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Номер 6, страница 86, 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 6, страница 86.
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" => 1119702 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "86" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/2-2-3zad-6" "field_display_title" => "6" "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 {#1214 #items: array:5 [ 0 => App\Models\Branch {#1226 #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 {#1230 #items: array:1 [ 0 => App\Models\Term {#1227 #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 {#1259 #items: array:1 [ 0 => App\Models\Book {#1229 #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 {#1231 #items: array:1 [ 0 => App\Models\Term {#1228 #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 {#1232 #items: array:1 [ 0 => App\Models\Term {#1233 #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 {#1234 #items: array:1 [ 0 => App\Models\Term {#1235 #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 {#1236 #items: array:3 [ 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: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 {#1238 #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 {#1239 #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 {#1240 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1241 #items: array:1 [ 0 => App\Models\Term {#1242 #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 {#1243 #items: array:1 [ 0 => App\Models\Term {#1244 #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 {#1245 #items: array:1 [ 0 => App\Models\Term {#1246 #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 {#1247 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1248 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1249 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1250 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1251 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1252 #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 {#1253 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1254 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1255 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1256 #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 {#1231} "field_class" => Illuminate\Database\Eloquent\Collection {#1232} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1234} "field_author" => Illuminate\Database\Eloquent\Collection {#1236} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1240} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1241} "field_country" => Illuminate\Database\Eloquent\Collection {#1243} "field_city" => Illuminate\Database\Eloquent\Collection {#1245} "field_series" => Illuminate\Database\Eloquent\Collection {#1247} "field_umk" => Illuminate\Database\Eloquent\Collection {#1248} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1249} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1250} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1251} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1252} "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 {#1253} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1254} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1255} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1256} "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 {#1257 #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 {#1230} "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 {#1259} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1257} ] #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 {#1258 #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 {#1263 #items: array:1 [ 0 => App\Models\Term {#1260 #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 {#1262 #items: array:1 [ 0 => App\Models\Book {#1229} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1261 #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 {#1263} "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 {#1262} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1261} ] #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 {#1264 #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 {#1268 #items: array:1 [ 0 => App\Models\Term {#1265 #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 {#1267 #items: array:1 [ 0 => App\Models\Book {#1229} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1308 #items: array:1 [ 0 => App\Models\Branch {#1266 #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 {#1270 #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 {#1271 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1300 #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 {#1270} "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 {#1271} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1300} ] #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 {#1268} "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 {#1267} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1308} ] #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 {#1306 #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 {#1307 #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 {#1309 #items: array:1 [ 0 => App\Models\Book {#1229} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1383 #items: array:1 [ 0 => App\Models\Branch {#1310 #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 {#1312 #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 {#1313 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1342 #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 {#1312} "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 {#1313} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1342} ] #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 {#1307} "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 {#1309} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1383} ] #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 {#1381 #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 {#1382 #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 {#1384 #items: array:1 [ 0 => App\Models\Book {#1229} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1393 #items: array:1 [ 0 => App\Models\Branch {#1392 #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 {#1394 #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 {#1395 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1396 #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 {#1394} "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 {#1395} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1396} ] #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 {#1382} "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 {#1384} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1393} ] #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 {#1210 #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" => 1300386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\Edition {#1209 #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 {#1207 #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 {#1206 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1205 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1204 #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 {#1207} "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 {#1206} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1205} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1204} "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" => "1119702" "type" => "task" ] "text" => "<p><strong>6. (3)</strong> Постройте графики функций и проведите исследование свойств функций по построенным графикам:</p><p><strong>а)</strong> $y=1+\cos(1.5x)$;</p><p><strong>б)</strong> $y=1.5\cos\left(\frac{\pi}{6}-x\right)+2$;</p><p><strong>в)</strong> $y=3\cos\left(\frac{x}{3}-\frac{\pi}{3}\right)-1$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1699" "height" => 339 "path" => "/media/algebra_10/baspasy-u19/0-2-2-3zad/6-1.webp?ts=1753534556" ] ] ] #original: array:7 [ "id" => 1300386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1200} "task" => array:2 [ "refs" => "1119702" "type" => "task" ] "text" => "<p><strong>6. (3)</strong> Постройте графики функций и проведите исследование свойств функций по построенным графикам:</p><p><strong>а)</strong> $y=1+\cos(1.5x)$;</p><p><strong>б)</strong> $y=1.5\cos\left(\frac{\pi}{6}-x\right)+2$;</p><p><strong>в)</strong> $y=3\cos\left(\frac{x}{3}-\frac{\pi}{3}\right)-1$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1699" "height" => 339 "path" => "/media/algebra_10/baspasy-u19/0-2-2-3zad/6-1.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 {#1203 #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" => 1551422 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1194 #items: array:1 [ 0 => App\Models\Edition {#1202 #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 {#1198 #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 {#1199 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1196 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1197 #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 {#1198} "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 {#1199} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1196} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1197} "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" => "1119702" "type" => "task" ] "text" => "<p><strong>а) $y=1+\cos1,5x$</strong></p><p>Построение графика функции $y = 1 + \cos(1,5x)$ выполняется путем преобразований графика базовой функции $y = \cos(x)$.</p><p>1. Сначала строим график $y = \cos(x)$.</p><p>2. Затем сжимаем его по горизонтали (к оси Oy) с коэффициентом 1,5. Получаем график функции $y = \cos(1,5x)$. Период этой функции будет $T = \frac{2\pi}{1,5} = \frac{4\pi}{3}$.</p><p>3. Наконец, сдвигаем полученный график на 1 единицу вверх вдоль оси Oy. Это дает нам искомый график $y = 1 + \cos(1,5x)$.</p><p><b>Исследование свойств функции по графику:</b></p><p>1. <b>Область определения:</b> множество всех действительных чисел, $D(y) = (-\infty; +\infty)$.</p><p>2. <b>Область значений:</b> так как $-1 \le \cos(1,5x) \le 1$, то $1-1 \le 1+\cos(1,5x) \le 1+1$. Следовательно, $E(y) = [0; 2]$.</p><p>3. <b>Периодичность:</b> функция является периодической с основным периодом $T = \frac{4\pi}{3}$.</p><p>4. <b>Четность:</b> функция четная, поскольку $y(-x) = 1 + \cos(1,5(-x)) = 1 + \cos(1,5x) = y(x)$. Ее график симметричен относительно оси Oy.</p><p>5. <b>Нули функции (точки пересечения с осью Ox):</b> $y=0 \implies 1 + \cos(1,5x) = 0 \implies \cos(1,5x) = -1$. Отсюда $1,5x = \pi + 2\pi n$, что дает $x = \frac{2\pi}{3} + \frac{4\pi n}{3}$, где $n \in \mathbb{Z}$.</p><p>6. <b>Промежутки знакопостоянства:</b> так как область значений $E(y) = [0; 2]$, функция неотрицательна на всей области определения, т.е. $y \ge 0$ при всех $x \in \mathbb{R}$.</p><p>7. <b>Промежутки монотонности</b> (на одном периоде, например $[0; \frac{4\pi}{3}]$):</p><p> - функция убывает на промежутке $[0 + \frac{4\pi n}{3}; \frac{2\pi}{3} + \frac{4\pi n}{3}]$, $n \in \mathbb{Z}$.</p><p> - функция возрастает на промежутке $[\frac{2\pi}{3} + \frac{4\pi n}{3}; \frac{4\pi}{3} + \frac{4\pi n}{3}]$, $n \in \mathbb{Z}$.</p><p>8. <b>Точки экстремума:</b></p><p> - точки максимума: $x_{max} = \frac{4\pi n}{3}$, $n \in \mathbb{Z}$. Значение в максимумах: $y_{max} = 2$.</p><p> - точки минимума: $x_{min} = \frac{2\pi}{3} + \frac{4\pi n}{3}$, $n \in \mathbb{Z}$. Значение в минимумах: $y_{min} = 0$.</p><p><strong>Ответ:</strong> Проведено исследование функции $y = 1 + \cos(1,5x)$, ее свойства описаны выше.</p><p><strong>б) $y=1,5\cos(\frac{\pi}{6}-x)+2$</strong></p><p>Используя свойство четности косинуса $\cos(-\alpha) = \cos(\alpha)$, можно переписать функцию в виде $y=1,5\cos(x-\frac{\pi}{6})+2$.</p><p>Построение графика функции выполняется путем преобразований графика $y = \cos(x)$.</p><p>1. Строим график $y = \cos(x)$.</p><p>2. Растягиваем его от оси Ox в 1,5 раза, получая $y = 1,5\cos(x)$. Амплитуда колебаний становится 1,5.</p><p>3. Сдвигаем полученный график вправо по оси Ox на $\frac{\pi}{6}$. Получаем $y = 1,5\cos(x - \frac{\pi}{6})$.</p><p>4. Сдвигаем график на 2 единицы вверх вдоль оси Oy. Получаем искомый график $y = 1,5\cos(x - \frac{\pi}{6}) + 2$.</p><p><b>Исследование свойств функции по графику:</b></p><p>1. <b>Область определения:</b> $D(y) = (-\infty; +\infty)$.</p><p>2. <b>Область значений:</b> $-1 \le \cos(x - \frac{\pi}{6}) \le 1 \implies -1,5 \le 1,5\cos(x - \frac{\pi}{6}) \le 1,5 \implies 2-1,5 \le 1,5\cos(x - \frac{\pi}{6}) + 2 \le 2+1,5$. Следовательно, $E(y) = [0,5; 3,5]$.</p><p>3. <b>Периодичность:</b> функция периодическая с основным периодом $T = \frac{2\pi}{|-1|} = 2\pi$.</p><p>4. <b>Четность:</b> функция является функцией общего вида (ни четной, ни нечетной), так как $y(-x) = 1,5\cos(\frac{\pi}{6}+x)+2$, что не равно ни $y(x)$, ни $-y(x)$.</p><p>5. <b>Нули функции:</b> $y=0 \implies 1,5\cos(\frac{\pi}{6} - x) = -2 \implies \cos(\frac{\pi}{6} - x) = -\frac{2}{1,5} = -\frac{4}{3}$. Так как значение косинуса не может быть меньше -1, уравнение не имеет решений. График не пересекает ось Ox.</p><p>6. <b>Промежутки знакопостоянства:</b> так как минимальное значение функции равно 0,5, функция положительна на всей области определения, $y > 0$ при всех $x \in \mathbb{R}$.</p><p>7. <b>Промежутки монотонности</b> (на одном периоде, например $[\frac{\pi}{6}; \frac{13\pi}{6}]$):</p><p> - функция убывает на промежутках $[\frac{\pi}{6} + 2\pi n; \frac{7\pi}{6} + 2\pi n]$, $n \in \mathbb{Z}$.</p><p> - функция возрастает на промежутках $[\frac{7\pi}{6} + 2\pi n; \frac{13\pi}{6} + 2\pi n]$, $n \in \mathbb{Z}$.</p><p>8. <b>Точки экстремума:</b></p><p> - точки максимума: $x_{max} = \frac{\pi}{6} + 2\pi n$, $n \in \mathbb{Z}$. Значение в максимумах: $y_{max} = 3,5$.</p><p> - точки минимума: $x_{min} = \frac{7\pi}{6} + 2\pi n$, $n \in \mathbb{Z}$. Значение в минимумах: $y_{min} = 0,5$.</p><p><strong>Ответ:</strong> Проведено исследование функции $y=1,5\cos(\frac{\pi}{6}-x)+2$, ее свойства описаны выше.</p><p><strong>в) $y=3\cos(\frac{x}{3}-\frac{\pi}{3})-1$</strong></p><p>Преобразуем аргумент косинуса: $y=3\cos(\frac{1}{3}(x-\pi))-1$.</p><p>Построение графика функции выполняется путем преобразований графика $y = \cos(x)$.</p><p>1. Строим график $y = \cos(x)$.</p><p>2. Растягиваем его по горизонтали (от оси Oy) с коэффициентом 3. Получаем $y=\cos(\frac{x}{3})$. Период становится $T = \frac{2\pi}{1/3} = 6\pi$.</p><p>3. Растягиваем график от оси Ox в 3 раза, получая $y = 3\cos(\frac{x}{3})$. Амплитуда становится 3.</p><p>4. Сдвигаем график вправо по оси Ox на $\pi$. Получаем $y = 3\cos(\frac{1}{3}(x - \pi))$.</p><p>5. Сдвигаем график на 1 единицу вниз вдоль оси Oy. Получаем искомый график $y = 3\cos(\frac{1}{3}(x - \pi)) - 1$.</p><p><b>Исследование свойств функции по графику:</b></p><p>1. <b>Область определения:</b> $D(y) = (-\infty; +\infty)$.</p><p>2. <b>Область значений:</b> $-1 \le \cos(\frac{x}{3}-\frac{\pi}{3}) \le 1 \implies -3 \le 3\cos(\frac{x}{3}-\frac{\pi}{3}) \le 3 \implies -3-1 \le 3\cos(\frac{x}{3}-\frac{\pi}{3})-1 \le 3-1$. Следовательно, $E(y) = [-4; 2]$.</p><p>3. <b>Периодичность:</b> функция периодическая с основным периодом $T = \frac{2\pi}{1/3} = 6\pi$.</p><p>4. <b>Четность:</b> функция является функцией общего вида (ни четной, ни нечетной), так как $y(-x) = 3\cos(-\frac{x}{3}-\frac{\pi}{3})-1 = 3\cos(\frac{x}{3}+\frac{\pi}{3})-1$, что не равно ни $y(x)$, ни $-y(x)$.</p><p>5. <b>Нули функции:</b> $y=0 \implies 3\cos(\frac{x}{3}-\frac{\pi}{3}) = 1 \implies \cos(\frac{x}{3}-\frac{\pi}{3}) = \frac{1}{3}$. Отсюда $\frac{x}{3}-\frac{\pi}{3} = \pm\arccos(\frac{1}{3}) + 2\pi n$, что дает $x = \pi \pm 3\arccos(\frac{1}{3}) + 6\pi n$, где $n \in \mathbb{Z}$.</p><p>6. <b>Промежутки знакопостоянства:</b></p><p> - $y > 0$ на интервалах $(\pi - 3\arccos(\frac{1}{3}) + 6\pi n; \pi + 3\arccos(\frac{1}{3}) + 6\pi n)$, $n \in \mathbb{Z}$.</p><p> - $y < 0$ на интервалах $(\pi + 3\arccos(\frac{1}{3}) + 6\pi n; 7\pi - 3\arccos(\frac{1}{3}) + 6\pi n)$, $n \in \mathbb{Z}$.</p><p>7. <b>Промежутки монотонности</b> (на одном периоде, например $[\pi; 7\pi]$):</p><p> - функция убывает на промежутках $[\pi + 6\pi n; 4\pi + 6\pi n]$, $n \in \mathbb{Z}$.</p><p> - функция возрастает на промежутках $[4\pi + 6\pi n; 7\pi + 6\pi n]$, $n \in \mathbb{Z}$.</p><p>8. <b>Точки экстремума:</b></p><p> - точки максимума: $x_{max} = \pi + 6\pi n$, $n \in \mathbb{Z}$. Значение в максимумах: $y_{max} = 2$.</p><p> - точки минимума: $x_{min} = 4\pi + 6\pi n$, $n \in \mathbb{Z}$. Значение в минимумах: $y_{min} = -4$.</p><p><strong>Ответ:</strong> Проведено исследование функции $y=3\cos(\frac{x}{3}-\frac{\pi}{3})-1$, ее свойства описаны выше.</p>" ] #original: array:6 [ "id" => 1551422 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1194} "task" => array:2 [ "refs" => "1119702" "type" => "task" ] "text" => "<p><strong>а) $y=1+\cos1,5x$</strong></p><p>Построение графика функции $y = 1 + \cos(1,5x)$ выполняется путем преобразований графика базовой функции $y = \cos(x)$.</p><p>1. Сначала строим график $y = \cos(x)$.</p><p>2. Затем сжимаем его по горизонтали (к оси Oy) с коэффициентом 1,5. Получаем график функции $y = \cos(1,5x)$. Период этой функции будет $T = \frac{2\pi}{1,5} = \frac{4\pi}{3}$.</p><p>3. Наконец, сдвигаем полученный график на 1 единицу вверх вдоль оси Oy. Это дает нам искомый график $y = 1 + \cos(1,5x)$.</p><p><b>Исследование свойств функции по графику:</b></p><p>1. <b>Область определения:</b> множество всех действительных чисел, $D(y) = (-\infty; +\infty)$.</p><p>2. <b>Область значений:</b> так как $-1 \le \cos(1,5x) \le 1$, то $1-1 \le 1+\cos(1,5x) \le 1+1$. Следовательно, $E(y) = [0; 2]$.</p><p>3. <b>Периодичность:</b> функция является периодической с основным периодом $T = \frac{4\pi}{3}$.</p><p>4. <b>Четность:</b> функция четная, поскольку $y(-x) = 1 + \cos(1,5(-x)) = 1 + \cos(1,5x) = y(x)$. Ее график симметричен относительно оси Oy.</p><p>5. <b>Нули функции (точки пересечения с осью Ox):</b> $y=0 \implies 1 + \cos(1,5x) = 0 \implies \cos(1,5x) = -1$. Отсюда $1,5x = \pi + 2\pi n$, что дает $x = \frac{2\pi}{3} + \frac{4\pi n}{3}$, где $n \in \mathbb{Z}$.</p><p>6. <b>Промежутки знакопостоянства:</b> так как область значений $E(y) = [0; 2]$, функция неотрицательна на всей области определения, т.е. $y \ge 0$ при всех $x \in \mathbb{R}$.</p><p>7. <b>Промежутки монотонности</b> (на одном периоде, например $[0; \frac{4\pi}{3}]$):</p><p> - функция убывает на промежутке $[0 + \frac{4\pi n}{3}; \frac{2\pi}{3} + \frac{4\pi n}{3}]$, $n \in \mathbb{Z}$.</p><p> - функция возрастает на промежутке $[\frac{2\pi}{3} + \frac{4\pi n}{3}; \frac{4\pi}{3} + \frac{4\pi n}{3}]$, $n \in \mathbb{Z}$.</p><p>8. <b>Точки экстремума:</b></p><p> - точки максимума: $x_{max} = \frac{4\pi n}{3}$, $n \in \mathbb{Z}$. Значение в максимумах: $y_{max} = 2$.</p><p> - точки минимума: $x_{min} = \frac{2\pi}{3} + \frac{4\pi n}{3}$, $n \in \mathbb{Z}$. Значение в минимумах: $y_{min} = 0$.</p><p><strong>Ответ:</strong> Проведено исследование функции $y = 1 + \cos(1,5x)$, ее свойства описаны выше.</p><p><strong>б) $y=1,5\cos(\frac{\pi}{6}-x)+2$</strong></p><p>Используя свойство четности косинуса $\cos(-\alpha) = \cos(\alpha)$, можно переписать функцию в виде $y=1,5\cos(x-\frac{\pi}{6})+2$.</p><p>Построение графика функции выполняется путем преобразований графика $y = \cos(x)$.</p><p>1. Строим график $y = \cos(x)$.</p><p>2. Растягиваем его от оси Ox в 1,5 раза, получая $y = 1,5\cos(x)$. Амплитуда колебаний становится 1,5.</p><p>3. Сдвигаем полученный график вправо по оси Ox на $\frac{\pi}{6}$. Получаем $y = 1,5\cos(x - \frac{\pi}{6})$.</p><p>4. Сдвигаем график на 2 единицы вверх вдоль оси Oy. Получаем искомый график $y = 1,5\cos(x - \frac{\pi}{6}) + 2$.</p><p><b>Исследование свойств функции по графику:</b></p><p>1. <b>Область определения:</b> $D(y) = (-\infty; +\infty)$.</p><p>2. <b>Область значений:</b> $-1 \le \cos(x - \frac{\pi}{6}) \le 1 \implies -1,5 \le 1,5\cos(x - \frac{\pi}{6}) \le 1,5 \implies 2-1,5 \le 1,5\cos(x - \frac{\pi}{6}) + 2 \le 2+1,5$. Следовательно, $E(y) = [0,5; 3,5]$.</p><p>3. <b>Периодичность:</b> функция периодическая с основным периодом $T = \frac{2\pi}{|-1|} = 2\pi$.</p><p>4. <b>Четность:</b> функция является функцией общего вида (ни четной, ни нечетной), так как $y(-x) = 1,5\cos(\frac{\pi}{6}+x)+2$, что не равно ни $y(x)$, ни $-y(x)$.</p><p>5. <b>Нули функции:</b> $y=0 \implies 1,5\cos(\frac{\pi}{6} - x) = -2 \implies \cos(\frac{\pi}{6} - x) = -\frac{2}{1,5} = -\frac{4}{3}$. Так как значение косинуса не может быть меньше -1, уравнение не имеет решений. График не пересекает ось Ox.</p><p>6. <b>Промежутки знакопостоянства:</b> так как минимальное значение функции равно 0,5, функция положительна на всей области определения, $y > 0$ при всех $x \in \mathbb{R}$.</p><p>7. <b>Промежутки монотонности</b> (на одном периоде, например $[\frac{\pi}{6}; \frac{13\pi}{6}]$):</p><p> - функция убывает на промежутках $[\frac{\pi}{6} + 2\pi n; \frac{7\pi}{6} + 2\pi n]$, $n \in \mathbb{Z}$.</p><p> - функция возрастает на промежутках $[\frac{7\pi}{6} + 2\pi n; \frac{13\pi}{6} + 2\pi n]$, $n \in \mathbb{Z}$.</p><p>8. <b>Точки экстремума:</b></p><p> - точки максимума: $x_{max} = \frac{\pi}{6} + 2\pi n$, $n \in \mathbb{Z}$. Значение в максимумах: $y_{max} = 3,5$.</p><p> - точки минимума: $x_{min} = \frac{7\pi}{6} + 2\pi n$, $n \in \mathbb{Z}$. Значение в минимумах: $y_{min} = 0,5$.</p><p><strong>Ответ:</strong> Проведено исследование функции $y=1,5\cos(\frac{\pi}{6}-x)+2$, ее свойства описаны выше.</p><p><strong>в) $y=3\cos(\frac{x}{3}-\frac{\pi}{3})-1$</strong></p><p>Преобразуем аргумент косинуса: $y=3\cos(\frac{1}{3}(x-\pi))-1$.</p><p>Построение графика функции выполняется путем преобразований графика $y = \cos(x)$.</p><p>1. Строим график $y = \cos(x)$.</p><p>2. Растягиваем его по горизонтали (от оси Oy) с коэффициентом 3. Получаем $y=\cos(\frac{x}{3})$. Период становится $T = \frac{2\pi}{1/3} = 6\pi$.</p><p>3. Растягиваем график от оси Ox в 3 раза, получая $y = 3\cos(\frac{x}{3})$. Амплитуда становится 3.</p><p>4. Сдвигаем график вправо по оси Ox на $\pi$. Получаем $y = 3\cos(\frac{1}{3}(x - \pi))$.</p><p>5. Сдвигаем график на 1 единицу вниз вдоль оси Oy. Получаем искомый график $y = 3\cos(\frac{1}{3}(x - \pi)) - 1$.</p><p><b>Исследование свойств функции по графику:</b></p><p>1. <b>Область определения:</b> $D(y) = (-\infty; +\infty)$.</p><p>2. <b>Область значений:</b> $-1 \le \cos(\frac{x}{3}-\frac{\pi}{3}) \le 1 \implies -3 \le 3\cos(\frac{x}{3}-\frac{\pi}{3}) \le 3 \implies -3-1 \le 3\cos(\frac{x}{3}-\frac{\pi}{3})-1 \le 3-1$. Следовательно, $E(y) = [-4; 2]$.</p><p>3. <b>Периодичность:</b> функция периодическая с основным периодом $T = \frac{2\pi}{1/3} = 6\pi$.</p><p>4. <b>Четность:</b> функция является функцией общего вида (ни четной, ни нечетной), так как $y(-x) = 3\cos(-\frac{x}{3}-\frac{\pi}{3})-1 = 3\cos(\frac{x}{3}+\frac{\pi}{3})-1$, что не равно ни $y(x)$, ни $-y(x)$.</p><p>5. <b>Нули функции:</b> $y=0 \implies 3\cos(\frac{x}{3}-\frac{\pi}{3}) = 1 \implies \cos(\frac{x}{3}-\frac{\pi}{3}) = \frac{1}{3}$. Отсюда $\frac{x}{3}-\frac{\pi}{3} = \pm\arccos(\frac{1}{3}) + 2\pi n$, что дает $x = \pi \pm 3\arccos(\frac{1}{3}) + 6\pi n$, где $n \in \mathbb{Z}$.</p><p>6. <b>Промежутки знакопостоянства:</b></p><p> - $y > 0$ на интервалах $(\pi - 3\arccos(\frac{1}{3}) + 6\pi n; \pi + 3\arccos(\frac{1}{3}) + 6\pi n)$, $n \in \mathbb{Z}$.</p><p> - $y < 0$ на интервалах $(\pi + 3\arccos(\frac{1}{3}) + 6\pi n; 7\pi - 3\arccos(\frac{1}{3}) + 6\pi n)$, $n \in \mathbb{Z}$.</p><p>7. <b>Промежутки монотонности</b> (на одном периоде, например $[\pi; 7\pi]$):</p><p> - функция убывает на промежутках $[\pi + 6\pi n; 4\pi + 6\pi n]$, $n \in \mathbb{Z}$.</p><p> - функция возрастает на промежутках $[4\pi + 6\pi n; 7\pi + 6\pi n]$, $n \in \mathbb{Z}$.</p><p>8. <b>Точки экстремума:</b></p><p> - точки максимума: $x_{max} = \pi + 6\pi n$, $n \in \mathbb{Z}$. Значение в максимумах: $y_{max} = 2$.</p><p> - точки минимума: $x_{min} = 4\pi + 6\pi n$, $n \in \mathbb{Z}$. Значение в минимумах: $y_{min} = -4$.</p><p><strong>Ответ:</strong> Проведено исследование функции $y=3\cos(\frac{x}{3}-\frac{\pi}{3})-1$, ее свойства описаны выше.</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" => "1119703" "type" => "task" ] "previous" => array:2 [ "refs" => "1119701" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Book {#1229} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\BookPage {#1212 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] 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№6 (с. 86)
Условие. №6 (с. 86)
скриншот условия
6. (3) Постройте графики функций и проведите исследование свойств функций по построенным графикам:
а) $y=1+\cos(1.5x)$;
б) $y=1.5\cos\left(\frac{\pi}{6}-x\right)+2$;
в) $y=3\cos\left(\frac{x}{3}-\frac{\pi}{3}\right)-1$.
Решение 2 (rus). №6 (с. 86)
а) $y=1+\cos1,5x$
Построение графика функции $y = 1 + \cos(1,5x)$ выполняется путем преобразований графика базовой функции $y = \cos(x)$.
1. Сначала строим график $y = \cos(x)$.
2. Затем сжимаем его по горизонтали (к оси Oy) с коэффициентом 1,5. Получаем график функции $y = \cos(1,5x)$. Период этой функции будет $T = \frac{2\pi}{1,5} = \frac{4\pi}{3}$.
3. Наконец, сдвигаем полученный график на 1 единицу вверх вдоль оси Oy. Это дает нам искомый график $y = 1 + \cos(1,5x)$.
Исследование свойств функции по графику:
1. Область определения: множество всех действительных чисел, $D(y) = (-\infty; +\infty)$.
2. Область значений: так как $-1 \le \cos(1,5x) \le 1$, то $1-1 \le 1+\cos(1,5x) \le 1+1$. Следовательно, $E(y) = [0; 2]$.
3. Периодичность: функция является периодической с основным периодом $T = \frac{4\pi}{3}$.
4. Четность: функция четная, поскольку $y(-x) = 1 + \cos(1,5(-x)) = 1 + \cos(1,5x) = y(x)$. Ее график симметричен относительно оси Oy.
5. Нули функции (точки пересечения с осью Ox): $y=0 \implies 1 + \cos(1,5x) = 0 \implies \cos(1,5x) = -1$. Отсюда $1,5x = \pi + 2\pi n$, что дает $x = \frac{2\pi}{3} + \frac{4\pi n}{3}$, где $n \in \mathbb{Z}$.
6. Промежутки знакопостоянства: так как область значений $E(y) = [0; 2]$, функция неотрицательна на всей области определения, т.е. $y \ge 0$ при всех $x \in \mathbb{R}$.
7. Промежутки монотонности (на одном периоде, например $[0; \frac{4\pi}{3}]$):
- функция убывает на промежутке $[0 + \frac{4\pi n}{3}; \frac{2\pi}{3} + \frac{4\pi n}{3}]$, $n \in \mathbb{Z}$.
- функция возрастает на промежутке $[\frac{2\pi}{3} + \frac{4\pi n}{3}; \frac{4\pi}{3} + \frac{4\pi n}{3}]$, $n \in \mathbb{Z}$.
8. Точки экстремума:
- точки максимума: $x_{max} = \frac{4\pi n}{3}$, $n \in \mathbb{Z}$. Значение в максимумах: $y_{max} = 2$.
- точки минимума: $x_{min} = \frac{2\pi}{3} + \frac{4\pi n}{3}$, $n \in \mathbb{Z}$. Значение в минимумах: $y_{min} = 0$.
Ответ: Проведено исследование функции $y = 1 + \cos(1,5x)$, ее свойства описаны выше.
б) $y=1,5\cos(\frac{\pi}{6}-x)+2$
Используя свойство четности косинуса $\cos(-\alpha) = \cos(\alpha)$, можно переписать функцию в виде $y=1,5\cos(x-\frac{\pi}{6})+2$.
Построение графика функции выполняется путем преобразований графика $y = \cos(x)$.
1. Строим график $y = \cos(x)$.
2. Растягиваем его от оси Ox в 1,5 раза, получая $y = 1,5\cos(x)$. Амплитуда колебаний становится 1,5.
3. Сдвигаем полученный график вправо по оси Ox на $\frac{\pi}{6}$. Получаем $y = 1,5\cos(x - \frac{\pi}{6})$.
4. Сдвигаем график на 2 единицы вверх вдоль оси Oy. Получаем искомый график $y = 1,5\cos(x - \frac{\pi}{6}) + 2$.
Исследование свойств функции по графику:
1. Область определения: $D(y) = (-\infty; +\infty)$.
2. Область значений: $-1 \le \cos(x - \frac{\pi}{6}) \le 1 \implies -1,5 \le 1,5\cos(x - \frac{\pi}{6}) \le 1,5 \implies 2-1,5 \le 1,5\cos(x - \frac{\pi}{6}) + 2 \le 2+1,5$. Следовательно, $E(y) = [0,5; 3,5]$.
3. Периодичность: функция периодическая с основным периодом $T = \frac{2\pi}{|-1|} = 2\pi$.
4. Четность: функция является функцией общего вида (ни четной, ни нечетной), так как $y(-x) = 1,5\cos(\frac{\pi}{6}+x)+2$, что не равно ни $y(x)$, ни $-y(x)$.
5. Нули функции: $y=0 \implies 1,5\cos(\frac{\pi}{6} - x) = -2 \implies \cos(\frac{\pi}{6} - x) = -\frac{2}{1,5} = -\frac{4}{3}$. Так как значение косинуса не может быть меньше -1, уравнение не имеет решений. График не пересекает ось Ox.
6. Промежутки знакопостоянства: так как минимальное значение функции равно 0,5, функция положительна на всей области определения, $y > 0$ при всех $x \in \mathbb{R}$.
7. Промежутки монотонности (на одном периоде, например $[\frac{\pi}{6}; \frac{13\pi}{6}]$):
- функция убывает на промежутках $[\frac{\pi}{6} + 2\pi n; \frac{7\pi}{6} + 2\pi n]$, $n \in \mathbb{Z}$.
- функция возрастает на промежутках $[\frac{7\pi}{6} + 2\pi n; \frac{13\pi}{6} + 2\pi n]$, $n \in \mathbb{Z}$.
8. Точки экстремума:
- точки максимума: $x_{max} = \frac{\pi}{6} + 2\pi n$, $n \in \mathbb{Z}$. Значение в максимумах: $y_{max} = 3,5$.
- точки минимума: $x_{min} = \frac{7\pi}{6} + 2\pi n$, $n \in \mathbb{Z}$. Значение в минимумах: $y_{min} = 0,5$.
Ответ: Проведено исследование функции $y=1,5\cos(\frac{\pi}{6}-x)+2$, ее свойства описаны выше.
в) $y=3\cos(\frac{x}{3}-\frac{\pi}{3})-1$
Преобразуем аргумент косинуса: $y=3\cos(\frac{1}{3}(x-\pi))-1$.
Построение графика функции выполняется путем преобразований графика $y = \cos(x)$.
1. Строим график $y = \cos(x)$.
2. Растягиваем его по горизонтали (от оси Oy) с коэффициентом 3. Получаем $y=\cos(\frac{x}{3})$. Период становится $T = \frac{2\pi}{1/3} = 6\pi$.
3. Растягиваем график от оси Ox в 3 раза, получая $y = 3\cos(\frac{x}{3})$. Амплитуда становится 3.
4. Сдвигаем график вправо по оси Ox на $\pi$. Получаем $y = 3\cos(\frac{1}{3}(x - \pi))$.
5. Сдвигаем график на 1 единицу вниз вдоль оси Oy. Получаем искомый график $y = 3\cos(\frac{1}{3}(x - \pi)) - 1$.
Исследование свойств функции по графику:
1. Область определения: $D(y) = (-\infty; +\infty)$.
2. Область значений: $-1 \le \cos(\frac{x}{3}-\frac{\pi}{3}) \le 1 \implies -3 \le 3\cos(\frac{x}{3}-\frac{\pi}{3}) \le 3 \implies -3-1 \le 3\cos(\frac{x}{3}-\frac{\pi}{3})-1 \le 3-1$. Следовательно, $E(y) = [-4; 2]$.
3. Периодичность: функция периодическая с основным периодом $T = \frac{2\pi}{1/3} = 6\pi$.
4. Четность: функция является функцией общего вида (ни четной, ни нечетной), так как $y(-x) = 3\cos(-\frac{x}{3}-\frac{\pi}{3})-1 = 3\cos(\frac{x}{3}+\frac{\pi}{3})-1$, что не равно ни $y(x)$, ни $-y(x)$.
5. Нули функции: $y=0 \implies 3\cos(\frac{x}{3}-\frac{\pi}{3}) = 1 \implies \cos(\frac{x}{3}-\frac{\pi}{3}) = \frac{1}{3}$. Отсюда $\frac{x}{3}-\frac{\pi}{3} = \pm\arccos(\frac{1}{3}) + 2\pi n$, что дает $x = \pi \pm 3\arccos(\frac{1}{3}) + 6\pi n$, где $n \in \mathbb{Z}$.
6. Промежутки знакопостоянства:
- $y > 0$ на интервалах $(\pi - 3\arccos(\frac{1}{3}) + 6\pi n; \pi + 3\arccos(\frac{1}{3}) + 6\pi n)$, $n \in \mathbb{Z}$.
- $y < 0$ на интервалах $(\pi + 3\arccos(\frac{1}{3}) + 6\pi n; 7\pi - 3\arccos(\frac{1}{3}) + 6\pi n)$, $n \in \mathbb{Z}$.
7. Промежутки монотонности (на одном периоде, например $[\pi; 7\pi]$):
- функция убывает на промежутках $[\pi + 6\pi n; 4\pi + 6\pi n]$, $n \in \mathbb{Z}$.
- функция возрастает на промежутках $[4\pi + 6\pi n; 7\pi + 6\pi n]$, $n \in \mathbb{Z}$.
8. Точки экстремума:
- точки максимума: $x_{max} = \pi + 6\pi n$, $n \in \mathbb{Z}$. Значение в максимумах: $y_{max} = 2$.
- точки минимума: $x_{min} = 4\pi + 6\pi n$, $n \in \mathbb{Z}$. Значение в минимумах: $y_{min} = -4$.
Ответ: Проведено исследование функции $y=3\cos(\frac{x}{3}-\frac{\pi}{3})-1$, ее свойства описаны выше.
Другие задания:
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