{"id":87616,"date":"2021-08-09T10:35:46","date_gmt":"2021-08-09T15:35:46","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=87616"},"modified":"2026-03-25T11:36:19","modified_gmt":"2026-03-25T16:36:19","slug":"manipulation-of-functions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/manipulation-of-functions\/","title":{"rendered":"Manipulation of Functions"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_Cp-E-RWU3QQ\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_Cp-E-RWU3QQ\" data-source-videoID=\"Cp-E-RWU3QQ\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Manipulation of Functions Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Manipulation of Functions\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_Cp-E-RWU3QQ:hover {cursor:pointer;} img#videoThumbnailImage_Cp-E-RWU3QQ {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1723-thumb-final-1-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_Cp-E-RWU3QQ\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_Cp-E-RWU3QQ\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Manipulation of Functions\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_Cp-E-RWU3QQ\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_Cp-E-RWU3QQ\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_Cp-E-RWU3QQ\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction Hk3_Function() {\n  var x = document.getElementById(\"Hk3\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"Hk3_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"Hk3\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Reviewing_Terms\" class=\"smooth-scroll\">Reviewing Terms<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Graphing_Inputs_and_Outputs\" class=\"smooth-scroll\">Graphing Inputs and Outputs<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Manipulating_a_Function\" class=\"smooth-scroll\">Manipulating a Function<\/a>\n<ul><\/li>\n<li class=\"toc-h3\"><a href=\"#Translation\" class=\"smooth-scroll\">Translation<\/a><\/li>\n<li class=\"toc-h3\"><a href=\"#Stretch_and_Compression\" class=\"smooth-scroll\">Stretch and Compression<\/a><\/li>\n<li class=\"toc-h3\"><a href=\"#Reflection\" class=\"smooth-scroll\">Reflection<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"toc-h2\"><a href=\"#Function_Manipulation_Practice_Questions\" class=\"smooth-scroll\">Function Manipulation Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hello, and welcome to this video about the manipulation of functions!<\/p>\n<p>Functions can be manipulated, or transformed, to move the graph around the coordinate plane. Today we\u2019ll learn how to translate, stretch, compress, and reflect the graphs of functions. <\/p>\n<h2><span id=\"Reviewing_Terms\" class=\"m-toc-anchor\"><\/span>Reviewing Terms<\/h2>\n<p>\nBefore we get started, let\u2019s review a few things.<\/p>\n<p>A <strong>function<\/strong> connects an input to an output. Its equation starts with <span style=\"font-style:normal; font-size:90%\">\\(f(x)=\\)<\/span>&#8230; where <span style=\"font-style:normal; font-size:90%\">\\(f\\)<\/span> is the function name, <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span> is the input, and the expression after the equal sign is the output. An example of this is <span style=\"font-style:normal; font-size:90%\">\\(f(x)=2x\\)<\/span>. With an input <span style=\"font-style:normal; font-size:90%\">\\((x)\\)<\/span> value of 4, the output would be 2(4), which equals 8. We can rewrite the function as <span style=\"font-style:normal; font-size:90%\">\\(f(4)=8\\)<\/span>. <\/p>\n<h2><span id=\"Graphing_Inputs_and_Outputs\" class=\"m-toc-anchor\"><\/span>Graphing Inputs and Outputs<\/h2>\n<p>\nThe input and output of a function can be graphed on a coordinate plane from an ordered pair that is written as (input, output), or (<span style=\"font-style:normal; font-size:90%\">\\(x, f(x)\\)<\/span>). <\/p>\n<p>Let\u2019s take a look at the graph for the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=2x\\)<\/span>:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89797\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulating-of-Functions-1.png\" alt=\"Image of the table and graph for f(x)= 2x\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulating-of-Functions-1.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulating-of-Functions-1-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulating-of-Functions-1-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulating-of-Functions-1-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulating-of-Functions-1-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>As you can see, this is a <strong>linear function<\/strong> because the graph forms a straight line. The <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-axis represents the input, <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>, and the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis represents the output, <span style=\"font-style:normal; font-size:90%\">\\(2x\\)<\/span>. <\/p>\n<p>Now let\u2019s take a look at the graph for the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span>:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89851\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-2.png\" alt=\"f(x) = x^2 shown on a graph.\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-2.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-2-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-2-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-2-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-2-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>This function is <strong>quadratic<\/strong> because its graph forms a u-shape, called a parabola. The <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-axis represents the input, <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>, and the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis represents the output, <span style=\"font-style:normal; font-size:90%\">\\(x^{2}\\)<\/span>. <\/p>\n<h2><span id=\"Manipulating_a_Function\" class=\"m-toc-anchor\"><\/span>Manipulating a Function<\/h2>\n<p>\nManipulating a function is the process of altering its equation to move the graph\u2019s position on the coordinate plane. There are lots of ways to manipulate functions.<\/p>\n<p>Let\u2019s start by discussing translations. <\/p>\n<h3><span id=\"Translation\" class=\"m-toc-anchor\"><\/span>Translation<\/h3>\n<p>\nTranslating the graph of a function means the entire graph shifts vertically or horizontally. Translations happen when constants are added or subtracted from the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>&#8211; or <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-values in the equation.<\/p>\n<p>For <strong>vertical translations<\/strong>, adding to the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-value shifts the graph up, and subtracting from the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-value shifts it down. <\/p>\n<p>For instance, the translation <span style=\"font-style:normal; font-size:90%\">\\(f(x)+3\\)<\/span> moves the graph three units up. Applied to our original function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=2x\\)<\/span>, the equation would read <span style=\"font-style:normal; font-size:90%\">\\(f(x)=2x+3\\)<\/span>. Let\u2019s examine how this would affect the graph of this function. Since the graph shifts three units up, the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-coordinate of the origin point increases by 3. Create a table reflecting this function. The first set of values in the table is 0 and 3 or (0,3). <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89860\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-3.png\" alt=\"Image of table x over 2x+3 plotted on a graph.\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-3.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-3-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-3-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-3-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-3-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>From here, write the rest of the values from the table as ordered pairs and graph them on the coordinate plane: (0, 3); (1, 5); (2, 7); (3, 9); (4, 11); and (5, 13). Each point in the new function <span style=\"font-style:normal; font-size:90%\">\\(2x+3\\)<\/span> (shown in red) is three units higher than the original function <span style=\"font-style:normal; font-size:90%\">\\(2x\\)<\/span> (shown in black). <\/p>\n<p><strong>Horizontal translations<\/strong> work a little bit differently. They are indicated inside the parentheses of the function. For horizontal translations, <em>adding<\/em> to the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value shifts the graph to the <em>left<\/em>, and <em>subtracting<\/em> from the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value shifts the graph to the <em>right<\/em>. In other words, horizontal shifts move in the opposite direction of the change. <\/p>\n<p>To illustrate this point, we will translate the graph of our original function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=2x\\)<\/span> five units to the right. First, rewrite the equation reflecting this translation by subtracting 5 from the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value inside parentheses: <span style=\"font-style:normal; font-size:90%\">\\(f(x)=2(x-5)\\)<\/span>. Next, complete a table showing the new values for this function: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89863\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-4.png\" alt=\"Image of table x over 2)x-5) plotted on a graph.\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-4.png 1918w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-4-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-4-1024x577.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-4-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-4-1536x865.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>Since the graph shifted five units to the right, the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-coordinate of the origin point increased by 5. Therefore, the first set of values in the tables are 5 and 0 or (5, 0). <\/p>\n<p>From here, write the rest of the values from the table as ordered pairs and graph them on the coordinate plane: (5, 0); (6, 2); (7, 4); (8, 6); (9, 8); and (10, 10). In the graph, the original function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=2x\\)<\/span> is graphed in black, and the new function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=2x-5\\)<\/span> is graphed in blue. The <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-axis represents the values for <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span> and the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis represents the values for <span style=\"font-style:normal; font-size:90%\">\\(2(x-5)\\)<\/span>. <\/p>\n<p>Let\u2019s take a look at one more translation. This time we\u2019ll translate the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=2x\\)<\/span> four units to the left and six units down. <\/p>\n<p>First, rewrite the equation to reflect this translation by subtracting 6 from the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-value outside the parentheses and adding 4 units to the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value inside the parentheses: <span style=\"font-style:normal; font-size:90%\">\\(f(x)=(2(x+4))-6\\)<\/span>. Next, complete a table showing the new values for this function: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89866\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-5.png\" alt=\"Image of table x over (2(x+4))-6 plotted on a graph.\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-5.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-5-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-5-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-5-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-5-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>Since the graph shifts four units to the left, the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-coordinate of the origin point decreased by 4. Since the graph shifts six units down, the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-coordinate of the origin point decreased by 6. The first set of values in the table are -4 and -6, or (-4, -6). <\/p>\n<p>From here, write the rest of the values from the table as ordered pairs and graph them on the coordinate plane: (-4, -6); (-3, -4); (-2, -2); (-1, 0); (0, 2); and (1, 4). On the coordinate plane, the original function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=2x\\)<\/span> is graphed in black, and the new function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=(2(x+4))-6\\)<\/span> is graphed in purple. The <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-axis represents the values for <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span> and the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis represents the values for <span style=\"font-style:normal; font-size:90%\">\\((2(x+4))-6\\)<\/span>. <\/p>\n<p>Now it\u2019s your turn. I\u2019m going to give you a chance to practice what you\u2019ve just learned about translations. When you see the practice problem, pause the video and solve it. Once you\u2019re ready, resume the video and we\u2019ll go over the answer together.<\/p>\n<p>Ready? Here we go. The function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=6x+8\\)<\/span> is shifted 3 units to the right and 5 units down. Write an equation to represent these changes. <\/p>\n<p>How\u2019d you do? Let\u2019s take a look at the problem together. Since the function shifts 3 units to the right, subtract 3 from the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value in parentheses. Rewrite the function as <span style=\"font-style:normal; font-size:90%\">\\(f(x)=6(x-3)+8\\)<\/span>. Next, subtract 5 from the entire function to move 5 units down. The correct equation is <span style=\"font-style:normal; font-size:90%\">\\(f(x)=(6(x-3)+8)-5\\)<\/span>. <\/p>\n<h3><span id=\"Stretch_and_Compression\" class=\"m-toc-anchor\"><\/span>Stretch and Compression<\/h3>\n<p>\nAnother way to manipulate the graphs of functions is by stretching or compressing them. Vertical stretching happens when the function as a whole gets multiplied by a constant (<span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span>) greater than 1, using the formula <span style=\"font-style:normal; font-size:90%\">\\(k\\times f(x)\\)<\/span>. <\/p>\n<p>Let\u2019s stretch the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span> vertically by the constant (<span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span>) 3. Start by graphing the original function, <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span>, shown as a solid line on the coordinate plane. When this function is multiplied by 3, the graph stretches vertically. The stretched graph appears as a dashed line on the coordinate plane. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89869\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-6.png\" alt=\"Image of f(x)=x^2 and K=3 plotted on a graph. \" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-6.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-6-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-6-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-6-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-6-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>A graph is stretched vertically, in relation to the graph of the original function, when the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-coordinates are multiplied by a value greater than 1.<\/p>\n<p>If the value of the constant (<span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span>) is greater than 0 but less than 1, then the graph is compressed vertically. Let\u2019s compress the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span> vertically by using <span style=\"font-style:normal; font-size:90%\">\\(\\frac{1}{2}\\)<\/span> for the constant (<span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span>). Start by graphing the original function, which is shown on the graph in red. When the function is multiplied by <span style=\"font-style:normal; font-size:90%\">\\(\\frac{1}{2}\\)<\/span>, the graph is compressed vertically. This is shown on the coordinate plane in green. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89872\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulations-fo-Functions-7.png\" alt=\"Image of f(x)=x^2 and k=2 plotted on a graph.\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulations-fo-Functions-7.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulations-fo-Functions-7-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulations-fo-Functions-7-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulations-fo-Functions-7-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulations-fo-Functions-7-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>Multiplying the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-coordinates by a value greater than 0 but less than 1 compresses the graph vertically in relation to the graph of the original function. <\/p>\n<p>A function can also stretch and compress horizontally. Horizontal compression happens when the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value is multiplied by a constant (<span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span>) greater than one, <span style=\"font-style:normal; font-size:90%\">\\(f(k\\times x)\\)<\/span>. <\/p>\n<p>Let\u2019s compress the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span> horizontally by the constant (<span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span>) 2. As always, start by graphing the original function, which is shown on the coordinate plane in red. The graph of <span style=\"font-style:normal; font-size:90%\">\\(f(x)=(2x)^{2}\\)<\/span>, shown in green, is a horizontal compression of <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span> by a factor of 2. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89875\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-8.png\" alt=\"Image of f(x)=x^2 and K=0.5 plotted on a graph.\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-8.png 1917w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-8-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-8-1024x577.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-8-768x433.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-8-1536x865.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>In relation to the graph of the original function, a graph is compressed when the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value is multiplied inside parentheses by a value greater than 1.<\/p>\n<p>A function stretches horizontally when the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value is multiplied by a constant (<span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span>), using the formula <span style=\"font-style:normal; font-size:90%\">\\(f(k\\times x)\\)<\/span>. The value of <span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span> must be greater than 0 but less than 1. <\/p>\n<p>Let\u2019s stretch the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span> horizontally by the constant (<span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span>) 0.5. The original function is graphed in red. The graph of <span style=\"font-style:normal; font-size:90%\">\\(f(x)=(0.5x)^{2}\\)<\/span>, which stretches the function horizontally by a factor of 2, is shown in green. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89878\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-9.png\" alt=\"Image of f(x)=x^2 and K=-1 plotted on a graph.\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3pxgrey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-9-300x168.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-9-1024x574.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-9-1536x861.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>Multiplying the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value inside parentheses by a value greater than 0 but less than 1 stretches the graph horizontally in relation to the graph of the original function. <\/p>\n<h3><span id=\"Reflection\" class=\"m-toc-anchor\"><\/span>Reflection<\/h3>\n<p>\nAnother way to manipulate functions is by reflecting them across the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>&#8211; or <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis. Reflections can be thought of as flipping the graph of a function over an axis. When reflected, the new graph is congruent to the original. <\/p>\n<p>Let\u2019s reflect the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span> across the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-axis. Using the formula <span style=\"font-style:normal; font-size:90%\">\\(k\\times f(x)\\)<\/span>, multiply the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)\\)<\/span> by a number (<span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span>) that is less than zero. In the example shown here, the value of <span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span> is -1:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89881\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-10.png\" alt=\"Image of f(x)=(x-2)^2 and k=-1 plotted on a graph.\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-10.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-10-300x168.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-10-1024x574.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-10-768x431.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-10-1536x862.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>The original function (shown in red) is reflected across the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-axis by multiplying the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-coordinates by -1. The reflection is shown in green. If the coordinate plane were folded along the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-axis, the two functions would align. <\/p>\n<p>Now let\u2019s reflect the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=(x-2)^{2}\\)<\/span> across the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis. Using the formula <span style=\"font-style:normal; font-size:90%\">\\(f(k\\times x)\\)<\/span>, multiply the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value inside parentheses by a number (<span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span>) that is less than zero. In the following example, the value of <span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span> is -1: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89887\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-11.png\" alt=\"Image of f(x)=X^2 and k+-4 plotted on a graph.\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-11.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-11-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-11-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-11-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Manipulation-of-Functions-11-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>The original function in red is reflected across the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis by multiplying the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value in parentheses by -1. The reflection is shown in green. If the coordinate plane were folded along the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis, the two functions would align. <\/p>\n<p>Now it\u2019s your turn. I have one last problem for you to analyze. It\u2019s a little more challenging, but I know you can handle it. <\/p>\n<p>Graphs of functions can be reflected across an axis, in addition to being stretched or compressed. <\/p>\n<p>Consider the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span>. Reflect this function across the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-axis and stretch it vertically by a factor of 4. Show your work in a table and on a graph. <\/p>\n<p>Now that you\u2019ve seen the problem, pause the video here and see if you can answer it yourself. When you\u2019re ready, resume the video and we\u2019ll go over the answers together. <\/p>\n<p>Now that you\u2019ve tried this problem yourself, let\u2019s go over it together. <\/p>\n<p>Consider the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span>. Reflect this function across the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-axis and stretch it vertically by a factor of 4. Show your work in a table and a graph. <\/p>\n<table class=\"ATable\">\n<tbody>\n<tr>\n<td><span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span><\/td>\n<td>-2<\/td>\n<td>-1<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<\/tbody>\n<tbody>\n<tr>\n<td><span style=\"font-style:normal; font-size:90%\">\\(f(x)\\)<\/span><\/td>\n<td>4<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>4<\/td>\n<\/tr>\n<\/tbody>\n<tbody>\n<tr>\n<td><span style=\"font-style:normal; font-size:90%\">\\(k\\times f(x)\\)<\/span><\/td>\n<td>-16<\/td>\n<td>-4<\/td>\n<td>0<\/td>\n<td>-4<\/td>\n<td>-16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nThe original function is shown in red and the new function is shown in blue. The function is stretched vertically and reflected across the <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-axis by multiplying each of the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-coordinates by -4. <\/p>\n<hr>\n<p>\nI hope this video about the manipulation of functions was helpful. Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Function_Manipulation_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Function Manipulation Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe graph of the function \\(y=f(x)\\) is shown on the coordinate plane below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-142237 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09076.png\" alt=\"a graph of a parabola\" width=\"515\" height=\"444\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09076.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09076-300x258.png 300w\" sizes=\"auto, (max-width: 515px) 100vw, 515px\" \/><\/p>\n<p>Which of the following is the graph of \\(g(x)=f\\left(x-4\\right)+2\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-1-1\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-142243\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09077.png\" alt=\"a graph of a parabola\" width=\"457\" height=\"393\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09077.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09077-300x258.png 300w\" sizes=\"auto, (max-width: 457px) 100vw, 457px\" \/>\r\n<\/div><div class=\"PQ\"  id=\"PQ-1-2\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-142246\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09078.png\" alt=\"a graph of a parabola\" width=\"455\" height=\"392\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09078.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09078-300x258.png 300w\" sizes=\"auto, (max-width: 455px) 100vw, 455px\" \/>\r\n<\/div><div class=\"PQ\"  id=\"PQ-1-3\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-142252\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09079.png\" alt=\"a graph of a parabola\" width=\"448\" height=\"388\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09079.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09079-300x259.png 300w\" sizes=\"auto, (max-width: 448px) 100vw, 448px\" \/>\r\n<\/div><div class=\"PQ\"  id=\"PQ-1-4\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-142240\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09080.png\" alt=\"a graph of a parabola\" width=\"443\" height=\"380\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09080.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09080-300x258.png 300w\" sizes=\"auto, (max-width: 443px) 100vw, 443px\" \/><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Let \\(h\\) and \\(k\\) be real numbers. We can translate the graph of the function \\(y=f(x)\\) the following ways to produce an equation for the graph of \\(g\\):<\/p>\n<ul>\n<li>If \\(g(x)=f\\left(x-h\\right)\\), then the graph of \\(g\\) is a horizontal shift of the graph of \\(f\\) that is \\(h\\) units to the left if \\(h\\lt0\\), and h units to the right if \\(h>0\\).<\/li>\n<li>If \\(g\\left(x\\right)=f\\left(x\\right)+k\\), then the graph of \\(g\\) is a vertical shift of the graph of \\(f\\) that is \\(k\\) units upwards if \\(k>0\\), and \\(k\\) units downwards if \\(k\\lt0\\).<\/li>\n<\/ul>\n<p>For the graph of \\(g(x)=f\\left(x-4\\right)+2\\), \\(h=4\\) and \\(k=2\\), which means the graph of \\(f\\) is shifted horizontally 4 units to the right, then shifted vertically 2 units upwards. The graph of \\(f\\) and the translated graph of \\(g\\) are shown in the coordinate plane below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-144535 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09081.png\" alt=\"a graph of two parabolas\" width=\"650\" height=\"532\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09081.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09081-300x246.png 300w\" sizes=\"auto, (max-width: 650px) 100vw, 650px\" \/><\/p>\n<p>While each point for the graph of \\(f\\) is translated 4 units to the right then 2 units upwards, the graph only explicitly shows how the vertex for \\(f\\) is translated to the vertex of \\(g\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nLet \\(f\\left(x\\right)=2x+3\\). If the graph of \\(y=g(x)\\) is the result of shifting the graph of \\(f\\) horizontally 5 units to the left, then vertically 6 units upwards, which of the following will be the equation for the graph of \\(g\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(g\\left(x\\right)=(2\\left(x-5\\right)+3)+6\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">\\(g\\left(x\\right)=(2\\left(x+5\\right)+3)+6\\)<\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\(g\\left(x\\right)=\\left(2\\left(x-6\\right)+3\\right)-5\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(g\\left(x\\right)=\\left(2\\left(x+6\\right)+3\\right)+5\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Let \\(a\\neq0\\), \\(h\\), and \\(k\\) be real numbers. We can transform the graph of the function \\(y=f(x)\\) the following ways to produce an equation for the graph of \\(g\\):<\/p>\n<ul>\n<li>If \\(g\\left(x\\right)=af(x)\\), then the graph of \\(g\\) is a vertical stretch of the graph of \\(f\\) if \\(a>0\\), and a vertical compression if \\(0\\lt a \\lt1\\). Furthermore, if \\(a\\lt0\\), the graph is reflected across the \\(x\\)-axis.<\/li>\n<li>If \\(g(x)=f\\left(x-h\\right)\\), then the graph of \\(g\\) is a horizontal shift of the graph of \\(f\\) that is \\(h\\) units to the left if \\(h\\lt0\\), and \\(h\\) units to the right if \\(h>0\\).<\/li>\n<li>If \\(g\\left(x\\right)=f\\left(x\\right)+k\\), then the graph of \\(g\\) is a vertical shift of the graph of \\(f\\) that is \\(k\\) units upwards if \\(k>0\\), and \\(k\\) units downwards if \\(k\\lt0\\).<\/li>\n<\/ul>\n<p>Since we are shifting \\(f\\) horizontally 5 units to the left to write the equation \\(g\\), the value for \\(h\\) must be negative 5 since \\(h\\lt0\\) for this type of transformation. So, the expression for \\(g\\) must contain \\(f\\left(x-(-5)\\right)=f(x+5)\\).<\/p>\n<p>Next, since we are shifting the graph of \\(f\\) vertically 6 units upwards, the value of \\(k\\) is positive, so \\(g\\left(x\\right)=f(x+5)+6=\\left(2\\left(x+5\\right)+3\\right)\\mathbf{+6}\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe graph of the function \\(f(x)=x^2\\) is shown on the coordinate plane below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-143173 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09082.png\" alt=\"a graph of a parabola\" width=\"475\" height=\"506\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09082.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09082-282x300.png 282w\" sizes=\"auto, (max-width: 475px) 100vw, 475px\" \/><\/p>\n<p>Which of the following is the graph of \\(g\\left(x\\right)=-2f(x)\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-145432\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09083.png\" alt=\"a graph of a parabola\" width=\"478\" height=\"515\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09083.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09083-279x300.png 279w\" sizes=\"auto, (max-width: 478px) 100vw, 478px\" \/>\r\n<\/div><div class=\"PQ\"  id=\"PQ-3-2\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-143182\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09084.png\" alt=\"a graph of a parabola\" width=\"476\" height=\"509\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09084.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09084-281x300.png 281w\" sizes=\"auto, (max-width: 476px) 100vw, 476px\" \/>\r\n<\/div><div class=\"PQ\"  id=\"PQ-3-3\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-143176\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09085.png\" alt=\"a graph of a parabola\" width=\"473\" height=\"504\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09085.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09085-282x300.png 282w\" sizes=\"auto, (max-width: 473px) 100vw, 473px\" \/>\r\n<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-4\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-144832\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09086.png\" alt=\"a graph of a parabola\" width=\"473\" height=\"504\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09086.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09086-282x300.png 282w\" sizes=\"auto, (max-width: 473px) 100vw, 473px\" \/><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Let \\(a\\neq0\\) be a real number. If \\(g(x)=ax^2\\), then the graph of \\(f(x)=x^2\\) is a vertical stretch if \\(a>0\\), and a vertical compression if \\(0\\lt a \\lt1\\). Furthermore, if \\(a\\lt0\\), the graph is reflected across the \\(y\\)-axis.<\/p>\n<p>Transforming \\(f(x)=x^2\\) with \\(a=-2\\), we get the graph of \\(g(x)=-2x^2\\), which is a vertical stretch for the graph of \\(f(x)=x^2\\) that is reflected across the \\(x\\)-axis shown in red below. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-144835 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09087.png\" alt=\"a graph of two parabolas\" width=\"443\" height=\"474\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09087.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09087-281x300.png 281w\" sizes=\"auto, (max-width: 443px) 100vw, 443px\" \/><\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nA basketball player plays for a basketball team that plays 20 games for a given season. For the first 10 games of the season, the basketball player scored the same number of points as the number of minutes he played for each game. For the last 10 games, he scored 3 more than twice the number of points he played. If \\(x\\) is the number of minutes played and \\(y\\) is the number of points scored, which of the following is a graph of the function that represents the number of points he scored for the last 10 games of the season?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-143185\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09088.png\" alt=\"a graph of a line\" width=\"481\" height=\"431\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09088.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09088-300x269.png 300w\" sizes=\"auto, (max-width: 481px) 100vw, 481px\" \/><\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-2\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-143188\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09089.png\" alt=\"a graph of a line\" width=\"481\" height=\"431\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09089.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09089-300x269.png 300w\" sizes=\"auto, (max-width: 481px) 100vw, 481px\" \/>\r\n<\/div><div class=\"PQ\"  id=\"PQ-4-3\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-143194\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09090.png\" alt=\"a graph of a line\" width=\"482\" height=\"432\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09090.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09090-300x269.png 300w\" sizes=\"auto, (max-width: 482px) 100vw, 482px\" \/>\r\n<\/div><div class=\"PQ\"  id=\"PQ-4-4\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-143191\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09091.png\" alt=\"a graph of a line\" width=\"483\" height=\"432\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09091.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09091-300x269.png 300w\" sizes=\"auto, (max-width: 483px) 100vw, 483px\" \/><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Since the basketball player scores as many points, \\(y\\), as minutes played, \\(x\\), for the first 10 games, we can write the number of points he scored as \\(y=x\\).<\/p>\n<p>For the last 10 games, the basketball player scored 3 more than twice the number of minutes he played the first ten games, so we can transform the graph of \\(y=x\\) to the graph of \\(y=2x+3\\).<\/p>\n<p>Let \\(a\\neq0\\) and \\(k\\) be real numbers. We can transform the graph of the function \\(y=x\\) to find the graph of a function that represents the number of points scored for the last 10 games using the following transformations:<\/p>\n<ul>\n<li>If \\(y=kx\\), then the graph of \\(y=x\\) is a vertical stretch if \\(a>0\\), and a vertical compression if \\(0\\lt a \\lt 1\\). Furthermore, if \\(a\\lt0\\), the graph is reflected across the \\(x\\)-axis.<\/li>\n<li>If \\(y=f(x)+k\\), then the graph of the function \\(y\\) is a vertical shift of the graph of \\(f\\) that is \\(k\\) units upwards if \\(k>0\\), and \\(k\\) units downwards if \\(k\\lt0\\).<\/li>\n<\/ul>\n<p>Transforming \\(y=x\\) with \\(a=2\\), we get the graph of \\(y=2x\\), which is a vertical stretch of the graph of \\(y=x\\) shown in red below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-144541 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09092.png\" alt=\"a graph of two intersecting lines\" width=\"509\" height=\"425\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09092.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09092-300x251.png 300w\" sizes=\"auto, (max-width: 509px) 100vw, 509px\" \/><\/p>\n<p>Next, transforming \\(y=2x\\) with \\(k=3\\), we get the graph of \\(y=2x+3\\), which is a vertical shift upwards 3 units of the graph of \\(y=2x\\) shown in red below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-145435 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09093.png\" alt=\"a graph of two intersecting lines\" width=\"512\" height=\"431\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09093.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09093-300x252.png 300w\" sizes=\"auto, (max-width: 512px) 100vw, 512px\" \/><\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe square of some number is transformed so the result is 2 less than one-half the negative of its square. If \\(y\\) is the square of some number, \\(x\\), then which of the following is the graph of the transformed result?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-143200\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09094.png\" alt=\"a graph of a parabola\" width=\"458\" height=\"466\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09094.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09094-295x300.png 295w\" sizes=\"auto, (max-width: 458px) 100vw, 458px\" \/>\r\n<\/div><div class=\"PQ\"  id=\"PQ-5-2\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-143197\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09095.png\" alt=\"a graph of a parabola\" width=\"455\" height=\"463\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09095.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09095-295x300.png 295w\" sizes=\"auto, (max-width: 455px) 100vw, 455px\" \/>\r\n\r\n<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-3\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-143206\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09096.png\" alt=\"a graph of a parabola\" width=\"459\" height=\"467\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09096.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09096-295x300.png 295w\" sizes=\"auto, (max-width: 459px) 100vw, 459px\" \/>\r\n\r\n<\/div><div class=\"PQ\"  id=\"PQ-5-4\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-143203\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09097.png\" alt=\"a graph of a parabola\" width=\"467\" height=\"475\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09097.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09097-295x300.png 295w\" sizes=\"auto, (max-width: 467px) 100vw, 467px\" \/><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>We can write the square of some number, \\(x\\), as \\(y=x^2\\). Transforming \\(y=x^2\\) so the result is 2 less than one-half the negative of its square, gives us the transformed function \\(y=-\\frac{1}{2}x^2-2\\).<\/p>\n<p>Let \\(a\\neq0\\) and \\(k\\) be real numbers. We can transform the graph of the function \\(y=x^2\\) to find the graph of the function that is the result of 2 less than one-half the negative of its square using the following transformations:<\/p>\n<ul>\n<li>If \\(y=kx\\), then the graph of \\(y=x^2\\) is a vertical stretch if \\(a>0\\), and a vertical compression if \\(0 \\lt a \\lt 1\\). Furthermore, if \\(a \\lt 0\\), the graph is reflected across the \\(x\\)-axis.<\/li>\n<li>If \\(y=f(x)+k\\), then the graph of the function \\(y\\) is a vertical shift of the graph of \\(f\\) that is \\(k\\) units upwards if \\(k>0\\), and a \\(k\\) units downwards if \\(k \\lt 0\\).<\/li>\n<\/ul>\n<p>Transforming \\(y=x^2\\) with \\(a=-\\frac{1}{2}\\), we get the graph of \\(y=-\\frac{1}{2}x^2\\), which is a vertical compression of \\(y=x^2\\) that is reflected across the \\(x\\)-axis shown in red below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-144544 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09098.png\" alt=\"a graph of two parabolas\" width=\"650\" height=\"531\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09098.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09098-300x245.png 300w\" sizes=\"auto, (max-width: 650px) 100vw, 650px\" \/><\/p>\n<p>Next, transforming the graph of \\(y=-\\frac{1}{2}x^2\\) with \\(k=-2\\), we get the graph of \\(y=-\\frac{1}{2}x^2-2\\) which is a vertical transformation down 2 units of the graph \\(y=-\\frac{1}{2}x^2\\) shown in red below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-144838 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09099.png\" alt=\"a graph of two parabolas\" width=\"650\" height=\"543\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09099.png 650w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/09\/I09099-300x251.png 300w\" sizes=\"auto, (max-width: 650px) 100vw, 650px\" \/><\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-ii\/\">Return to Algebra II Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra II Videos<\/p>\n","protected":false},"author":1,"featured_media":100717,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-87616","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-functions-and-their-graphs-videos","7":"page_category-video-pages-for-study-course-sidebar-ad","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 4.9.9 - aioseo.com -->\n\t<meta name=\"description\" content=\"Manipulation of functions means transforming (moving) them across a coordinate plane. 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