Receiving Helpdesk

how do you stretch or shrink a graph

by Dr. Ayden Stamm V Published 3 years ago Updated 2 years ago

How do you stretch or shrink a graph?

  • When by either f (x) or x is multiplied by a number, functions can “stretch” or “shrink” vertically or horizontally, respectively, when graphed.
  • In general, a vertical stretch is given by the equation y=bf (x) y = b f ( x ) .
  • In general, a horizontal stretch is given by the equation y=f (cx) y = f ( c x ) .

We can also stretch and shrink the graph of a function. To stretch or shrink the graph in the y direction, multiply or divide the output by a constant. 2f (x) is stretched in the y direction by a factor of 2, and f (x) is shrunk in the y direction by a factor of 2 (or stretched by a factor of ).

Full Answer

How to stretch or shrink the graph in the x direction?

To stretch or shrink the graph in the x direction, divide or multiply the input by a constant. As in translating, when we change the input, the function changes to compensate. Thus, dividing the input by a constant stretches the function in the x direction, and multiplying the input by a constant shrinks...

How do you shrink a graph by a factor of 3?

If g(x) = f (3x): For any given output, the input of g is one-third the input of f, so the graph is shrunk horizontally by a factor of 3.

How do you shrink a function in x direction?

Thus, dividing the input by a constant stretches the function in the x direction, and multiplying the input by a constant shrinks the function in the x direction. f (x) is stretched in the x direction by a factor of 2, and f (2x) is shrunk in the x direction by a factor of 2 (or stretched by a factor of frac12 ).

How do you stretch horizontally by a factor of K?

Now let's stretch horizontally by a factor of k. That can be done by replacing x with x/k this time: y = f (x/k) When thought of in that way, the two approaches are the same--only the variable replaced changes depending on if you dilate vertically or horizontally.

How do you stretch and shrink a graph?

0:297:57Pre-Calculus - Applying stretching and shrinking transformations - YouTubeYouTubeStart of suggested clipEnd of suggested clipIf you multiply by a number larger than one on the outside this will stretch it vertically. So itMoreIf you multiply by a number larger than one on the outside this will stretch it vertically. So it makes it look taller in a lot of cases.

How do you know when to stretch or shrink a graph?

2:385:04Vertically Stretching and Shrinking Graphs - YouTubeYouTubeStart of suggested clipEnd of suggested clipSo again we know the graph of something. Then. If you take that real number and multiply it timesMoreSo again we know the graph of something. Then. If you take that real number and multiply it times your original function you're going to get a vertical stretch. If that real if the absolute value of

What does it mean to stretch a graph?

When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression.

How does a graph get stretched?

When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression.

How do you know if its a horizontal stretch or shrink?

3:159:32Function Transformations: Horizontal and Vertical Stretches ... - YouTubeYouTubeStart of suggested clipEnd of suggested clipIf we have f of bx where b is greater than one this will compress the graph of f of x horizontally.MoreIf we have f of bx where b is greater than one this will compress the graph of f of x horizontally.

How do you find a horizontal stretch or shrink?

A horizontal stretch or shrink by a factor of 1/k means that the point (x, y) on the graph of f(x) is transformed to the point (x/k, y) on the graph of g(x). Consider the following base functions, (1) f (x) = x2 - 3, (2) g(x) = cos (x).

What is a shrink in math?

Also, a vertical stretch/shrink by a factor of k means that the point (x, y) on the graph of f (x) is transformed to the point (x, ky) on the graph of g(x).

What is a transformation that shrinks or stretches a figure?

Dilation. a transformation that shrinks or stretches a figure.

How do you compress and stretch a function?

In math terms, you can stretch or compress a function horizontally by multiplying x by some number before any other operations. To stretch the function, multiply by a fraction between 0 and 1. To compress the function, multiply by some number greater than 1.

How do you shrink a graph horizontally?

To shrink or compress horizontally by a factor of c, replace y = f(x) with y = f(cx). Note that if |c|<1, that's the same as scaling, or stretching, by a factor of 1/c.

How do you stretch a horizontal graph?

We can only horizontally stretch a graph by a factor of 1/a when the input value is also increased by a. When f(x) is stretched horizontally to f(ax), multiply the x-coordinates by a. Retain the y-intercepts' position. The resulting function will have the same range but may have a different domain.

What is a stretch in math?

A stretch or compression is a function transformation that makes a graph narrower or wider. stretching. Stretching a graph means to make the graph narrower or wider.

Translating (shifting) a graph

Translation means moving an object without rotation, and can be described as “sliding”. In describing transformations of graphs, some textbooks use the formal term “translate”, while others use an informal term like “shift”.

A note on reflection

None of these discussions went deeper into reflections than a brief mention in the first question. I will just add here that you can think of a reflection as a “stretch by a factor of -1”. That is, it just reverses direction.

Horizontal shifts: A closer look

The horizontal transformations, involving x, confuse many students. Here is a question from 2002 about just that:

Horizontal stretches

In general, everything we do with x will be the opposite of what you might expect, for this same reason. This is true not only of horizontal shifts, but of horizontal stretching as well, which we haven’t seen yet. Here is a question specifically about that issue, from 2004:

Why horizontal transformations are backward

Functional Transformations Explained and Undone Why do horizontal transformations behave in an opposite manner? For example, why does f (x + c) shift a graph c units to the left? Likewise, to shrink or stretch horizontally, i.e., f (cx), why do you divide the x coordinate by c?

What do you call it?

Stretching Definitions, and Compressing The original function is y = f (x). Given a new function y = f (cx).

image

Translating (Shifting) A Graph

Shifting and Stretching

  • Here is another very similar question from 2001: This time we have a vertical translation, a horizontal translation, and a vertical dilation. I chose to illustrate each concept with sample graphs, with only brief explanation of why they do what they do: Here are some actual graphs corresponding to the three above: [shift up 2]: Here, the point (2, ...
See more on themathdoctors.org

A Note on Reflection

  • None of these discussions went deeper into reflections than a brief mention in the first question. I will just add here that you can think of a reflection as a “stretch by a factor of -1”. That is, it just reverses direction. So a vertical reflection (reflection in the x-axis) is accomplished by , which changes the sign of y; and a horizontal reflection (reflection in the y-axis) is accomplished by , w…
See more on themathdoctors.org

Horizontal Shifts: A Closer Look

  • The horizontal transformations, involving x, confuse many students. Here is a question from 2002 about just that: I referred to the last answer, and gave a little more detail: We have to add c to x to compensate for the fact that it will be decreased by c before being fed into function f. So replacing with in the function moves every point to the right by cunits.
See more on themathdoctors.org

Horizontal Stretches

  • In general, everything we do with xwill be the opposite of what you might expect, for this same reason. This is true not only of horizontal shifts, but of horizontal stretching as well, which we haven’t seen yet. Here is a question specifically about that issue, from 2004: Here is an example, based on our function above: The dotted graph is f(2x), compressed (shrunk) by a factor of 1/2 h…
See more on themathdoctors.org

What Do You Call It?

  • In 2014, we got a very different question, asking about the terminology of stretches: If you don’t see what Jason is concerned about, notice that for , his book calls it a stretch by a factor of , where I would call it 2; and it calls f(2x) a compression by a factor of 2, which could also be called a factor of (since that is what coordinates are multiplied by). Is the book wrong? I had observed …
See more on themathdoctors.org

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 1 2 3 4 5 6 7 8 9
8.3.21PHP Version1.83sRequest Duration2MBMemory UsageGET {post}Route
  • warninglog[06:23:09] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\QueryFormatter:...
  • warninglog[06:23:09] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\QueryFormatter:...
  • warninglog[06:23:09] LOG.warning: Callables of the form ["Swift_SmtpTransport", "Swift_Transport_EsmtpTranspor...
  • warninglog[06:23:09] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\SimpleFormatter...
  • warninglog[06:23:09] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\SimpleFormatter...
  • warninglog[06:23:09] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[06:23:09] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • Booting (14.92ms)
  • Application (1.82s)
  • 1 x Application (99.16%)
    1.82s
    1 x Booting (0.81%)
    14.92ms
    7 templates were rendered
    • themes.DevBlog.content.post (resources/views/themes/DevBlog/content/post.blade.php)34blade
      Params
      0
      post
      1
      postContent
      2
      author
      3
      updated_at
      4
      bing_rich_snippet_text
      5
      bing_rich_snippet_link
      6
      bing_related_keywords
      7
      google_related_keywords
      8
      bing_news_title
      9
      bing_news_description
      10
      bing_videos
      11
      bing_images
      12
      bing_search_result_title
      13
      bing_search_result_description
      14
      bing_search_result_url
      15
      bing_paa_questions
      16
      bing_paa_answers
      17
      bing_slider_faq_questions
      18
      bing_slider_faq_answers
      19
      bing_pop_faq_questions
      20
      bing_pop_faq_answers
      21
      bing_tab_faq_questions
      22
      bing_tab_faq_answers
      23
      google_faq_questions
      24
      google_faq_answers
      25
      google_rich_snippet
      26
      google_search_result
      27
      indexedArray
      28
      total_images
      29
      total_videos
      30
      settings
      31
      url_current
      32
      menus
      33
      sidebar
    • themes.DevBlog.layouts.master (resources/views/themes/DevBlog/layouts/master.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.head (resources/views/themes/DevBlog/panels/head.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.header (resources/views/themes/DevBlog/panels/header.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.navbar (resources/views/themes/DevBlog/panels/navbar.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.footer (resources/views/themes/DevBlog/panels/footer.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.scripts (resources/views/themes/DevBlog/panels/scripts.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    uri
    GET {post}
    middleware
    web, checkdate
    as
    post.show
    controller
    App\Http\Controllers\Frontend\json_data\PostController@show
    namespace
    where
    file
    app/Http/Controllers/Frontend/json_data/PostController.php:18-166
    7 statements were executed1.79s
    • select * from `posts` where `published_at` <= '2025-06-08 06:23:09' and `slug` = 'how-do-you-stretch-or-shrink-a-graph' and `posts`.`deleted_at` is null limit 1
      4.47ms/app/Providers/RouteServiceProvider.php:54receivinghelpdeskask
      Metadata
      Bindings
      • 0. 2025-06-08 06:23:09
      • 1. how-do-you-stretch-or-shrink-a-graph
      Backtrace
      • 15. /app/Providers/RouteServiceProvider.php:54
      • 18. /vendor/laravel/framework/src/Illuminate/Routing/Router.php:842
      • 19. Route binding:39
      • 20. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 21. /vendor/laravel/framework/src/Illuminate/Foundation/Http/Middleware/VerifyCsrfToken.php:78
    • select * from `json_post_contents` where `json_post_contents`.`post_id` = 179156 and `json_post_contents`.`post_id` is not null and `rewrite_id` = 0
      10.43msmiddleware::checkdate:30receivinghelpdeskask
      Metadata
      Bindings
      • 0. 179156
      • 1. 0
      Backtrace
      • 19. middleware::checkdate:30
      • 20. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 21. /vendor/laravel/jetstream/src/Http/Middleware/ShareInertiaData.php:61
      • 22. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 23. /vendor/laravel/framework/src/Illuminate/Routing/Middleware/SubstituteBindings.php:50
    • select * from `nova_menu_menus` where `slug` = 'header' limit 1
      690μs/vendor/outl1ne/nova-menu-builder/src/helpers.php:32receivinghelpdeskask
      Metadata
      Bindings
      • 0. header
      Backtrace
      • 15. /vendor/outl1ne/nova-menu-builder/src/helpers.php:32
      • 17. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 18. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 19. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
      • 20. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
    • select * from `nova_menu_menu_items` where `nova_menu_menu_items`.`menu_id` = 1 and `nova_menu_menu_items`.`menu_id` is not null and `parent_id` is null order by `parent_id` asc, `order` asc, `name` asc
      470μs/vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35receivinghelpdeskask
      Metadata
      Bindings
      • 0. 1
      Backtrace
      • 19. /vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35
      • 20. /vendor/outl1ne/nova-menu-builder/src/helpers.php:33
      • 22. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 23. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 24. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
    • select * from `nova_menu_menu_items` where `nova_menu_menu_items`.`parent_id` in (1) order by `order` asc
      600μs/vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35receivinghelpdeskask
      Metadata
      Backtrace
      • 24. /vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35
      • 25. /vendor/outl1ne/nova-menu-builder/src/helpers.php:33
      • 27. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 28. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 29. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
    • select `id`, `post_title`, `slug` from `posts` where `status` = 'publish' and `posts`.`deleted_at` is null order by RAND() limit 10
      1.77s/app/View/Composers/SidebarView.php:22receivinghelpdeskask
      Metadata
      Bindings
      • 0. publish
      Backtrace
      • 14. /app/View/Composers/SidebarView.php:22
      • 15. /app/View/Composers/SidebarView.php:12
      • 16. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:124
      • 17. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:162
      • 20. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:177
    • select * from `fake_users` where `fake_users`.`id` = 33358 limit 1
      2.2msview::2dd102cf0462e89a4d4d8bc77355d767652bf9aa:15receivinghelpdeskask
      Metadata
      Bindings
      • 0. 33358
      Backtrace
      • 21. view::2dd102cf0462e89a4d4d8bc77355d767652bf9aa:15
      • 23. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:108
      • 24. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 25. /vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
      • 26. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:61
    App\Models\FakeUser
    1
    Outl1ne\MenuBuilder\Models\MenuItem
    1
    Outl1ne\MenuBuilder\Models\Menu
    1
    App\Models\JsonPostContent
    1
    App\Models\Post
    11
        _token
        u85QzFX4NDGbfEgMrhS8kF0PrZjVASGWslogvtQY
        _previous
        array:1 [ "url" => "https://receivinghelpdesk.com/ask/how-do-you-stretch-or-shrink-a-graph" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /how-do-you-stretch-or-shrink-a-graph
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:25 [ "cookie" => array:1 [ 0 => "XSRF-TOKEN=eyJpdiI6IkNOYkpIV01sdHgwMCtsOFNHT2QwS3c9PSIsInZhbHVlIjoic1hnUjlJeks5MC94NG5zTVAvNHQ1RFQwMzA3aGQ5WkhrOHl1ZFBDZVB5NDJKL2cvcFE0aExXY0lsOC9YQUUzWi9kcnNTNThiMVNYWVB2RHFWWXlYY0IvWGgycWdialFwZEZVQWgxSGRqMjZsM1ZMUTAvbWZ5MnN0bktnOGUyUjciLCJtYWMiOiI0NDQ4NjQ5MjlkN2QyNzM5MTM0YWRjNDljMzRkY2UyNTYwY2RlZjY2YWFhOGNiNmQzNDlmYWVmNWU1NTJkZWEwIiwidGFnIjoiIn0%3D; askhelpdesk_session=eyJpdiI6IkQ4amp4b3ppT3hVVXhMaWFPdTV2Z0E9PSIsInZhbHVlIjoiTStLL3owNWZMYnBUOU4zZFRaZ281M2MxWCs1cy9vS1l2dWp0THkwK2NHdEZDdFpwQTMzbU1JMlBieXQxU1BqTTJnZ0xNUDNWU0V0R2lVbUVkTGVjY3I4VnFtUTltZFhzT1Z6b1VTbU03VzR5WmplRmhLZ1JBTFhaS0ZnajVQMisiLCJtYWMiOiI4NzFhODUxOWZiNjg5ZWM3YmEyNmE2OGVhZmNmYTdlNWI3NmZhMzEwMzdlN2E5YzhjM2U5ZTRkMzYyMmQ0MjIxIiwidGFnIjoiIn0%3D; _pk_id.64.7c30=1a9fb7309e185dd2.1749343989.; _pk_ses.64.7c30=1XSRF-TOKEN=eyJpdiI6IkNOYkpIV01sdHgwMCtsOFNHT2QwS3c9PSIsInZhbHVlIjoic1hnUjlJeks5MC94NG5zTVAvNHQ1RFQwMzA3aGQ5WkhrOHl1ZFBDZVB5NDJKL2cvcFE0aExXY0lsOC9YQUUzWi9kcnNTN" ] "cf-ipcountry" => array:1 [ 0 => "US" ] "cf-connecting-ip" => array:1 [ 0 => "216.73.216.214" ] "cdn-loop" => array:1 [ 0 => "cloudflare; loops=1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""Chromium";v="130", "HeadlessChrome";v="130", "Not?A_Brand";v="99"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] "accept-encoding" => array:1 [ 0 => "gzip, br" ] "cf-ray" => array:1 [ 0 => "94c4691c8b97f602-ORD" ] "priority" => array:1 [ 0 => "u=0, i" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "cf-visitor" => array:1 [ 0 => "{"scheme":"https"}" ] "connection" => array:1 [ 0 => "close" ] "x-forwarded-proto" => array:1 [ 0 => "https" ] "x-forwarded-for" => array:1 [ 0 => "216.73.216.214, 172.70.131.215" ] "x-server-addr" => array:1 [ 0 => "154.12.239.204" ] "host" => array:1 [ 0 => "receivinghelpdesk.com" ] ]
        request_server
        0 of 0
        array:56 [ "USER" => "runcloud" "HOME" => "/home/runcloud" "SCRIPT_NAME" => "/ask/index.php" "REQUEST_URI" => "/ask/how-do-you-stretch-or-shrink-a-graph" "QUERY_STRING" => "" "REQUEST_METHOD" => "GET" "SERVER_PROTOCOL" => "HTTP/1.0" "GATEWAY_INTERFACE" => "CGI/1.1" "REDIRECT_URL" => "/ask/how-do-you-stretch-or-shrink-a-graph" "REMOTE_PORT" => "45786" "SCRIPT_FILENAME" => "/home/runcloud/webapps/ReceivingHelpDesk/ask/index.php" "SERVER_ADMIN" => "you@example.com" "CONTEXT_DOCUMENT_ROOT" => "/home/runcloud/webapps/ReceivingHelpDesk/" "CONTEXT_PREFIX" => "" "REQUEST_SCHEME" => "http" "DOCUMENT_ROOT" => "/home/runcloud/webapps/ReceivingHelpDesk/" "REMOTE_ADDR" => "172.70.131.215" "SERVER_PORT" => "80" "SERVER_ADDR" => "127.0.0.1" "SERVER_NAME" => "receivinghelpdesk.com" "SERVER_SOFTWARE" => "Apache/2.4.63 (Unix) OpenSSL/1.1.1f" "SERVER_SIGNATURE" => "" "LD_LIBRARY_PATH" => "/RunCloud/Packages/apache2-rc/lib" "PATH" => "/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin" "HTTP_COOKIE" => "XSRF-TOKEN=eyJpdiI6IkNOYkpIV01sdHgwMCtsOFNHT2QwS3c9PSIsInZhbHVlIjoic1hnUjlJeks5MC94NG5zTVAvNHQ1RFQwMzA3aGQ5WkhrOHl1ZFBDZVB5NDJKL2cvcFE0aExXY0lsOC9YQUUzWi9kcnNTNThiMVNYWVB2RHFWWXlYY0IvWGgycWdialFwZEZVQWgxSGRqMjZsM1ZMUTAvbWZ5MnN0bktnOGUyUjciLCJtYWMiOiI0NDQ4NjQ5MjlkN2QyNzM5MTM0YWRjNDljMzRkY2UyNTYwY2RlZjY2YWFhOGNiNmQzNDlmYWVmNWU1NTJkZWEwIiwidGFnIjoiIn0%3D; askhelpdesk_session=eyJpdiI6IkQ4amp4b3ppT3hVVXhMaWFPdTV2Z0E9PSIsInZhbHVlIjoiTStLL3owNWZMYnBUOU4zZFRaZ281M2MxWCs1cy9vS1l2dWp0THkwK2NHdEZDdFpwQTMzbU1JMlBieXQxU1BqTTJnZ0xNUDNWU0V0R2lVbUVkTGVjY3I4VnFtUTltZFhzT1Z6b1VTbU03VzR5WmplRmhLZ1JBTFhaS0ZnajVQMisiLCJtYWMiOiI4NzFhODUxOWZiNjg5ZWM3YmEyNmE2OGVhZmNmYTdlNWI3NmZhMzEwMzdlN2E5YzhjM2U5ZTRkMzYyMmQ0MjIxIiwidGFnIjoiIn0%3D; _pk_id.64.7c30=1a9fb7309e185dd2.1749343989.; _pk_ses.64.7c30=1XSRF-TOKEN=eyJpdiI6IkNOYkpIV01sdHgwMCtsOFNHT2QwS3c9PSIsInZhbHVlIjoic1hnUjlJeks5MC94NG5zTVAvNHQ1RFQwMzA3aGQ5WkhrOHl1ZFBDZVB5NDJKL2cvcFE0aExXY0lsOC9YQUUzWi9kcnNTN" "HTTP_CF_IPCOUNTRY" => "US" "HTTP_CF_CONNECTING_IP" => "216.73.216.214" "HTTP_CDN_LOOP" => "cloudflare; loops=1" "HTTP_SEC_FETCH_MODE" => "navigate" "HTTP_SEC_FETCH_SITE" => "none" "HTTP_ACCEPT" => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" "HTTP_USER_AGENT" => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" "HTTP_UPGRADE_INSECURE_REQUESTS" => "1" "HTTP_SEC_CH_UA_PLATFORM" => ""Windows"" "HTTP_SEC_CH_UA_MOBILE" => "?0" "HTTP_SEC_CH_UA" => ""Chromium";v="130", "HeadlessChrome";v="130", "Not?A_Brand";v="99"" "HTTP_CACHE_CONTROL" => "no-cache" "HTTP_PRAGMA" => "no-cache" "HTTP_ACCEPT_ENCODING" => "gzip, br" "HTTP_CF_RAY" => "94c4691c8b97f602-ORD" "HTTP_PRIORITY" => "u=0, i" "HTTP_SEC_FETCH_DEST" => "document" "HTTP_SEC_FETCH_USER" => "?1" "HTTP_CF_VISITOR" => "{"scheme":"https"}" "HTTP_CONNECTION" => "close" "HTTP_X_FORWARDED_PROTO" => "https" "HTTP_X_FORWARDED_FOR" => "216.73.216.214, 172.70.131.215" "HTTP_X_SERVER_ADDR" => "154.12.239.204" "HTTP_HOST" => "receivinghelpdesk.com" "HTTPS" => "on" "REDIRECT_STATUS" => "200" "REDIRECT_HTTPS" => "on" "FCGI_ROLE" => "RESPONDER" "PHP_SELF" => "/ask/index.php" "REQUEST_TIME_FLOAT" => 1749343989.2184 "REQUEST_TIME" => 1749343989 ]
        request_cookies
        0 of 0
        array:4 [ "XSRF-TOKEN" => "u85QzFX4NDGbfEgMrhS8kF0PrZjVASGWslogvtQY" "askhelpdesk_session" => "LKJBgwxwJ8UfqZ0DQh8WSgnD58qwS3VBNyTwidoL" "_pk_id_64_7c30" => null "_pk_ses_64_7c30" => null ]
        response_headers
        0 of 0
        array:7 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "private, must-revalidate" ] "date" => array:1 [ 0 => "Sun, 08 Jun 2025 00:53:09 GMT" ] "pragma" => array:1 [ 0 => "no-cache" ] "expires" => array:1 [ 0 => -1 ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IjJvMHIya2ZVaEdRc3Y0RnpTdGppSXc9PSIsInZhbHVlIjoic0cvellOUFRHM0oxV2pDMWMxd21BTk5uUmQwdk0rVW96NlZycmVCU0x1QysrMXlUY1lLLzRGWUhqNkg2SnBLVWFCQnlVYmZyZlVtZ3pUMS8xbmhtcU9acGlNSWRnRjhBbWh4Y1hGeFBwTllaK0R3WTd1RFArVVRsMmZPUlBESzciLCJtYWMiOiJlOWU3ZWEwMWUxZDc1ZDA5OGFhMzg1NjAyODJiYzU4N2EzMjkwNTRhZTIyYzRkZjg3ZDE4ZmJmNWM5MjM1Mjk3IiwidGFnIjoiIn0%3D; expires=Sun, 08-Jun-2025 02:53:11 GMT; Max-Age=7200; path=/; samesite=laxXSRF-TOKEN=eyJpdiI6IjJvMHIya2ZVaEdRc3Y0RnpTdGppSXc9PSIsInZhbHVlIjoic0cvellOUFRHM0oxV2pDMWMxd21BTk5uUmQwdk0rVW96NlZycmVCU0x1QysrMXlUY1lLLzRGWUhqNkg2SnBLVWFCQnlVY" 1 => "askhelpdesk_session=eyJpdiI6IjVoTkRiOXZIMzJFTjFIMlBlSXVjS3c9PSIsInZhbHVlIjoiVVRaY1VLS1ViOXZRMmY3eUZMSmh0cWVQWkJoNkhkelhpK3BvUytacE1DRnZ6Tmh3ZkthSVh1eXRYazdmYkNzNzJxTDAwTzM3Y2ZCWXhMODRpM0ZEcUFyOUV3Y1o1ODBvOFJ3dENhQWVoNmZBcmtzMktZeEJ4WHFjUkU1c2RBcTkiLCJtYWMiOiI3OWIyMDY1Mjg3ODE2NDhiOTI1ZmFjYTg3NTE0M2ZmYWVjMDZlZDU4MmU3NzE5NTY2NWVhMTVkZDdiMWExNjg4IiwidGFnIjoiIn0%3D; expires=Sun, 08-Jun-2025 02:53:11 GMT; Max-Age=7200; path=/; httponly; samesite=laxaskhelpdesk_session=eyJpdiI6IjVoTkRiOXZIMzJFTjFIMlBlSXVjS3c9PSIsInZhbHVlIjoiVVRaY1VLS1ViOXZRMmY3eUZMSmh0cWVQWkJoNkhkelhpK3BvUytacE1DRnZ6Tmh3ZkthSVh1eXRYazdmYkNz" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IjJvMHIya2ZVaEdRc3Y0RnpTdGppSXc9PSIsInZhbHVlIjoic0cvellOUFRHM0oxV2pDMWMxd21BTk5uUmQwdk0rVW96NlZycmVCU0x1QysrMXlUY1lLLzRGWUhqNkg2SnBLVWFCQnlVYmZyZlVtZ3pUMS8xbmhtcU9acGlNSWRnRjhBbWh4Y1hGeFBwTllaK0R3WTd1RFArVVRsMmZPUlBESzciLCJtYWMiOiJlOWU3ZWEwMWUxZDc1ZDA5OGFhMzg1NjAyODJiYzU4N2EzMjkwNTRhZTIyYzRkZjg3ZDE4ZmJmNWM5MjM1Mjk3IiwidGFnIjoiIn0%3D; expires=Sun, 08-Jun-2025 02:53:11 GMT; path=/XSRF-TOKEN=eyJpdiI6IjJvMHIya2ZVaEdRc3Y0RnpTdGppSXc9PSIsInZhbHVlIjoic0cvellOUFRHM0oxV2pDMWMxd21BTk5uUmQwdk0rVW96NlZycmVCU0x1QysrMXlUY1lLLzRGWUhqNkg2SnBLVWFCQnlVY" 1 => "askhelpdesk_session=eyJpdiI6IjVoTkRiOXZIMzJFTjFIMlBlSXVjS3c9PSIsInZhbHVlIjoiVVRaY1VLS1ViOXZRMmY3eUZMSmh0cWVQWkJoNkhkelhpK3BvUytacE1DRnZ6Tmh3ZkthSVh1eXRYazdmYkNzNzJxTDAwTzM3Y2ZCWXhMODRpM0ZEcUFyOUV3Y1o1ODBvOFJ3dENhQWVoNmZBcmtzMktZeEJ4WHFjUkU1c2RBcTkiLCJtYWMiOiI3OWIyMDY1Mjg3ODE2NDhiOTI1ZmFjYTg3NTE0M2ZmYWVjMDZlZDU4MmU3NzE5NTY2NWVhMTVkZDdiMWExNjg4IiwidGFnIjoiIn0%3D; expires=Sun, 08-Jun-2025 02:53:11 GMT; path=/; httponlyaskhelpdesk_session=eyJpdiI6IjVoTkRiOXZIMzJFTjFIMlBlSXVjS3c9PSIsInZhbHVlIjoiVVRaY1VLS1ViOXZRMmY3eUZMSmh0cWVQWkJoNkhkelhpK3BvUytacE1DRnZ6Tmh3ZkthSVh1eXRYazdmYkNz" ] ]
        session_attributes
        0 of 0
        array:4 [ "_token" => "u85QzFX4NDGbfEgMrhS8kF0PrZjVASGWslogvtQY" "_previous" => array:1 [ "url" => "https://receivinghelpdesk.com/ask/how-do-you-stretch-or-shrink-a-graph" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]