{"id":158374,"date":"2024-10-28T08:50:34","date_gmt":"2024-10-28T08:50:34","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=158374"},"modified":"2024-10-28T08:50:37","modified_gmt":"2024-10-28T08:50:37","slug":"chris-wanted-to-transform-the-graph-of-the-parent-function-y-cot-x-by-horizontally-compressing-it-so-that-it-has-a-period-of-2-%cf%80-units","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/28\/chris-wanted-to-transform-the-graph-of-the-parent-function-y-cot-x-by-horizontally-compressing-it-so-that-it-has-a-period-of-2-%cf%80-units\/","title":{"rendered":"Chris wanted to transform the graph of the parent function Y= cot (x) by horizontally compressing it so that it has a period of 2\/\u03c0 units"},"content":{"rendered":"\n

Chris wanted to transform the graph of the parent function Y= cot (x) by horizontally compressing it so that it has a period of 2\/\u03c0 units, horizontally Terslating it \u03c0\/4 units to the right, and vertically translating it 1 unit up. To do so, he graphed the function y= cot (2x-\u03c0\/4)+1 as shown. What did he do wrong?\u200b<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To analyze Chris’s transformation of the cotangent function, let\u2019s break down the required transformations step-by-step:<\/p>\n\n\n\n

    \n
  1. Parent Function<\/strong>: The parent function is (y = \\cot(x)), which has a period of (\\pi) (i.e., it repeats every (\\pi) units).<\/li>\n\n\n\n
  2. Desired Period<\/strong>: Chris wants to transform the graph to have a period of (\\frac{2}{\\pi}). The formula for the period of the cotangent function after horizontal scaling is given by:
    [
    \\text{Period} = \\frac{\\pi}{|b|}
    ]
    where (b) is the coefficient of (x) in the function (y = \\cot(bx)). To achieve a period of (\\frac{2}{\\pi}), we set up the equation:
    [
    \\frac{\\pi}{|b|} = \\frac{2}{\\pi}
    ]
    Solving for (b):
    [
    |b| = \\frac{\\pi^2}{2}
    ]
    Therefore, Chris should use (b = \\frac{\\pi^2}{2}) in the function to achieve the desired period.<\/li>\n\n\n\n
  3. Horizontal Translation<\/strong>: Chris intended to shift the graph (\\frac{\\pi}{4}) units to the right. This is done by replacing (x) with ((x – \\frac{\\pi}{4})). So the correct form, before vertical translation, is:
    [
    y = \\cot\\left(\\frac{\\pi^2}{2}\\left(x – \\frac{\\pi}{4}\\right)\\right)
    ]<\/li>\n\n\n\n
  4. Vertical Translation<\/strong>: The vertical translation of 1 unit up is correctly represented by adding 1 to the function, resulting in:
    [
    y = \\cot\\left(\\frac{\\pi^2}{2}\\left(x – \\frac{\\pi}{4}\\right)\\right) + 1
    ]<\/li>\n<\/ol>\n\n\n\n

    What Chris Did Wrong<\/h3>\n\n\n\n

    Chris’s function (y = \\cot(2x – \\frac{\\pi}{4}) + 1) has a coefficient (b = 2), which means the period he derived is:
    [
    \\text{Period} = \\frac{\\pi}{2} \\text{ (not } \\frac{2}{\\pi}\\text{)}
    ]
    Moreover, the horizontal translation is improperly calculated since ((2x – \\frac{\\pi}{4})) does not correctly shift the cotangent function right by (\\frac{\\pi}{4}) units.<\/p>\n\n\n\n

    Correct Function<\/h3>\n\n\n\n

    The correct transformation of the cotangent function to achieve a period of (\\frac{2}{\\pi}), a right shift of (\\frac{\\pi}{4}), and a vertical shift of 1 unit would be:
    [
    y = \\cot\\left(\\frac{\\pi^2}{2}\\left(x – \\frac{\\pi}{4}\\right)\\right) + 1
    ]<\/p>\n\n\n\n

    By using this function, Chris would correctly implement the transformations he intended.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Chris wanted to transform the graph of the parent function Y= cot (x) by horizontally compressing it so that it has a period of 2\/\u03c0 units, horizontally Terslating it \u03c0\/4 units to the right, and vertically translating it 1 unit up. To do so, he graphed the function y= cot (2x-\u03c0\/4)+1 as shown. What did […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-158374","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/158374","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=158374"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/158374\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=158374"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=158374"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=158374"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}