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Номер 12, страница 5 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Упражнения для повторения курса алгебры 7—9 классов - номер 12, страница 5.
App\Models\Task {#1024 // 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" => 1113602 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-12" "field_display_title" => "12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#335 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#341 #items: array:1 [ 0 => App\Models\Branch {#1023 #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" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#331 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "4" "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 {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #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" => 4293 "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 {#1032 #items: array:1 [ 0 => App\Models\Term {#1034 #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 {#1035 #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" => 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 {#1037 #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:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #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 {#1039 #items: array:2 [ 0 => App\Models\Term {#1040 #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" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "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 {#1041 #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" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "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 {#1042 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Term {#1044 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Term {#1046 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:1 [ 0 => App\Models\Term {#1048 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1049 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1058 #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" => 4293 "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 {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1042} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1043} "field_country" => Illuminate\Database\Eloquent\Collection {#1045} "field_city" => Illuminate\Database\Eloquent\Collection {#1047} "field_series" => Illuminate\Database\Eloquent\Collection {#1049} "field_umk" => Illuminate\Database\Eloquent\Collection {#1050} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1051} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1053} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1055} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1056} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1058} "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 {#1059 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#331} "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" 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=> Illuminate\Database\Eloquent\Collection {#1061 #items: array:3 [ 0 => App\Models\Element {#1062 #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" => 1290662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Edition {#1064 #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" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1065} "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 {#1067} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1068} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1069} "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" => "1113602" "type" => "task" ] "text" => "<p><strong>12.</strong> Постройте график функции и найдите множество значений:</p><p><strong>а)</strong> $y = x^2 - 8x;$</p><p><strong>б)</strong> $y = -x^2 + 7x;$</p><p><strong>в)</strong> $y = \sqrt{x+1} + 1;$</p><p><strong>г)</strong> $y = 2 - \sqrt{x+2};$</p><p><strong>д)</strong> $y = x^2 - |x|;$</p><p><strong>е)</strong> $y = -x^2 + |x|.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "12-1.jpg" "alt" => null "width" => "2573" "height" => 523 "path" => "/media/algebra_10/Abylkasymova-u/0-00/12-1.webp?ts=1753262978" ] ] ] #original: array:7 [ "id" => 1290662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1063} "task" => array:2 [ "refs" => "1113602" "type" => "task" ] "text" => "<p><strong>12.</strong> Постройте график функции и найдите множество значений:</p><p><strong>а)</strong> $y = x^2 - 8x;$</p><p><strong>б)</strong> $y = -x^2 + 7x;$</p><p><strong>в)</strong> $y = \sqrt{x+1} + 1;$</p><p><strong>г)</strong> $y = 2 - \sqrt{x+2};$</p><p><strong>д)</strong> $y = x^2 - |x|;$</p><p><strong>е)</strong> $y = -x^2 + |x|.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "12-1.jpg" "alt" => null "width" => "2573" "height" => 523 "path" => "/media/algebra_10/Abylkasymova-u/0-00/12-1.webp?ts=1753262978" ] ] ] #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 {#1070 #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" => 1291176 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Edition {#1072 #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" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1074 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1073} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1075} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1076} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1077} "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" => "1113602" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "12-1.jpg" "alt" => null "width" => "1756" "height" => 3018 "path" => "/media/algebra_10/Abylkasymova-u/1-00/12-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "12-2.jpg" "alt" => null "width" => "1609" "height" => 3225 "path" => "/media/algebra_10/Abylkasymova-u/1-00/12-2.webp?ts=1753263138" ] 2 => array:5 [ "name" => "12-3.jpg" "alt" => null "width" => "1466" "height" => 984 "path" => "/media/algebra_10/Abylkasymova-u/1-00/12-3.webp?ts=1753263138" ] ] ] #original: array:6 [ "id" => 1291176 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1071} "task" => array:2 [ "refs" => "1113602" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "12-1.jpg" "alt" => null "width" => "1756" "height" => 3018 "path" => "/media/algebra_10/Abylkasymova-u/1-00/12-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "12-2.jpg" "alt" => null "width" => "1609" "height" => 3225 "path" => "/media/algebra_10/Abylkasymova-u/1-00/12-2.webp?ts=1753263138" ] 2 => array:5 [ "name" => "12-3.jpg" "alt" => null "width" => "1466" "height" => 984 "path" => "/media/algebra_10/Abylkasymova-u/1-00/12-3.webp?ts=1753263138" ] ] ] #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\Element {#1078 #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" => 1549305 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Edition {#1080 #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" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Term {#1082 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1081} "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 {#1083} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1084} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1085} "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" => "1113602" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Функция $y = x^2 - 8x$ является квадратичной. Ее график — парабола. Коэффициент при $x^2$ равен $1$, что больше нуля, поэтому ветви параболы направлены вверх.</p><p>Для построения графика найдем координаты вершины параболы по формуле $x_в = -b/(2a)$, $y_в = y(x_в)$. $x_в = -(-8) / (2 \cdot 1) = 4$.</p><p>$y_в = (4)^2 - 8(4) = 16 - 32 = -16$.</p><p>Вершина параболы находится в точке $(4; -16)$.</p><p>Найдем точки пересечения с осями координат: При $x = 0$, $y = 0$. Точка пересечения с осью OY: $(0; 0)$. При $y = 0$, $x^2 - 8x = 0 \Rightarrow x(x-8)=0$, откуда $x_1=0$, $x_2=8$. Точки пересечения с осью OX: $(0; 0)$ и $(8; 0)$. График — парабола с вершиной в точке $(4; -16)$, проходящая через точки $(0; 0)$ и $(8; 0)$.</p><p>Поскольку ветви параболы направлены вверх, наименьшее значение функции достигается в вершине. Следовательно, множество значений функции (или область значений) — это все значения $y$, начиная от $-16$ и до бесконечности.</p><p><strong>Ответ:</strong> $E(y) = [-16; +\infty)$.</p><p><strong>б)</strong></p><p>Функция $y = -x^2 + 7x$ является квадратичной. Ее график — парабола. Коэффициент при $x^2$ равен $-1$, что меньше нуля, поэтому ветви параболы направлены вниз.</p><p>Найдем координаты вершины параболы: $x_в = -7 / (2 \cdot (-1)) = 3.5$.</p><p>$y_в = -(3.5)^2 + 7(3.5) = -12.25 + 24.5 = 12.25$.</p><p>Вершина параболы находится в точке $(3.5; 12.25)$.</p><p>Найдем точки пересечения с осями координат: При $x = 0$, $y = 0$. Точка пересечения с осью OY: $(0; 0)$. При $y = 0$, $-x^2 + 7x = 0 \Rightarrow x(-x+7)=0$, откуда $x_1=0$, $x_2=7$. Точки пересечения с осью OX: $(0; 0)$ и $(7; 0)$. График — парабола с вершиной в точке $(3.5; 12.25)$, проходящая через точки $(0; 0)$ и $(7; 0)$.</p><p>Поскольку ветви параболы направлены вниз, наибольшее значение функции достигается в вершине. Следовательно, множество значений функции — это все значения $y$ от минус бесконечности до $12.25$ включительно.</p><p><strong>Ответ:</strong> $E(y) = (-\infty; 12.25]$.</p><p><strong>в)</strong></p><p>Функция $y = \sqrt{x+1} + 1$. График этой функции можно получить из графика функции $y = \sqrt{x}$ путем сдвига на 1 единицу влево по оси OX и на 1 единицу вверх по оси OY.</p><p>Область определения функции: $x+1 \ge 0 \Rightarrow x \ge -1$.</p><p>Начальная точка графика находится при $x=-1$. $y(-1) = \sqrt{-1+1} + 1 = 1$. Точка $(-1; 1)$.</p><p>Найдем еще одну точку для построения: при $x=3$, $y=\sqrt{3+1}+1=3$. Точка $(3; 3)$. График представляет собой ветвь параболы, выходящую из точки $(-1; 1)$ и идущую вправо и вверх.</p><p>Значение выражения $\sqrt{x+1}$ всегда неотрицательно, т.е. $\sqrt{x+1} \ge 0$. Тогда $y = \sqrt{x+1} + 1 \ge 0+1=1$. Наименьшее значение функции равно $1$.</p><p><strong>Ответ:</strong> $E(y) = [1; +\infty)$.</p><p><strong>г)</strong></p><p>Функция $y = 2 - \sqrt{x+2}$. График этой функции можно получить из графика $y = \sqrt{x}$ следующими преобразованиями: сдвиг на 2 единицы влево, симметричное отражение относительно оси OX, и сдвиг на 2 единицы вверх.</p><p>Область определения функции: $x+2 \ge 0 \Rightarrow x \ge -2$.</p><p>Начальная точка графика находится при $x=-2$. $y(-2) = 2 - \sqrt{-2+2} = 2$. Точка $(-2; 2)$.</p><p>Найдем еще одну точку для построения: при $x=2$, $y = 2 - \sqrt{2+2} = 2 - 2 = 0$. Точка $(2; 0)$. График представляет собой ветвь параболы, выходящую из точки $(-2; 2)$ и идущую вправо и вниз.</p><p>Значение выражения $\sqrt{x+2} \ge 0$, следовательно $-\sqrt{x+2} \le 0$. Тогда $y = 2 - \sqrt{x+2} \le 2-0 = 2$. Наибольшее значение функции равно $2$.</p><p><strong>Ответ:</strong> $E(y) = (-\infty; 2]$.</p><p><strong>д)</strong></p><p>Функция $y = x^2 - |x|$. Это четная функция, так как $y(-x) = (-x)^2 - |-x| = x^2 - |x| = y(x)$. Ее график симметричен относительно оси OY.</p><p>Раскроем модуль:</p><p>1) Если $x \ge 0$, то $|x|=x$, и функция принимает вид $y = x^2 - x$. Это парабола с ветвями вверх. Вершина: $x_в = -(-1)/2 = 0.5$, $y_в = (0.5)^2 - 0.5 = 0.25 - 0.5 = -0.25$. Вершина $(0.5; -0.25)$.</p><p>2) Если $x < 0$, то $|x|=-x$, и функция принимает вид $y = x^2 + x$. Это парабола с ветвями вверх. Вершина: $x_в = -1/2 = -0.5$, $y_в = (-0.5)^2 - 0.5 = 0.25 - 0.5 = -0.25$. Вершина $(-0.5; -0.25)$.</p><p>График состоит из двух частей парабол, симметричных относительно оси OY. В точке $(0,0)$ они соединяются. Минимальное значение функции достигается в двух точках: $(0.5; -0.25)$ и $(-0.5; -0.25)$. Это значение равно $-0.25$.</p><p><strong>Ответ:</strong> $E(y) = [-0.25; +\infty)$.</p><p><strong>е)</strong></p><p>Функция $y = -x^2 + |x|$. Это четная функция, так как $y(-x) = -(-x)^2 + |-x| = -x^2 + |x| = y(x)$. Ее график симметричен относительно оси OY.</p><p>Раскроем модуль:</p><p>1) Если $x \ge 0$, то $|x|=x$, и функция принимает вид $y = -x^2 + x$. Это парабола с ветвями вниз. Вершина: $x_в = -1/(2 \cdot (-1)) = 0.5$, $y_в = -(0.5)^2 + 0.5 = -0.25 + 0.5 = 0.25$. Вершина $(0.5; 0.25)$.</p><p>2) Если $x < 0$, то $|x|=-x$, и функция принимает вид $y = -x^2 - x$. Это парабола с ветвями вниз. Вершина: $x_в = -(-1)/(2 \cdot (-1)) = -0.5$, $y_в = -(-0.5)^2 - (-0.5) = -0.25 + 0.5 = 0.25$. Вершина $(-0.5; 0.25)$.</p><p>График состоит из двух частей парабол, симметричных относительно оси OY. В точке $(0,0)$ они соединяются. Максимальное значение функции достигается в двух точках: $(0.5; 0.25)$ и $(-0.5; 0.25)$. Это значение равно $0.25$.</p><p><strong>Ответ:</strong> $E(y) = (-\infty; 0.25]$.</p>" ] #original: array:6 [ "id" => 1549305 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079} "task" => array:2 [ "refs" => "1113602" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Функция $y = x^2 - 8x$ является квадратичной. Ее график — парабола. Коэффициент при $x^2$ равен $1$, что больше нуля, поэтому ветви параболы направлены вверх.</p><p>Для построения графика найдем координаты вершины параболы по формуле $x_в = -b/(2a)$, $y_в = y(x_в)$. $x_в = -(-8) / (2 \cdot 1) = 4$.</p><p>$y_в = (4)^2 - 8(4) = 16 - 32 = -16$.</p><p>Вершина параболы находится в точке $(4; -16)$.</p><p>Найдем точки пересечения с осями координат: При $x = 0$, $y = 0$. Точка пересечения с осью OY: $(0; 0)$. При $y = 0$, $x^2 - 8x = 0 \Rightarrow x(x-8)=0$, откуда $x_1=0$, $x_2=8$. Точки пересечения с осью OX: $(0; 0)$ и $(8; 0)$. График — парабола с вершиной в точке $(4; -16)$, проходящая через точки $(0; 0)$ и $(8; 0)$.</p><p>Поскольку ветви параболы направлены вверх, наименьшее значение функции достигается в вершине. Следовательно, множество значений функции (или область значений) — это все значения $y$, начиная от $-16$ и до бесконечности.</p><p><strong>Ответ:</strong> $E(y) = [-16; +\infty)$.</p><p><strong>б)</strong></p><p>Функция $y = -x^2 + 7x$ является квадратичной. Ее график — парабола. Коэффициент при $x^2$ равен $-1$, что меньше нуля, поэтому ветви параболы направлены вниз.</p><p>Найдем координаты вершины параболы: $x_в = -7 / (2 \cdot (-1)) = 3.5$.</p><p>$y_в = -(3.5)^2 + 7(3.5) = -12.25 + 24.5 = 12.25$.</p><p>Вершина параболы находится в точке $(3.5; 12.25)$.</p><p>Найдем точки пересечения с осями координат: При $x = 0$, $y = 0$. Точка пересечения с осью OY: $(0; 0)$. При $y = 0$, $-x^2 + 7x = 0 \Rightarrow x(-x+7)=0$, откуда $x_1=0$, $x_2=7$. Точки пересечения с осью OX: $(0; 0)$ и $(7; 0)$. График — парабола с вершиной в точке $(3.5; 12.25)$, проходящая через точки $(0; 0)$ и $(7; 0)$.</p><p>Поскольку ветви параболы направлены вниз, наибольшее значение функции достигается в вершине. Следовательно, множество значений функции — это все значения $y$ от минус бесконечности до $12.25$ включительно.</p><p><strong>Ответ:</strong> $E(y) = (-\infty; 12.25]$.</p><p><strong>в)</strong></p><p>Функция $y = \sqrt{x+1} + 1$. График этой функции можно получить из графика функции $y = \sqrt{x}$ путем сдвига на 1 единицу влево по оси OX и на 1 единицу вверх по оси OY.</p><p>Область определения функции: $x+1 \ge 0 \Rightarrow x \ge -1$.</p><p>Начальная точка графика находится при $x=-1$. $y(-1) = \sqrt{-1+1} + 1 = 1$. Точка $(-1; 1)$.</p><p>Найдем еще одну точку для построения: при $x=3$, $y=\sqrt{3+1}+1=3$. Точка $(3; 3)$. График представляет собой ветвь параболы, выходящую из точки $(-1; 1)$ и идущую вправо и вверх.</p><p>Значение выражения $\sqrt{x+1}$ всегда неотрицательно, т.е. $\sqrt{x+1} \ge 0$. Тогда $y = \sqrt{x+1} + 1 \ge 0+1=1$. Наименьшее значение функции равно $1$.</p><p><strong>Ответ:</strong> $E(y) = [1; +\infty)$.</p><p><strong>г)</strong></p><p>Функция $y = 2 - \sqrt{x+2}$. График этой функции можно получить из графика $y = \sqrt{x}$ следующими преобразованиями: сдвиг на 2 единицы влево, симметричное отражение относительно оси OX, и сдвиг на 2 единицы вверх.</p><p>Область определения функции: $x+2 \ge 0 \Rightarrow x \ge -2$.</p><p>Начальная точка графика находится при $x=-2$. $y(-2) = 2 - \sqrt{-2+2} = 2$. Точка $(-2; 2)$.</p><p>Найдем еще одну точку для построения: при $x=2$, $y = 2 - \sqrt{2+2} = 2 - 2 = 0$. Точка $(2; 0)$. График представляет собой ветвь параболы, выходящую из точки $(-2; 2)$ и идущую вправо и вниз.</p><p>Значение выражения $\sqrt{x+2} \ge 0$, следовательно $-\sqrt{x+2} \le 0$. Тогда $y = 2 - \sqrt{x+2} \le 2-0 = 2$. Наибольшее значение функции равно $2$.</p><p><strong>Ответ:</strong> $E(y) = (-\infty; 2]$.</p><p><strong>д)</strong></p><p>Функция $y = x^2 - |x|$. Это четная функция, так как $y(-x) = (-x)^2 - |-x| = x^2 - |x| = y(x)$. Ее график симметричен относительно оси OY.</p><p>Раскроем модуль:</p><p>1) Если $x \ge 0$, то $|x|=x$, и функция принимает вид $y = x^2 - x$. Это парабола с ветвями вверх. Вершина: $x_в = -(-1)/2 = 0.5$, $y_в = (0.5)^2 - 0.5 = 0.25 - 0.5 = -0.25$. Вершина $(0.5; -0.25)$.</p><p>2) Если $x < 0$, то $|x|=-x$, и функция принимает вид $y = x^2 + x$. Это парабола с ветвями вверх. Вершина: $x_в = -1/2 = -0.5$, $y_в = (-0.5)^2 - 0.5 = 0.25 - 0.5 = -0.25$. Вершина $(-0.5; -0.25)$.</p><p>График состоит из двух частей парабол, симметричных относительно оси OY. В точке $(0,0)$ они соединяются. Минимальное значение функции достигается в двух точках: $(0.5; -0.25)$ и $(-0.5; -0.25)$. Это значение равно $-0.25$.</p><p><strong>Ответ:</strong> $E(y) = [-0.25; +\infty)$.</p><p><strong>е)</strong></p><p>Функция $y = -x^2 + |x|$. Это четная функция, так как $y(-x) = -(-x)^2 + |-x| = -x^2 + |x| = y(x)$. Ее график симметричен относительно оси OY.</p><p>Раскроем модуль:</p><p>1) Если $x \ge 0$, то $|x|=x$, и функция принимает вид $y = -x^2 + x$. Это парабола с ветвями вниз. Вершина: $x_в = -1/(2 \cdot (-1)) = 0.5$, $y_в = -(0.5)^2 + 0.5 = -0.25 + 0.5 = 0.25$. Вершина $(0.5; 0.25)$.</p><p>2) Если $x < 0$, то $|x|=-x$, и функция принимает вид $y = -x^2 - x$. Это парабола с ветвями вниз. Вершина: $x_в = -(-1)/(2 \cdot (-1)) = -0.5$, $y_в = -(-0.5)^2 - (-0.5) = -0.25 + 0.5 = 0.25$. Вершина $(-0.5; 0.25)$.</p><p>График состоит из двух частей парабол, симметричных относительно оси OY. В точке $(0,0)$ они соединяются. Максимальное значение функции достигается в двух точках: $(0.5; 0.25)$ и $(-0.5; 0.25)$. Это значение равно $0.25$.</p><p><strong>Ответ:</strong> $E(y) = (-\infty; 0.25]$.</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" => "1113603" "type" => "task" ] "previous" => array:2 [ "refs" => "1113601" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\BookPage {#1100 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false 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Illuminate\Database\Eloquent\Collection {#1126} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1127} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1128} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1129} "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 } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1130 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1141 #items: array:1 [ 0 => App\Models\Element {#1140 #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" => 1261767 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:1 [ 0 => App\Models\Edition {#1142 …30} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "3154" "height" => 4601 "path" => "/media/algebra_10/Abylkasymova-u/0-1/5-1.webp?ts=1752147445" ] ] ] #original: array:6 [ "id" => 1261767 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1150} "book_page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "3154" "height" => 4601 …1 ] ] ] #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" => "1098343" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098341" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1270 #items: array:8 [ 0 => App\Models\Task {#1284 #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" => 1113596 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1288 #items: array:1 [ …1] #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 {#1287 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1321 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1319 #items: array:1 [ …1] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1323 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113597" "type" => "task" ] "previous" => array:2 [ "refs" => "1113595" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1329 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113596 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1288} "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 {#1287} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1321} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1319} "content" => Illuminate\Database\Eloquent\Collection {#1323} "next" => array:2 [ "refs" => "1113597" "type" => "task" ] "previous" => array:2 [ "refs" => "1113595" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1329} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #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\Task {#1322 #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" => 1113597 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1320 #items: array:1 [ …1] #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 {#1328 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1330 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1326 #items: array:1 [ …1] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1325 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113598" "type" => "task" ] "previous" => array:2 [ "refs" => "1113596" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1355 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113597 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1320} "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 {#1328} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1330} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1326} "content" => Illuminate\Database\Eloquent\Collection {#1325} "next" => array:2 [ "refs" => "1113598" "type" => "task" ] "previous" => array:2 [ "refs" => "1113596" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1355} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #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\Task {#1327 #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" => 1113598 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1324 #items: array:1 [ …1] #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 {#1354 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1356 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1352 #items: array:1 [ …1] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1350 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113599" "type" => "task" ] "previous" => array:2 [ "refs" => "1113597" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1369 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113598 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1324} "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 {#1354} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1356} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1352} "content" => Illuminate\Database\Eloquent\Collection {#1350} "next" => array:2 [ "refs" => "1113599" "type" => "task" ] "previous" => array:2 [ "refs" => "1113597" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1369} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #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\Task {#1353 #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" => 1113599 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1349 #items: array:1 [ …1] #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 {#1368 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1370 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1366 #items: array:1 [ …1] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1365 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113600" "type" => "task" ] "previous" => array:2 [ "refs" => "1113598" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1383 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113599 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1349} "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 {#1368} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1370} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1366} "content" => Illuminate\Database\Eloquent\Collection {#1365} "next" => array:2 [ "refs" => "1113600" "type" => "task" ] "previous" => array:2 [ "refs" => "1113598" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1383} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #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\Task {#1367 #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" => 1113600 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-10" "field_display_title" => "10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1364 #items: array:1 [ …1] #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 {#1382 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1384 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1380 #items: array:1 [ …1] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1379 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113601" "type" => "task" ] "previous" => array:2 [ "refs" => "1113599" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1397 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113600 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-10" "field_display_title" => "10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1364} "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 {#1382} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1384} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1380} "content" => Illuminate\Database\Eloquent\Collection {#1379} "next" => array:2 [ "refs" => "1113601" "type" => "task" ] "previous" => array:2 [ "refs" => "1113599" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1397} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1381 #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" => 1113601 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-11" "field_display_title" => "11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1378 #items: array:1 [ …1] #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 {#1396 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1398 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1394 #items: array:1 [ …1] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1420 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113602" "type" => "task" ] "previous" => array:2 [ "refs" => "1113600" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1427 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113601 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-11" "field_display_title" => "11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1378} "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 {#1396} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1398} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1394} "content" => Illuminate\Database\Eloquent\Collection {#1420} "next" => array:2 [ "refs" => "1113602" "type" => "task" ] "previous" => array:2 [ "refs" => "1113600" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1427} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 6 => App\Models\Task {#1411 #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" => 1113602 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-12" "field_display_title" => "12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1412 #items: array:1 [ …1] #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 {#1413 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1418 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1417 #items: array:1 [ …1] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1442 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113603" "type" => "task" ] "previous" => array:2 [ "refs" => "1113601" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1443 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113602 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-12" "field_display_title" => "12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1412} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null 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№12 (с. 5)
Условие. №12 (с. 5)
скриншот условия
12. Постройте график функции и найдите множество значений:
а) $y = x^2 - 8x;$
б) $y = -x^2 + 7x;$
в) $y = \sqrt{x+1} + 1;$
г) $y = 2 - \sqrt{x+2};$
д) $y = x^2 - |x|;$
е) $y = -x^2 + |x|.$
Решение. №12 (с. 5)
Решение 2. №12 (с. 5)
а)
Функция $y = x^2 - 8x$ является квадратичной. Ее график — парабола. Коэффициент при $x^2$ равен $1$, что больше нуля, поэтому ветви параболы направлены вверх.
Для построения графика найдем координаты вершины параболы по формуле $x_в = -b/(2a)$, $y_в = y(x_в)$. $x_в = -(-8) / (2 \cdot 1) = 4$.
$y_в = (4)^2 - 8(4) = 16 - 32 = -16$.
Вершина параболы находится в точке $(4; -16)$.
Найдем точки пересечения с осями координат: При $x = 0$, $y = 0$. Точка пересечения с осью OY: $(0; 0)$. При $y = 0$, $x^2 - 8x = 0 \Rightarrow x(x-8)=0$, откуда $x_1=0$, $x_2=8$. Точки пересечения с осью OX: $(0; 0)$ и $(8; 0)$. График — парабола с вершиной в точке $(4; -16)$, проходящая через точки $(0; 0)$ и $(8; 0)$.
Поскольку ветви параболы направлены вверх, наименьшее значение функции достигается в вершине. Следовательно, множество значений функции (или область значений) — это все значения $y$, начиная от $-16$ и до бесконечности.
Ответ: $E(y) = [-16; +\infty)$.
б)
Функция $y = -x^2 + 7x$ является квадратичной. Ее график — парабола. Коэффициент при $x^2$ равен $-1$, что меньше нуля, поэтому ветви параболы направлены вниз.
Найдем координаты вершины параболы: $x_в = -7 / (2 \cdot (-1)) = 3.5$.
$y_в = -(3.5)^2 + 7(3.5) = -12.25 + 24.5 = 12.25$.
Вершина параболы находится в точке $(3.5; 12.25)$.
Найдем точки пересечения с осями координат: При $x = 0$, $y = 0$. Точка пересечения с осью OY: $(0; 0)$. При $y = 0$, $-x^2 + 7x = 0 \Rightarrow x(-x+7)=0$, откуда $x_1=0$, $x_2=7$. Точки пересечения с осью OX: $(0; 0)$ и $(7; 0)$. График — парабола с вершиной в точке $(3.5; 12.25)$, проходящая через точки $(0; 0)$ и $(7; 0)$.
Поскольку ветви параболы направлены вниз, наибольшее значение функции достигается в вершине. Следовательно, множество значений функции — это все значения $y$ от минус бесконечности до $12.25$ включительно.
Ответ: $E(y) = (-\infty; 12.25]$.
в)
Функция $y = \sqrt{x+1} + 1$. График этой функции можно получить из графика функции $y = \sqrt{x}$ путем сдвига на 1 единицу влево по оси OX и на 1 единицу вверх по оси OY.
Область определения функции: $x+1 \ge 0 \Rightarrow x \ge -1$.
Начальная точка графика находится при $x=-1$. $y(-1) = \sqrt{-1+1} + 1 = 1$. Точка $(-1; 1)$.
Найдем еще одну точку для построения: при $x=3$, $y=\sqrt{3+1}+1=3$. Точка $(3; 3)$. График представляет собой ветвь параболы, выходящую из точки $(-1; 1)$ и идущую вправо и вверх.
Значение выражения $\sqrt{x+1}$ всегда неотрицательно, т.е. $\sqrt{x+1} \ge 0$. Тогда $y = \sqrt{x+1} + 1 \ge 0+1=1$. Наименьшее значение функции равно $1$.
Ответ: $E(y) = [1; +\infty)$.
г)
Функция $y = 2 - \sqrt{x+2}$. График этой функции можно получить из графика $y = \sqrt{x}$ следующими преобразованиями: сдвиг на 2 единицы влево, симметричное отражение относительно оси OX, и сдвиг на 2 единицы вверх.
Область определения функции: $x+2 \ge 0 \Rightarrow x \ge -2$.
Начальная точка графика находится при $x=-2$. $y(-2) = 2 - \sqrt{-2+2} = 2$. Точка $(-2; 2)$.
Найдем еще одну точку для построения: при $x=2$, $y = 2 - \sqrt{2+2} = 2 - 2 = 0$. Точка $(2; 0)$. График представляет собой ветвь параболы, выходящую из точки $(-2; 2)$ и идущую вправо и вниз.
Значение выражения $\sqrt{x+2} \ge 0$, следовательно $-\sqrt{x+2} \le 0$. Тогда $y = 2 - \sqrt{x+2} \le 2-0 = 2$. Наибольшее значение функции равно $2$.
Ответ: $E(y) = (-\infty; 2]$.
д)
Функция $y = x^2 - |x|$. Это четная функция, так как $y(-x) = (-x)^2 - |-x| = x^2 - |x| = y(x)$. Ее график симметричен относительно оси OY.
Раскроем модуль:
1) Если $x \ge 0$, то $|x|=x$, и функция принимает вид $y = x^2 - x$. Это парабола с ветвями вверх. Вершина: $x_в = -(-1)/2 = 0.5$, $y_в = (0.5)^2 - 0.5 = 0.25 - 0.5 = -0.25$. Вершина $(0.5; -0.25)$.
2) Если $x < 0$, то $|x|=-x$, и функция принимает вид $y = x^2 + x$. Это парабола с ветвями вверх. Вершина: $x_в = -1/2 = -0.5$, $y_в = (-0.5)^2 - 0.5 = 0.25 - 0.5 = -0.25$. Вершина $(-0.5; -0.25)$.
График состоит из двух частей парабол, симметричных относительно оси OY. В точке $(0,0)$ они соединяются. Минимальное значение функции достигается в двух точках: $(0.5; -0.25)$ и $(-0.5; -0.25)$. Это значение равно $-0.25$.
Ответ: $E(y) = [-0.25; +\infty)$.
е)
Функция $y = -x^2 + |x|$. Это четная функция, так как $y(-x) = -(-x)^2 + |-x| = -x^2 + |x| = y(x)$. Ее график симметричен относительно оси OY.
Раскроем модуль:
1) Если $x \ge 0$, то $|x|=x$, и функция принимает вид $y = -x^2 + x$. Это парабола с ветвями вниз. Вершина: $x_в = -1/(2 \cdot (-1)) = 0.5$, $y_в = -(0.5)^2 + 0.5 = -0.25 + 0.5 = 0.25$. Вершина $(0.5; 0.25)$.
2) Если $x < 0$, то $|x|=-x$, и функция принимает вид $y = -x^2 - x$. Это парабола с ветвями вниз. Вершина: $x_в = -(-1)/(2 \cdot (-1)) = -0.5$, $y_в = -(-0.5)^2 - (-0.5) = -0.25 + 0.5 = 0.25$. Вершина $(-0.5; 0.25)$.
График состоит из двух частей парабол, симметричных относительно оси OY. В точке $(0,0)$ они соединяются. Максимальное значение функции достигается в двух точках: $(0.5; 0.25)$ и $(-0.5; 0.25)$. Это значение равно $0.25$.
Ответ: $E(y) = (-\infty; 0.25]$.
Другие задания:
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