{"id":1826,"date":"2024-06-10T20:47:55","date_gmt":"2024-06-10T20:47:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1826"},"modified":"2025-01-19T02:40:46","modified_gmt":"2025-01-19T02:40:46","slug":"introduction-to-quadratic-functions-and-parabolas-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-quadratic-functions-and-parabolas-fresh-take\/","title":{"raw":"Introduction to Quadratic Functions and Parabolas: Fresh Take","rendered":"Introduction to Quadratic Functions and Parabolas: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Identify quadratic functions in both general and standard form<\/li>\r\n \t<li>Determine the domain and range of a quadratic function by recognizing whether the vertex represents a maximum or minimum point<\/li>\r\n \t<li>Recognize key features of a parabola's graph: vertex, axis of symmetry, y-intercept, and minimum or maximum value<\/li>\r\n \t<li>Create graphs of quadratic functions using tables and transformations<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Equations of Quadratic Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nQuadratic functions graph into a curve called a <strong>parabola<\/strong>. The direction in which the parabola opens is determined by the sign of the coefficient [latex]a[\/latex] in the quadratic function's general form [latex]f(x)=ax^2+bx+c[\/latex]. If [latex]a&gt;0[\/latex], the parabola opens upward like a smile, and if [latex]a&lt;0[\/latex], it opens downward like a frown.\r\n\r\nThere are two forms of equations quadratics can be represented in:\r\n<ul>\r\n \t<li><strong>General Form<\/strong>: [latex]f(x) = ax^2 + bx + c[\/latex], where [latex]a[\/latex], [latex]b[\/latex]&gt;, and [latex]c[\/latex] are constants and [latex]a \\neq 0[\/latex].<\/li>\r\n \t<li><strong>Standard Form<\/strong>: [latex]f(x) = a(x - h)^2 + k[\/latex], where [latex](h, k)[\/latex] is the vertex of the parabola.<\/li>\r\n<\/ul>\r\nHere are some key features of quadratic functions:\r\n<ul>\r\n \t<li><strong>Vertex<\/strong>: The turning point of the parabola, which can be a minimum or maximum.<\/li>\r\n \t<li><strong>Axis of Symmetry<\/strong>: A vertical line that divides the parabola into two mirror images, given by [latex]x = -\\frac{b}{2a}[\/latex].<\/li>\r\n \t<li><strong>[latex]Y[\/latex]-Intercept<\/strong>: The point where the parabola crosses the [latex]y[\/latex]-axis, which is at [latex]f(0)[\/latex].<\/li>\r\n \t<li><strong>[latex]X[\/latex]-Intercepts<\/strong>: Points where the parabola crosses the [latex]x[\/latex]-axis, found by setting [latex]f(x) = 0[\/latex].<\/li>\r\n<\/ul>\r\n<strong>Step-by-Step: Finding the Vertex<\/strong>\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>From the general form, identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].<\/li>\r\n \t<li>Calculate [latex]h[\/latex] using [latex]h = -\\frac{b}{2a}[\/latex].<\/li>\r\n \t<li>Determine [latex]k[\/latex] by evaluating [latex]f(h)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Given the equation [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex], write the equation in general form and then in standard form.[reveal-answer q=\"713769\"]Show Solution[\/reveal-answer][hidden-answer a=\"713769\"][latex]g\\left(x\\right)={x}^{2}-6x+13[\/latex] in general form; [latex]g\\left(x\\right)={\\left(x - 3\\right)}^{2}+4[\/latex] in standard form[\/hidden-answer]<\/section><section class=\"textbox example\">Using an online graphing calculator, plot the function [latex]f\\left(x\\right)=2\\left(x-h\\right)^2+k[\/latex].Change the values of [latex]h[\/latex] and [latex]k[\/latex] to examine how changing the location of the vertex [latex](h,k)[\/latex] of a parabola also changes the axis of symmetry. Notice that when you move [latex]k[\/latex] independently of [latex]h[\/latex], you are only moving the vertical location of the vertex. Experiment with values between [latex]-10[\/latex] and [latex]10[\/latex].The vertex of a parabola is the location of either the maximum or minimum value of the parabola. If [latex]a&gt;0[\/latex], the parabola opens upward and the parabola has a minimum value of [latex]k[\/latex] at [latex]x=h[\/latex]. If [latex]a&lt;0[\/latex], the parabola opens downward, and the parabola has a maximum value of [latex]k[\/latex] at [latex]x=h[\/latex]. In this case, the vertex is the location of the minimum value of the function because [latex]a=2[\/latex].<\/section>\r\n<h2>Finding the Domain and Range of a Quadratic Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nQuadratic functions form parabolas on a graph, and these parabolas have specific domains and ranges that dictate their possible values.\r\n<ul>\r\n \t<li><strong>Domain Details<\/strong>: The domain of a quadratic function includes all real numbers. There are no restrictions on the [latex]x[\/latex]-values that can be input into the function.<\/li>\r\n \t<li><strong>Range Rules<\/strong>: The range of a quadratic function depends on the direction the parabola opens. If the parabola opens upwards, the range is all [latex]y[\/latex]-values greater than or equal to the vertex's [latex]y[\/latex]-value. If it opens downwards, the range is all [latex]y[\/latex]-values less than or equal to the vertex's [latex]y[\/latex]-value.<\/li>\r\n<\/ul>\r\nThe vertex can tell you a lot about the range of a quadratic function.\r\n<ul>\r\n \t<li><strong>Positive 'a' Value<\/strong>: For [latex]f(x) = ax^2 + bx + c[\/latex] with a positive '[latex]a[\/latex]', the parabola opens upwards. The vertex represents the minimum [latex]y[\/latex]-value, so the range is [latex]y \\geq f(-\\frac{b}{2a})[\/latex].<\/li>\r\n \t<li><strong>Negative 'a' Value<\/strong>: For [latex]f(x) = ax^2 + bx + c[\/latex] with a negative '[latex]a[\/latex]', the parabola opens downwards. The vertex represents the maximum [latex]y[\/latex]-value, so the range is [latex]y \\leq f(-\\frac{b}{2a})[\/latex].<\/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-accaghea-qPobPhqy3A4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/qPobPhqy3A4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-accaghea-qPobPhqy3A4\" 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=11328531&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-accaghea-qPobPhqy3A4&vembed=0&video_id=qPobPhqy3A4&video_target=tpm-plugin-accaghea-qPobPhqy3A4'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+Determine+the+Domain+and+Range+of+a+Quadratic+Function+in+Standard+Form+-+f(x)+%3D+(x+%E2%88%92+3)2+%2B+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Determine the Domain and Range of a Quadratic Function in Standard Form: f(x) = (x \u2212 3)2 + 2\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\">Find the domain and range of [latex]f\\left(x\\right)=2{\\left(x-\\dfrac{4}{7}\\right)}^{2}+\\dfrac{8}{11}[\/latex].[reveal-answer q=\"307368\"]Show Solution[\/reveal-answer][hidden-answer a=\"307368\"]The domain is all real numbers. The range is [latex]f\\left(x\\right)\\ge \\dfrac{8}{11}[\/latex], or [latex]\\left[\\dfrac{8}{11},\\infty \\right)[\/latex].[\/hidden-answer]<\/section>\r\n<h2>Key Features of a Parabola's Graph<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nA parabola is a U-shaped curve that reflects the behavior of objects under the influence of gravity or the shape of satellite dishes capturing signals. It's a visual representation of a quadratic function, showcasing how variables interact to create a curve that opens upwards or downwards.\r\n<ul>\r\n \t<li><strong>Vertex<\/strong>: The vertex is the peak or the lowest point of the parabola, serving as a critical point that determines the maximum or minimum value of the quadratic function.<\/li>\r\n \t<li><strong>Axis of Symmetry<\/strong>: This is the vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. It's a guide to finding the vertex and understanding the parabola's symmetry.<\/li>\r\n \t<li><strong>Zeros<\/strong>: These are the points where the parabola intersects the x-axis, also known as solutions or roots of the quadratic equation, where [latex]y=0[\/latex].<\/li>\r\n \t<li><strong>[latex]Y[\/latex]-Intercept<\/strong>: The point where the parabola crosses the y-axis, indicating the value of [latex]y[\/latex] when [latex]x=0[\/latex].<\/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-bbgegcbb-KRwb4YhQPwA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/KRwb4YhQPwA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbgegcbb-KRwb4YhQPwA\" 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=12844472&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bbgegcbb-KRwb4YhQPwA&vembed=0&video_id=KRwb4YhQPwA&video_target=tpm-plugin-bbgegcbb-KRwb4YhQPwA'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/%CA%95%E2%80%A2%E1%B4%A5%E2%80%A2%CA%94+Quadratic+Functions+-+Explained%2C+Simplified+and+Made+Easy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c\u0295\u2022\u1d25\u2022\u0294 Quadratic Functions - Explained, Simplified and Made Easy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>The Graph of a Quadratic Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nCreating a graph of a function is one way to understand the relationship between the inputs and outputs of that function. Creating a graph can be done by choosing values for [latex]x[\/latex], finding the corresponding [latex]y[\/latex] values, and plotting them. However, it helps to understand the basic shape of the function. Knowing how changes to the basic function equation affect the graph is also helpful.\r\n\r\nThe shape of a quadratic function is a parabola. Parabolas have the equation [latex]f(x)=ax^{2}+bx+c[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers and [latex]a\\ne0[\/latex]. The value of [latex]a[\/latex] determines the width and the direction of the parabola, while the vertex depends on the values of [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]. The vertex is [latex] \\displaystyle \\left( \\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a} \\right) \\right)[\/latex].\r\n\r\nIn the following video, we show an example of plotting a quadratic function using a table of values.\r\n\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-cafbfhdb-wYfEzOJugS8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/wYfEzOJugS8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cafbfhdb-wYfEzOJugS8\" 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=12844473&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cafbfhdb-wYfEzOJugS8&vembed=0&video_id=wYfEzOJugS8&video_target=tpm-plugin-cafbfhdb-wYfEzOJugS8'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Graph+a+Quadratic+Function+Using+a+Table+of+Values_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Graph a Quadratic Function Using a Table of Values\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>The following video shows another example of plotting a quadratic function using the vertex.\r\n\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-adbeecge-leYhH_-3rVo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/leYhH_-3rVo?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-adbeecge-leYhH_-3rVo\" 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=12844474&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-adbeecge-leYhH_-3rVo&vembed=0&video_id=leYhH_-3rVo&video_target=tpm-plugin-adbeecge-leYhH_-3rVo'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+a+Quadratic+Function+Using+a+Table+of+Value+and+the+Vertex_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a Quadratic Function Using a Table of Value and the Vertex\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\u00a0<\/strong>\r\n\r\nThe vertex of a parabola is like the peak of a mountain or the bottom of a valley, depending on which way it opens. It's the most defining feature, determining the lowest or highest point on the graph. In the equation [latex]f(x) = a(x-h)^2 + k[\/latex], the coordinates [latex](h, k)[\/latex] pinpoint the vertex's location on the Cartesian plane. <strong>Shifting the Graph<\/strong>\r\n<ul>\r\n \t<li><strong>Vertical Shifts:<\/strong> The [latex]k[\/latex] value dictates the vertical shift. If [latex]k &gt; 0[\/latex], the graph moves upward; if [latex]k &lt; 0[\/latex], it moves downward. Think of it as lifting or pressing down on the curve.<\/li>\r\n \t<li><strong>Horizontal Shifts:<\/strong> The [latex]h[\/latex] value controls the horizontal shift. A positive [latex]h[\/latex] shifts the graph to the right, while a negative [latex]h[\/latex] shifts it to the left. The direction is opposite to the sign inside the parentheses due to the equation's structure.<\/li>\r\n<\/ul>\r\n<strong>Stretch or Compress the Graph<\/strong>\r\n<ul>\r\n \t<li><strong>Vertical Stretch\/Compression:<\/strong> The coefficient [latex]a[\/latex] affects the parabola's width. If [latex]|a| &gt; 1[\/latex], the graph becomes narrower; if [latex]|a| &lt; 1[\/latex], it becomes wider. It's like zooming in and out on the curve vertically.<\/li>\r\n<\/ul>\r\n<strong>Quick Tips: Applying Transformations<\/strong>\r\n<ul>\r\n \t<li>To vertically shift the graph, modify the [latex]k[\/latex] value accordingly.<\/li>\r\n \t<li>To horizontally shift the graph, change the [latex]h[\/latex] value, keeping in mind the inverse relationship with the direction of the shift.<\/li>\r\n \t<li>To alter the width of the graph, adjust the [latex]a[\/latex] value, which stretches or compresses the parabola.<\/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-cbhheead-hvyH-WJtMpc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/hvyH-WJtMpc?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cbhheead-hvyH-WJtMpc\" 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=12844475&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cbhheead-hvyH-WJtMpc&vembed=0&video_id=hvyH-WJtMpc&video_target=tpm-plugin-cbhheead-hvyH-WJtMpc'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Quadratic+Function+Transformations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cQuadratic Function Transformations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify quadratic functions in both general and standard form<\/li>\n<li>Determine the domain and range of a quadratic function by recognizing whether the vertex represents a maximum or minimum point<\/li>\n<li>Recognize key features of a parabola&#8217;s graph: vertex, axis of symmetry, y-intercept, and minimum or maximum value<\/li>\n<li>Create graphs of quadratic functions using tables and transformations<\/li>\n<\/ul>\n<\/section>\n<h2>Equations of Quadratic Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Quadratic functions graph into a curve called a <strong>parabola<\/strong>. The direction in which the parabola opens is determined by the sign of the coefficient [latex]a[\/latex] in the quadratic function&#8217;s general form [latex]f(x)=ax^2+bx+c[\/latex]. If [latex]a>0[\/latex], the parabola opens upward like a smile, and if [latex]a<0[\/latex], it opens downward like a frown.\n\nThere are two forms of equations quadratics can be represented in:\n\n\n<ul>\n<li><strong>General Form<\/strong>: [latex]f(x) = ax^2 + bx + c[\/latex], where [latex]a[\/latex], [latex]b[\/latex]&gt;, and [latex]c[\/latex] are constants and [latex]a \\neq 0[\/latex].<\/li>\n<li><strong>Standard Form<\/strong>: [latex]f(x) = a(x - h)^2 + k[\/latex], where [latex](h, k)[\/latex] is the vertex of the parabola.<\/li>\n<\/ul>\n<p>Here are some key features of quadratic functions:<\/p>\n<ul>\n<li><strong>Vertex<\/strong>: The turning point of the parabola, which can be a minimum or maximum.<\/li>\n<li><strong>Axis of Symmetry<\/strong>: A vertical line that divides the parabola into two mirror images, given by [latex]x = -\\frac{b}{2a}[\/latex].<\/li>\n<li><strong>[latex]Y[\/latex]-Intercept<\/strong>: The point where the parabola crosses the [latex]y[\/latex]-axis, which is at [latex]f(0)[\/latex].<\/li>\n<li><strong>[latex]X[\/latex]-Intercepts<\/strong>: Points where the parabola crosses the [latex]x[\/latex]-axis, found by setting [latex]f(x) = 0[\/latex].<\/li>\n<\/ul>\n<p><strong>Step-by-Step: Finding the Vertex<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>From the general form, identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].<\/li>\n<li>Calculate [latex]h[\/latex] using [latex]h = -\\frac{b}{2a}[\/latex].<\/li>\n<li>Determine [latex]k[\/latex] by evaluating [latex]f(h)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Given the equation [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex], write the equation in general form and then in standard form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q713769\">Show Solution<\/button><\/p>\n<div id=\"q713769\" class=\"hidden-answer\" style=\"display: none\">[latex]g\\left(x\\right)={x}^{2}-6x+13[\/latex] in general form; [latex]g\\left(x\\right)={\\left(x - 3\\right)}^{2}+4[\/latex] in standard form<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Using an online graphing calculator, plot the function [latex]f\\left(x\\right)=2\\left(x-h\\right)^2+k[\/latex].Change the values of [latex]h[\/latex] and [latex]k[\/latex] to examine how changing the location of the vertex [latex](h,k)[\/latex] of a parabola also changes the axis of symmetry. Notice that when you move [latex]k[\/latex] independently of [latex]h[\/latex], you are only moving the vertical location of the vertex. Experiment with values between [latex]-10[\/latex] and [latex]10[\/latex].The vertex of a parabola is the location of either the maximum or minimum value of the parabola. If [latex]a>0[\/latex], the parabola opens upward and the parabola has a minimum value of [latex]k[\/latex] at [latex]x=h[\/latex]. If [latex]a<0[\/latex], the parabola opens downward, and the parabola has a maximum value of [latex]k[\/latex] at [latex]x=h[\/latex]. In this case, the vertex is the location of the minimum value of the function because [latex]a=2[\/latex].<\/section>\n<h2>Finding the Domain and Range of a Quadratic Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Quadratic functions form parabolas on a graph, and these parabolas have specific domains and ranges that dictate their possible values.<\/p>\n<ul>\n<li><strong>Domain Details<\/strong>: The domain of a quadratic function includes all real numbers. There are no restrictions on the [latex]x[\/latex]-values that can be input into the function.<\/li>\n<li><strong>Range Rules<\/strong>: The range of a quadratic function depends on the direction the parabola opens. If the parabola opens upwards, the range is all [latex]y[\/latex]-values greater than or equal to the vertex&#8217;s [latex]y[\/latex]-value. If it opens downwards, the range is all [latex]y[\/latex]-values less than or equal to the vertex&#8217;s [latex]y[\/latex]-value.<\/li>\n<\/ul>\n<p>The vertex can tell you a lot about the range of a quadratic function.<\/p>\n<ul>\n<li><strong>Positive &#8216;a&#8217; Value<\/strong>: For [latex]f(x) = ax^2 + bx + c[\/latex] with a positive &#8216;[latex]a[\/latex]&#8216;, the parabola opens upwards. The vertex represents the minimum [latex]y[\/latex]-value, so the range is [latex]y \\geq f(-\\frac{b}{2a})[\/latex].<\/li>\n<li><strong>Negative &#8216;a&#8217; Value<\/strong>: For [latex]f(x) = ax^2 + bx + c[\/latex] with a negative &#8216;[latex]a[\/latex]&#8216;, the parabola opens downwards. The vertex represents the maximum [latex]y[\/latex]-value, so the range is [latex]y \\leq f(-\\frac{b}{2a})[\/latex].<\/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-accaghea-qPobPhqy3A4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/qPobPhqy3A4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-accaghea-qPobPhqy3A4\" 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=11328531&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-accaghea-qPobPhqy3A4&#38;vembed=0&#38;video_id=qPobPhqy3A4&#38;video_target=tpm-plugin-accaghea-qPobPhqy3A4\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+Determine+the+Domain+and+Range+of+a+Quadratic+Function+in+Standard+Form+-+f(x)+%3D+(x+%E2%88%92+3)2+%2B+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Determine the Domain and Range of a Quadratic Function in Standard Form: f(x) = (x \u2212 3)2 + 2\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Find the domain and range of [latex]f\\left(x\\right)=2{\\left(x-\\dfrac{4}{7}\\right)}^{2}+\\dfrac{8}{11}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q307368\">Show Solution<\/button><\/p>\n<div id=\"q307368\" class=\"hidden-answer\" style=\"display: none\">The domain is all real numbers. The range is [latex]f\\left(x\\right)\\ge \\dfrac{8}{11}[\/latex], or [latex]\\left[\\dfrac{8}{11},\\infty \\right)[\/latex].<\/div>\n<\/div>\n<\/section>\n<h2>Key Features of a Parabola&#8217;s Graph<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A parabola is a U-shaped curve that reflects the behavior of objects under the influence of gravity or the shape of satellite dishes capturing signals. It&#8217;s a visual representation of a quadratic function, showcasing how variables interact to create a curve that opens upwards or downwards.<\/p>\n<ul>\n<li><strong>Vertex<\/strong>: The vertex is the peak or the lowest point of the parabola, serving as a critical point that determines the maximum or minimum value of the quadratic function.<\/li>\n<li><strong>Axis of Symmetry<\/strong>: This is the vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. It&#8217;s a guide to finding the vertex and understanding the parabola&#8217;s symmetry.<\/li>\n<li><strong>Zeros<\/strong>: These are the points where the parabola intersects the x-axis, also known as solutions or roots of the quadratic equation, where [latex]y=0[\/latex].<\/li>\n<li><strong>[latex]Y[\/latex]-Intercept<\/strong>: The point where the parabola crosses the y-axis, indicating the value of [latex]y[\/latex] when [latex]x=0[\/latex].<\/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-bbgegcbb-KRwb4YhQPwA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/KRwb4YhQPwA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbgegcbb-KRwb4YhQPwA\" 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=12844472&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bbgegcbb-KRwb4YhQPwA&#38;vembed=0&#38;video_id=KRwb4YhQPwA&#38;video_target=tpm-plugin-bbgegcbb-KRwb4YhQPwA\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/%CA%95%E2%80%A2%E1%B4%A5%E2%80%A2%CA%94+Quadratic+Functions+-+Explained%2C+Simplified+and+Made+Easy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c\u0295\u2022\u1d25\u2022\u0294 Quadratic Functions &#8211; Explained, Simplified and Made Easy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>The Graph of a Quadratic Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Creating a graph of a function is one way to understand the relationship between the inputs and outputs of that function. Creating a graph can be done by choosing values for [latex]x[\/latex], finding the corresponding [latex]y[\/latex] values, and plotting them. However, it helps to understand the basic shape of the function. Knowing how changes to the basic function equation affect the graph is also helpful.<\/p>\n<p>The shape of a quadratic function is a parabola. Parabolas have the equation [latex]f(x)=ax^{2}+bx+c[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers and [latex]a\\ne0[\/latex]. The value of [latex]a[\/latex] determines the width and the direction of the parabola, while the vertex depends on the values of [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]. The vertex is [latex]\\displaystyle \\left( \\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a} \\right) \\right)[\/latex].<\/p>\n<p>In the following video, we show an example of plotting a quadratic function using a table of values.<\/p>\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-cafbfhdb-wYfEzOJugS8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/wYfEzOJugS8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cafbfhdb-wYfEzOJugS8\" 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=12844473&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cafbfhdb-wYfEzOJugS8&#38;vembed=0&#38;video_id=wYfEzOJugS8&#38;video_target=tpm-plugin-cafbfhdb-wYfEzOJugS8\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Graph+a+Quadratic+Function+Using+a+Table+of+Values_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Graph a Quadratic Function Using a Table of Values\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>The following video shows another example of plotting a quadratic function using the vertex.<\/p>\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-adbeecge-leYhH_-3rVo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/leYhH_-3rVo?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-adbeecge-leYhH_-3rVo\" 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=12844474&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-adbeecge-leYhH_-3rVo&#38;vembed=0&#38;video_id=leYhH_-3rVo&#38;video_target=tpm-plugin-adbeecge-leYhH_-3rVo\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+a+Quadratic+Function+Using+a+Table+of+Value+and+the+Vertex_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a Quadratic Function Using a Table of Value and the Vertex\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\u00a0<\/strong><\/p>\n<p>The vertex of a parabola is like the peak of a mountain or the bottom of a valley, depending on which way it opens. It&#8217;s the most defining feature, determining the lowest or highest point on the graph. In the equation [latex]f(x) = a(x-h)^2 + k[\/latex], the coordinates [latex](h, k)[\/latex] pinpoint the vertex&#8217;s location on the Cartesian plane. <strong>Shifting the Graph<\/strong><\/p>\n<ul>\n<li><strong>Vertical Shifts:<\/strong> The [latex]k[\/latex] value dictates the vertical shift. If [latex]k > 0[\/latex], the graph moves upward; if [latex]k < 0[\/latex], it moves downward. Think of it as lifting or pressing down on the curve.<\/li>\n<li><strong>Horizontal Shifts:<\/strong> The [latex]h[\/latex] value controls the horizontal shift. A positive [latex]h[\/latex] shifts the graph to the right, while a negative [latex]h[\/latex] shifts it to the left. The direction is opposite to the sign inside the parentheses due to the equation&#8217;s structure.<\/li>\n<\/ul>\n<p><strong>Stretch or Compress the Graph<\/strong><\/p>\n<ul>\n<li><strong>Vertical Stretch\/Compression:<\/strong> The coefficient [latex]a[\/latex] affects the parabola&#8217;s width. If [latex]|a| > 1[\/latex], the graph becomes narrower; if [latex]|a| < 1[\/latex], it becomes wider. It&#8217;s like zooming in and out on the curve vertically.<\/li>\n<\/ul>\n<p><strong>Quick Tips: Applying Transformations<\/strong><\/p>\n<ul>\n<li>To vertically shift the graph, modify the [latex]k[\/latex] value accordingly.<\/li>\n<li>To horizontally shift the graph, change the [latex]h[\/latex] value, keeping in mind the inverse relationship with the direction of the shift.<\/li>\n<li>To alter the width of the graph, adjust the [latex]a[\/latex] value, which stretches or compresses the parabola.<\/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-cbhheead-hvyH-WJtMpc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/hvyH-WJtMpc?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cbhheead-hvyH-WJtMpc\" 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=12844475&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cbhheead-hvyH-WJtMpc&#38;vembed=0&#38;video_id=hvyH-WJtMpc&#38;video_target=tpm-plugin-cbhheead-hvyH-WJtMpc\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Quadratic+Function+Transformations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cQuadratic Function Transformations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":12,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"How to Determine the Domain and 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