{"id":839,"date":"2025-07-15T20:55:07","date_gmt":"2025-07-15T20:55:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=839"},"modified":"2026-03-09T07:28:09","modified_gmt":"2026-03-09T07:28:09","slug":"quadratic-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/quadratic-functions-fresh-take\/","title":{"raw":"Quadratic Functions: Fresh Take","rendered":"Quadratic Functions: Fresh Take"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify key characteristics of parabolas from the graph.<\/li>\r\n \t<li>Understand how the graph of a parabola is related to its quadratic function.<\/li>\r\n \t<li>Draw the graph of a quadratic function.<\/li>\r\n \t<li>Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3>Key Features of a Parabola<\/h3>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\nThe graph of a quadratic function is a U-shaped curve called a parabola.\r\n\r\nEvery parabola has these important features:\r\n<ul>\r\n \t<li><strong>Vertex:<\/strong> The turning point\u2014either the highest or lowest point on the graph<\/li>\r\n \t<li><strong>Axis of symmetry:<\/strong> A vertical line through the vertex that divides the parabola into two mirror images<\/li>\r\n \t<li><strong>y-intercept:<\/strong> Where the parabola crosses the y-axis<\/li>\r\n \t<li><strong>x-intercepts (zeros\/roots):<\/strong> Where the parabola crosses the x-axis (if it does)<\/li>\r\n \t<li><strong>Direction of opening:<\/strong> Whether the parabola opens upward or downward<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox recall\">The x-intercepts are also called zeros, roots, or solutions of the quadratic function. These are the x-values where [latex]y = 0[\/latex]. Not all parabolas have x-intercepts!<\/div>\r\n<div class=\"textbox proTip\">The axis of symmetry is always a vertical line passing through the vertex. If the vertex has x-coordinate [latex]h[\/latex], the axis of symmetry is [latex]x = h[\/latex].<\/div>\r\n<h2>Domain and Range of Quadratic Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\nFor quadratic functions:\r\n<ul>\r\n \t<li>The <strong>domain is always all real numbers<\/strong> - you can input any value for x<\/li>\r\n \t<li>The <strong>range depends on the vertex<\/strong> and whether the parabola opens up or down<\/li>\r\n<\/ul>\r\nIf the parabola opens <strong>upward<\/strong> ([latex]a &gt; 0[\/latex]):\r\n<ul>\r\n \t<li>The vertex is the minimum point<\/li>\r\n \t<li>Range: [latex]y \\geq k[\/latex] or [latex][k, \\infty)[\/latex] where [latex]k[\/latex] is the y-coordinate of the vertex<\/li>\r\n<\/ul>\r\nIf the parabola opens <strong>downward<\/strong> ([latex]a &lt; 0[\/latex]):\r\n<ul>\r\n \t<li>The vertex is the maximum point<\/li>\r\n \t<li>Range: [latex]y \\leq k[\/latex] or [latex](-\\infty, k][\/latex] where [latex]k[\/latex] is the y-coordinate of the vertex<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox questionHelp\">\r\n\r\n<strong>Question Help: Finding Domain and Range<\/strong>\r\n<ol>\r\n \t<li>The domain is always all real numbers: [latex](-\\infty, \\infty)[\/latex].<\/li>\r\n \t<li>Determine whether [latex]a[\/latex] is positive or negative.<\/li>\r\n \t<li>If [latex]a &gt; 0[\/latex], the parabola opens upward and has a minimum at the vertex.<\/li>\r\n \t<li>If [latex]a &lt; 0[\/latex], the parabola opens downward and has a maximum at the vertex.<\/li>\r\n \t<li>Find the y-coordinate of the vertex, [latex]k[\/latex].<\/li>\r\n \t<li>Write the range:\r\n<ul>\r\n \t<li>Minimum: [latex][k, \\infty)[\/latex]<\/li>\r\n \t<li>Maximum: [latex](-\\infty, k][\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-beffbfbf-nmMwY-CpBMY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/nmMwY-CpBMY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-beffbfbf-nmMwY-CpBMY\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660092&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-beffbfbf-nmMwY-CpBMY&vembed=0&video_id=nmMwY-CpBMY&video_target=tpm-plugin-beffbfbf-nmMwY-CpBMY'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Finding+Domain+%26+Range+-+with+Parabolas_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding Domain &amp; Range - with Parabolas\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<div class=\"textbox proTip\">The range always depends on the y-coordinate of the vertex. Find the vertex first, then determine if it's a maximum or minimum based on the sign of [latex]a[\/latex].<\/div>\r\n<h2>Forms of Quadratic Functions<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\r\n<\/strong>\r\nQuadratic functions can be written in different forms, each useful for different purposes:\r\n<strong>\r\nGeneral Form:<\/strong> [latex]f(x) = ax^2 + bx + c[\/latex]\r\n<ul>\r\n \t<li>Shows the y-intercept directly: [latex]c[\/latex]<\/li>\r\n \t<li>Useful for identifying basic properties<\/li>\r\n<\/ul>\r\n<strong>Standard Form (Vertex Form):<\/strong> [latex]f(x) = a(x - h)^2 + k[\/latex]\r\n<ul>\r\n \t<li>Shows the vertex directly: [latex](h, k)[\/latex]<\/li>\r\n \t<li>Best for graphing and transformations<\/li>\r\n<\/ul>\r\n<strong>Key properties from the equation:<\/strong>\r\n<ul>\r\n \t<li>If [latex]a &gt; 0[\/latex], parabola opens upward<\/li>\r\n \t<li>If [latex]a &lt; 0[\/latex], parabola opens downward<\/li>\r\n \t<li>The axis of symmetry is [latex]x = -\\frac{b}{2a}[\/latex] (general form) or [latex]x = h[\/latex] (standard form)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Finding the Vertex<\/h3>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\r\n<\/strong>\r\nWhen a quadratic is in general form [latex]f(x) = ax^2 + bx + c[\/latex], you can find the vertex using these formulas:\r\n<strong>x-coordinate of vertex:<\/strong> [latex]h = -\\frac{b}{2a}[\/latex]<\/div>\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bhhchacb-WTk4m6Ovv9s\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/WTk4m6Ovv9s?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bhhchacb-WTk4m6Ovv9s\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660093&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bhhchacb-WTk4m6Ovv9s&vembed=0&video_id=WTk4m6Ovv9s&video_target=tpm-plugin-bhhchacb-WTk4m6Ovv9s'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Find+the+Vertex+of+the+Parabola+in+Standard+Form-Example+1_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFind the Vertex of the Parabola in Standard Form-Example 1\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Transformations of Quadratic Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\nUnderstanding how changes to the equation affect the graph helps you sketch parabolas quickly without making tables.\r\n<strong>Standard Form:<\/strong> [latex]f(x) = a(x - h)^2 + k[\/latex]\r\n<ul>\r\n \t<li><strong>Changing [latex]k[\/latex]:<\/strong> Shifts the parabola vertically\r\n<ul>\r\n \t<li>[latex]k &gt; 0[\/latex]: shifts up<\/li>\r\n \t<li>[latex]k &lt; 0[\/latex]: shifts down<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Changing [latex]h[\/latex]:<\/strong> Shifts the parabola horizontally\r\n<ul>\r\n \t<li>[latex]h &gt; 0[\/latex]: shifts right<\/li>\r\n \t<li>[latex]h &lt; 0[\/latex]: shifts left<\/li>\r\n \t<li>Remember: [latex](x - h)[\/latex] means opposite direction!<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Changing [latex]a[\/latex]:<\/strong> Affects width and direction\r\n<ul>\r\n \t<li>[latex]|a| &gt; 1[\/latex]: narrower (vertical stretch)<\/li>\r\n \t<li>[latex]0 &lt; |a| &lt; 1[\/latex]: wider (vertical compression)<\/li>\r\n \t<li>[latex]a &lt; 0[\/latex]: opens downward (reflection over x-axis)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bhgebgbf-H8OQtxSRe_k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/H8OQtxSRe_k?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bhgebgbf-H8OQtxSRe_k\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660094&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bhgebgbf-H8OQtxSRe_k&vembed=0&video_id=H8OQtxSRe_k&video_target=tpm-plugin-bhgebgbf-H8OQtxSRe_k'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Graphing+Quadratic+Functions+Using+Transformations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing Quadratic Functions Using Transformations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<div class=\"textbox recall\">The negative sign in [latex](x - h)[\/latex] is part of the formula! If [latex]h = 3[\/latex], you write [latex](x - 3)[\/latex], which shifts RIGHT. If [latex]h = -2[\/latex], you write [latex](x - (-2)) = (x + 2)[\/latex], which shifts LEFT.<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify key characteristics of parabolas from the graph.<\/li>\n<li>Understand how the graph of a parabola is related to its quadratic function.<\/li>\n<li>Draw the graph of a quadratic function.<\/li>\n<li>Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/li>\n<\/ul>\n<\/section>\n<h3>Key Features of a Parabola<\/h3>\n<\/div>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>The graph of a quadratic function is a U-shaped curve called a parabola.<\/p>\n<p>Every parabola has these important features:<\/p>\n<ul>\n<li><strong>Vertex:<\/strong> The turning point\u2014either the highest or lowest point on the graph<\/li>\n<li><strong>Axis of symmetry:<\/strong> A vertical line through the vertex that divides the parabola into two mirror images<\/li>\n<li><strong>y-intercept:<\/strong> Where the parabola crosses the y-axis<\/li>\n<li><strong>x-intercepts (zeros\/roots):<\/strong> Where the parabola crosses the x-axis (if it does)<\/li>\n<li><strong>Direction of opening:<\/strong> Whether the parabola opens upward or downward<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox recall\">The x-intercepts are also called zeros, roots, or solutions of the quadratic function. These are the x-values where [latex]y = 0[\/latex]. Not all parabolas have x-intercepts!<\/div>\n<div class=\"textbox proTip\">The axis of symmetry is always a vertical line passing through the vertex. If the vertex has x-coordinate [latex]h[\/latex], the axis of symmetry is [latex]x = h[\/latex].<\/div>\n<h2>Domain and Range of Quadratic Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>For quadratic functions:<\/p>\n<ul>\n<li>The <strong>domain is always all real numbers<\/strong> &#8211; you can input any value for x<\/li>\n<li>The <strong>range depends on the vertex<\/strong> and whether the parabola opens up or down<\/li>\n<\/ul>\n<p>If the parabola opens <strong>upward<\/strong> ([latex]a > 0[\/latex]):<\/p>\n<ul>\n<li>The vertex is the minimum point<\/li>\n<li>Range: [latex]y \\geq k[\/latex] or [latex][k, \\infty)[\/latex] where [latex]k[\/latex] is the y-coordinate of the vertex<\/li>\n<\/ul>\n<p>If the parabola opens <strong>downward<\/strong> ([latex]a < 0[\/latex]):\n\n\n<ul>\n<li>The vertex is the maximum point<\/li>\n<li>Range: [latex]y \\leq k[\/latex] or [latex](-\\infty, k][\/latex] where [latex]k[\/latex] is the y-coordinate of the vertex<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox questionHelp\">\n<p><strong>Question Help: Finding Domain and Range<\/strong><\/p>\n<ol>\n<li>The domain is always all real numbers: [latex](-\\infty, \\infty)[\/latex].<\/li>\n<li>Determine whether [latex]a[\/latex] is positive or negative.<\/li>\n<li>If [latex]a > 0[\/latex], the parabola opens upward and has a minimum at the vertex.<\/li>\n<li>If [latex]a < 0[\/latex], the parabola opens downward and has a maximum at the vertex.<\/li>\n<li>Find the y-coordinate of the vertex, [latex]k[\/latex].<\/li>\n<li>Write the range:\n<ul>\n<li>Minimum: [latex][k, \\infty)[\/latex]<\/li>\n<li>Maximum: [latex](-\\infty, k][\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-beffbfbf-nmMwY-CpBMY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/nmMwY-CpBMY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-beffbfbf-nmMwY-CpBMY\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660092&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-beffbfbf-nmMwY-CpBMY&#38;vembed=0&#38;video_id=nmMwY-CpBMY&#38;video_target=tpm-plugin-beffbfbf-nmMwY-CpBMY\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Finding+Domain+%26+Range+-+with+Parabolas_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding Domain &amp; Range &#8211; with Parabolas\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<div class=\"textbox proTip\">The range always depends on the y-coordinate of the vertex. Find the vertex first, then determine if it&#8217;s a maximum or minimum based on the sign of [latex]a[\/latex].<\/div>\n<h2>Forms of Quadratic Functions<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<br \/>\n<\/strong><br \/>\nQuadratic functions can be written in different forms, each useful for different purposes:<br \/>\n<strong><br \/>\nGeneral Form:<\/strong> [latex]f(x) = ax^2 + bx + c[\/latex]<\/p>\n<ul>\n<li>Shows the y-intercept directly: [latex]c[\/latex]<\/li>\n<li>Useful for identifying basic properties<\/li>\n<\/ul>\n<p><strong>Standard Form (Vertex Form):<\/strong> [latex]f(x) = a(x - h)^2 + k[\/latex]<\/p>\n<ul>\n<li>Shows the vertex directly: [latex](h, k)[\/latex]<\/li>\n<li>Best for graphing and transformations<\/li>\n<\/ul>\n<p><strong>Key properties from the equation:<\/strong><\/p>\n<ul>\n<li>If [latex]a > 0[\/latex], parabola opens upward<\/li>\n<li>If [latex]a < 0[\/latex], parabola opens downward<\/li>\n<li>The axis of symmetry is [latex]x = -\\frac{b}{2a}[\/latex] (general form) or [latex]x = h[\/latex] (standard form)<\/li>\n<\/ul>\n<\/div>\n<h3>Finding the Vertex<\/h3>\n<div class=\"textbox shaded\"><strong>The Main Idea<br \/>\n<\/strong><br \/>\nWhen a quadratic is in general form [latex]f(x) = ax^2 + bx + c[\/latex], you can find the vertex using these formulas:<br \/>\n<strong>x-coordinate of vertex:<\/strong> [latex]h = -\\frac{b}{2a}[\/latex]<\/div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bhhchacb-WTk4m6Ovv9s\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/WTk4m6Ovv9s?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bhhchacb-WTk4m6Ovv9s\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660093&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bhhchacb-WTk4m6Ovv9s&#38;vembed=0&#38;video_id=WTk4m6Ovv9s&#38;video_target=tpm-plugin-bhhchacb-WTk4m6Ovv9s\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Find+the+Vertex+of+the+Parabola+in+Standard+Form-Example+1_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFind the Vertex of the Parabola in Standard Form-Example 1\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Transformations of Quadratic Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>Understanding how changes to the equation affect the graph helps you sketch parabolas quickly without making tables.<br \/>\n<strong>Standard Form:<\/strong> [latex]f(x) = a(x - h)^2 + k[\/latex]<\/p>\n<ul>\n<li><strong>Changing [latex]k[\/latex]:<\/strong> Shifts the parabola vertically\n<ul>\n<li>[latex]k > 0[\/latex]: shifts up<\/li>\n<li>[latex]k < 0[\/latex]: shifts down<\/li>\n<\/ul>\n<\/li>\n<li><strong>Changing [latex]h[\/latex]:<\/strong> Shifts the parabola horizontally\n<ul>\n<li>[latex]h > 0[\/latex]: shifts right<\/li>\n<li>[latex]h < 0[\/latex]: shifts left<\/li>\n<li>Remember: [latex](x - h)[\/latex] means opposite direction!<\/li>\n<\/ul>\n<\/li>\n<li><strong>Changing [latex]a[\/latex]:<\/strong> Affects width and direction\n<ul>\n<li>[latex]|a| > 1[\/latex]: narrower (vertical stretch)<\/li>\n<li>[latex]0 < |a| < 1[\/latex]: wider (vertical compression)<\/li>\n<li>[latex]a < 0[\/latex]: opens downward (reflection over x-axis)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bhgebgbf-H8OQtxSRe_k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/H8OQtxSRe_k?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bhgebgbf-H8OQtxSRe_k\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660094&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bhgebgbf-H8OQtxSRe_k&#38;vembed=0&#38;video_id=H8OQtxSRe_k&#38;video_target=tpm-plugin-bhgebgbf-H8OQtxSRe_k\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Graphing+Quadratic+Functions+Using+Transformations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing Quadratic Functions Using Transformations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<div class=\"textbox recall\">The negative sign in [latex](x - h)[\/latex] is part of the formula! If [latex]h = 3[\/latex], you write [latex](x - 3)[\/latex], which shifts RIGHT. If [latex]h = -2[\/latex], you write [latex](x - (-2)) = (x + 2)[\/latex], which shifts LEFT.<\/div>\n","protected":false},"author":13,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Finding Domain & Range - with Parabolas\",\"author\":\"Ginger Hampton\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/nmMwY-CpBMY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Find the Vertex of the Parabola in Standard Form-Example 1\",\"author\":\"\",\"organization\":\"Ms. Milkosky the Mathematician\",\"url\":\"https:\/\/youtu.be\/WTk4m6Ovv9s\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Graphing Quadratic Functions Using Transformations\",\"author\":\"\",\"organization\":\"The Organic Chemistry Tutor\",\"url\":\"https:\/\/youtu.be\/H8OQtxSRe_k\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Finding Domain & Range - 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