $k \in \mathbb{Z}$. Максимальное значение: $y_{max} = 3$.
Точки минимума: $x_{min} = -\frac{\pi}{4} + \pi k = \frac{3\pi}{4} + \pi n$ (при $n=k-1$), $k \in \mathbb{Z}$. Минимальное значение: $y_{min} = -1$.
Ответ: Функция $y = 2\sin(2x) + 1$ — периодическая ($T=\pi$) функция общего вида, определенная на всей числовой оси, с областью значений $[-1, 3]$. Нули функции: $x = (-1)^{k+1}\frac{\pi}{12} + \frac{\pi k}{2}$. Возрастает на $(-\frac{\pi}{4} + \pi k; \frac{\pi}{4} + \pi k)$, убывает на $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$. Точки максимума $x = \frac{\pi}{4} + \pi k$ ($y_{max}=3$), точки минимума $x = \frac{3\pi}{4} + \pi k$ ($y_{min}=-1$), $k \in \mathbb{Z}$.
б) $y = \sin(2x - \frac{\pi}{3})$
Построение графика:
График функции $y = \sin(2x - \frac{\pi}{3})$ получается из графика $y = \sin x$. Удобно представить функцию в виде $y = \sin(2(x - \frac{\pi}{6}))$. 1. Сжатие графика $y = \sin x$ вдоль оси Ох в 2 раза, получаем $y = \sin(2x)$. Период становится $T = \pi$. 2. Сдвиг полученного графика $y = \sin(2x)$ вправо вдоль оси Ох на $\frac{\pi}{6}$. Это преобразование дает итоговый график $y = \sin(2(x - \frac{\pi}{6}))$.
Исследование свойств функции:
1. Область определения: $D(y) = (-\infty; +\infty)$.
2. Область значений: $E(y) = [-1; 1]$.
3. Четность, нечетность: Функция общего вида, так как $y(-x) = \sin(-2x - \frac{\pi}{3}) = -\sin(2x + \frac{\pi}{3}) \neq \pm y(x)$.
4. Периодичность: Периодическая с наименьшим положительным периодом $T = \frac{2\pi}{2} = \pi$.
5. Нули функции: $y=0$ при $\sin(2x - \frac{\pi}{3}) = 0$. Решения: $2x - \frac{\pi}{3} = \pi k$, откуда $x = \frac{\pi}{6} + \frac{\pi k}{2}$, где $k \in \mathbb{Z}$.
6. Промежутки знакопостоянства:
$y > 0$ при $x \in (\frac{\pi}{6} + \pi k; \frac{2\pi}{3} + \pi k)$, $k \in \mathbb{Z}$.
$y < 0$ при $x \in (\frac{2\pi}{3} + \pi k; \frac{7\pi}{6} + \pi k)$, $k \in \mathbb{Z}$.
7. Промежутки монотонности:
Производная $y' = 2\cos(2x - \frac{\pi}{3})$. Функция возрастает при $\cos(2x - \frac{\pi}{3}) > 0$, то есть на промежутках $(-\frac{\pi}{12} + \pi k; \frac{5\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.
Функция убывает при $\cos(2x - \frac{\pi}{3}) < 0$, то есть на промежутках $(\frac{5\pi}{12} + \pi k; \frac{11\pi}{12} + \pi k)$, $k \in \mathbb{Z}$.
8. Экстремумы:
Точки максимума (где $\sin(2x - \frac{\pi}{3}) = 1$): $x_{max} = \frac{5\pi}{12} + \pi k$, $k \in \mathbb{Z}$. Максимальное значение: $y_{max} = 1$.
Точки минимума (где $\sin(2x - \frac{\pi}{3}) = -1$): $x_{min} = -\frac{\pi}{12} + \pi k = \frac{11\pi}{12} + \pi n$, $k \in \mathbb{Z}$. Минимальное значение: $y_{min} = -1$.
Ответ: Функция $y = \sin(2x - \frac{\pi}{3})$ — периодическая ($T=\pi$) функция общего вида, $D(y)=\mathbb{R}$, $E(y)=[-1, 1]$. Нули: $x = \frac{\pi}{6} + \frac{\pi k}{2}$. Возрастает на $(-\frac{\pi}{12} + \pi k; \frac{5\pi}{12} + \pi k)$, убывает на $(\frac{5\pi}{12} + \pi k; \frac{11\pi}{12} + \pi k)$. Точки максимума $x = \frac{5\pi}{12} + \pi k$ ($y_{max}=1$), точки минимума $x = \frac{11\pi}{12} + \pi k$ ($y_{min}=-1$), $k \in \mathbb{Z}$.
в) $y = |\sin(x - \frac{\pi}{4})|$
Построение графика:
График функции $y = |\sin(x - \frac{\pi}{4})|$ получается из графика $y = \sin x$ в два шага: 1. Сдвиг графика $y = \sin x$ вправо вдоль оси Ох на $\frac{\pi}{4}$. Получаем график функции $y = \sin(x - \frac{\pi}{4})$. 2. Применение операции взятия модуля: $y = |\sin(x - \frac{\pi}{4})|$. Часть графика, которая находится ниже оси Ох (где $y < 0$), симметрично отражается относительно оси Ох вверх.
Исследование свойств функции:
1. Область определения: $D(y) = (-\infty; +\infty)$.
2. Область значений: $E(y) = [0; 1]$, так как модуль — неотрицательная величина, а максимальное значение синуса по модулю равно 1.
3. Четность, нечетность: Функция общего вида, так как $y(-x) = |\sin(-x - \frac{\pi}{4})| = |\sin(x + \frac{\pi}{4})| \neq \pm y(x)$.
4. Периодичность: Функция периодическая. Период функции $y=\sin(x-\frac{\pi}{4})$ равен $2\pi$. При взятии модуля период функции $y=|\sin(ax+b)|$ становится в два раза меньше, поэтому наименьший положительный период $T = \frac{2\pi}{2} = \pi$.
5. Нули функции: $y=0$ при $\sin(x - \frac{\pi}{4}) = 0$. Решения: $x - \frac{\pi}{4} = \pi k$, откуда $x = \frac{\pi}{4} + \pi k$, где $k \in \mathbb{Z}$.
6. Промежутки знакопостоянства:
$y > 0$ при всех $x$, кроме нулей функции, то есть при $x \neq \frac{\pi}{4} + \pi k$, $k \in \mathbb{Z}$.
Функция не принимает отрицательных значений ($y \ge 0$).
7. Промежутки монотонности:
Функция возрастает на промежутках $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.
Функция убывает на промежутках $(\frac{3\pi}{4} + \pi k; \frac{5\pi}{4} + \pi k)$, $k \in \mathbb{Z}$.
8. Экстремумы:
Точки максимума (где $|\sin(x - \frac{\pi}{4})| = 1$): $x_{max} = \frac{3\pi}{4} + \pi k$, $k \in \mathbb{Z}$. Максимальное значение: $y_{max} = 1$.
Точки минимума (нули функции): $x_{min} = \frac{\pi}{4} + \pi k$, $k \in \mathbb{Z}$. Минимальное значение: $y_{min} = 0$.
Ответ: Функция $y = |\sin(x - \frac{\pi}{4})|$ — периодическая ($T=\pi$) функция общего вида, $D(y)=\mathbb{R}$, $E(y)=[0, 1]$. Нули: $x = \frac{\pi}{4} + \pi k$. Функция неотрицательна. Возрастает на $(\frac{\pi}{4} + \pi k; \frac{3\pi}{4} + \pi k)$, убывает на $(\frac{3\pi}{4} + \pi k; \frac{5\pi}{4} + \pi k)$. Точки максимума $x = \frac{3\pi}{4} + \pi k$ ($y_{max}=1$), точки минимума $x = \frac{\pi}{4} + \pi k$ ($y_{min}=0$), $k \in \mathbb{Z}$.
Другие задания:
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