{"id":4326,"date":"2013-06-29T03:59:44","date_gmt":"2013-06-29T03:59:44","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=4326"},"modified":"2026-03-28T10:57:22","modified_gmt":"2026-03-28T15:57:22","slug":"domain-and-range-of-quadratic-functions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/domain-and-range-of-quadratic-functions\/","title":{"rendered":"Domain and Range of Quadratic Functions"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_UrqOGlZljHM\" 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_UrqOGlZljHM\" data-source-videoID=\"UrqOGlZljHM\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Domain and Range of Quadratic Functions Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Domain and Range of Quadratic Functions\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_UrqOGlZljHM:hover {cursor:pointer;} img#videoThumbnailImage_UrqOGlZljHM {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/1149-domain-and-range-of-quadratic-equations.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_UrqOGlZljHM\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_UrqOGlZljHM\" 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\",\"Domain and Range of Quadratic Functions\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_UrqOGlZljHM\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_UrqOGlZljHM\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_UrqOGlZljHM\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction aj8_Function() {\n  var x = document.getElementById(\"aj8\");\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=\"aj8_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=\"aj8\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Domain_and_Range\" class=\"smooth-scroll\">Domain and Range<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Structure_of_a_Function\" class=\"smooth-scroll\">Structure of a Function<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#How_to_Find_the_Domain_of_a_Quadratic_Function\" class=\"smooth-scroll\">How to Find the Domain of a Quadratic Function<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#How_to_Find_the_Range_of_a_Quadratic_Function\" class=\"smooth-scroll\">How to Find the Range of a Quadratic Function<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Using_Algebra_to_Find_Domain_and_Range\" class=\"smooth-scroll\">Using Algebra to Find Domain and Range<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Review\" class=\"smooth-scroll\">Review<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Frequently_Asked_Questions\" class=\"smooth-scroll\">Frequently Asked Questions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Quadratic_Function_Domain_and_Range_PDF\" class=\"smooth-scroll\">Quadratic Function Domain and Range PDF<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Domain_and_Range_of_Quadratic_Functions_Practice_Questions\" class=\"smooth-scroll\">Domain and Range of Quadratic Functions 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=\"FAQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"FAQs\">FAQs<\/label><input id=\"factsheet\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"factsheet\">Fact Sheet<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video about the domain and range of quadratic functions! In this video, we will explore how the structure of quadratic functions defines their domains and ranges and how to determine the domain and range of a quadratic function.<\/p>\n<p>Let\u2019s get started!<\/p>\n<h2><span id=\"Domain_and_Range\" class=\"m-toc-anchor\"><\/span>Domain and Range<\/h2>\n<p>\nBefore we begin, let\u2019s quickly revisit the terms domain and range.<\/p>\n<h3><span id=\"Domain_of_a_Quadratic_Function\" class=\"m-toc-anchor\"><\/span>Domain of a Quadratic Function<\/h3>\n<p>\nThe domain of a function is the set of all possible inputs.<\/p>\n<h3><span id=\"Range_of_a_Quadratic_Function\" class=\"m-toc-anchor\"><\/span>Range of a Quadratic Function<\/h3>\n<p>\nThe range of a function is the set of all possible outputs.<\/p>\n<h2><span id=\"Structure_of_a_Function\" class=\"m-toc-anchor\"><\/span>Structure of a Function<\/h2>\n<p>\nThe structure of a function determines its domain and range. Some functions, such as linear functions (e.g., <span style=\"font-style:normal; font-size:90%\">\\(f(x)=2x+1\\)<\/span>), have domains and ranges of all real numbers because any number can be input and a unique output can always be produced. On the other hand, functions with restrictions such as fractions or square roots may have limited domains and ranges (e.g., <span style=\"font-style:normal; font-size:90%\">\\(f(x)=\\frac{1}{2x}\\); \\(x\\neq 0\\)<\/span> because the denominator of a fraction cannot be 0).<\/p>\n<p>Let\u2019s see how the structure of quadratic functions defines and helps us determine their domains and ranges.<\/p>\n<p>Here\u2019s the graph of <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span>.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/06\/function-300x300.jpg\" alt=\"\" width=\"500\" height=\"500\" class=\"alignnone size-medium wp-image-84388\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/06\/function-300x300.jpg 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/06\/function-150x150.jpg 150w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/06\/function.jpg 533w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p>Quadratic functions together can be called a <strong>family<\/strong>, and this particular function the <strong>parent<\/strong>, because this is the most basic quadratic function (i.e., not transformed in any way). We can use this function to begin generalizing domains and ranges of quadratic functions.<\/p>\n<p>To determine the domain and range of any function on a graph, the general idea is to assume that they are both real numbers, then look for places where no values exist.<\/p>\n<h2><span id=\"How_to_Find_the_Domain_of_a_Quadratic_Function\" class=\"m-toc-anchor\"><\/span>How to Find the Domain of a Quadratic Function<\/h2>\n<p>\nLet\u2019s talk about domain first. Since domain is about inputs, we are only concerned with what the graph looks like horizontally. To see the domain, let\u2019s move from left-to-right along the \\(x\\)-axis looking for places where the graph doesn\u2019t exist.<\/p>\n<p>As you can see, there are no places where the graph doesn\u2019t exist horizontally. The domain of this function is all real numbers. In fact, the domain of all quadratic functions is all real numbers!<\/p>\n<h2><span id=\"How_to_Find_the_Range_of_a_Quadratic_Function\" class=\"m-toc-anchor\"><\/span>How to Find the Range of a Quadratic Function<\/h2>\n<p>\nNow for the range. We\u2019ll use a similar approach, but now we are only concerned with what the graph looks like vertically.<\/p>\n<p>As you can see, outputs only exist for \\(y\\)-values that are greater than or equal to 0. In other words, there are no outputs below the \\(x\\)-axis. We would say the range is all real numbers greater than or equal to 0.<\/p>\n<p>Let\u2019s generalize our findings with a few more graphs.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/09\/droqf-1-02.svg\" alt=\"\" width=\"310.2\" height=\"349.8\" class=\"aligncenter size-full wp-image-198722\"  role=\"img\" \/><\/p>\n<p>The range for this graph is all real numbers greater than or equal to 2.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/09\/droqf-2-03.svg\" alt=\"\" width=\"308.484\" height=\"398.772\" class=\"aligncenter size-full wp-image-198725\"  role=\"img\" \/> <\/p>\n<p>The range here is all real numbers less than or equal to 5.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/09\/droqf-4-03.svg\" alt=\"\" width=\"308.484\" height=\"398.772\" class=\"aligncenter size-full wp-image-198731\"  role=\"img\" \/><\/p>\n<p>The range for this one is all real numbers less than or equal to -2.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/09\/droqf-5-03.svg\" alt=\"\" width=\"311.6\" height=\"402.8\" class=\"aligncenter size-full wp-image-198734\"  role=\"img\" \/><\/p>\n<p>And the range for this graph is all real numbers greater than or equal to -3.<\/p>\n<p>As you can see, the turning point, or <strong>vertex<\/strong>, is part of what determines the range. The other is the direction the parabola opens. If a quadratic function opens up, then the range is all real numbers greater than or equal to the \\(y\\)-coordinate of the range. If a quadratic function opens down, then the range is all real numbers less than or equal to the \\(y\\)-coordinate of the range.<\/p>\n<p>Graphs can be helpful, but we often need algebra to determine the range of quadratic functions. Sometimes, we are only given an equation and other times the graph is not precise enough to be able to accurately read the range.<\/p>\n<h2><span id=\"Using_Algebra_to_Find_Domain_and_Range\" class=\"m-toc-anchor\"><\/span>Using Algebra to Find Domain and Range<\/h2>\n<p>\nSo let\u2019s look at finding the domain and range algebraically. There are three main forms of quadratic equations. Our goals here are to determine which way the function opens and find the \\(y\\)-coordinate of the vertex.<\/p>\n<h3><span id=\"Standard_Form\" class=\"m-toc-anchor\"><\/span>Standard Form<\/h3>\n<p>\nWhen the quadratic functions are in standard form, they generally look like this:<\/p>\n<div class=\"examplesentence\">\\(f(x)=ax^{2}+bx+c\\)<\/div>\n<p>\n&nbsp;<br \/>\nIf \\(a\\) is positive, the function opens up; if it\u2019s negative, the function opens down. In this form, the \\(y\\)-coordinate of the vertex is found by evaluating <span style=\"font-style:normal; font-size:90%\">\\(f(\\frac{-b}{2a})\\)<\/span>. For example, consider this function: <span style=\"font-style:normal; font-size:90%\"><\/p>\n<p style=\"text-align: center;\">\\(fx=-2x^2+8x-3\\)<\/span><\/p>\n<p>So we&#8217;re gonna do: <span style=\"font-style:normal; font-size:100%\">\\(\\frac{-b}{2a}=\\frac{-8}{2(-2)}=\\frac{-8}{-4}=2\\)<\/span><\/p>\n<p>Then, we plug this in: <span style=\"font-style:normal; font-size:90%\">\\(f(2)=-2(2)^{2}+8(2)-3=-8+16-3=5\\)<\/span><\/p>\n<p>\\(a\\) is negative, so the range is all real numbers less than or equal to 5.<\/p>\n<h3><span id=\"Vertex_Form\" class=\"m-toc-anchor\"><\/span>Vertex Form<\/h3>\n<p>\nWhen quadratic equations are in vertex form, they generally look like this:<\/p>\n<div class=\"examplesentence\">\\(f(x)=a(x-h)^2+k\\)<\/div>\n<p>\n&nbsp;<br \/>\nAs with standard form, if \\(a\\) is positive, the function opens up; if it\u2019s negative, the function opens down. The vertex is given by the coordinates <span style=\"font-style:normal; font-size:90%\">\\((h,k)\\)<\/span>, so all we need to consider is the <span style=\"font-style:normal; font-size:90%\">\\(k\\)<\/span>. For example, consider the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=3(x+4)^2-6\\)<\/span>. \\(a\\) is positive and the vertex is at <span style=\"font-style:normal; font-size:90%\">\\((-4,-6)\\)<\/span>, so the range is all real numbers greater than or equal to <span style=\"font-style:normal; font-size:90%\">\\(-6\\)<\/span>.<\/p>\n<h3><span id=\"Factored_Form\" class=\"m-toc-anchor\"><\/span>Factored Form<\/h3>\n<p>\nSometimes quadratic functions are defined using factored form as a way to easily identify their roots. For example:<\/p>\n<div class=\"examplesentence\">\\(f(x)=a(x-b)(x-c)\\)<\/div>\n<p>\n&nbsp;<br \/>\nAs with other forms, if \\(a\\) is positive, the function opens up; if it\u2019s negative, the function opens down. One way to use this form is to multiply the terms to get an equation in standard form, then apply the first method we saw. We can also apply the fact that quadratic functions are symmetric to find the vertex. We know the roots, and therefore, the locations of the \\(x\\)-intercepts. Horizontally, the vertex is halfway between them.<\/p>\n<p>Once we know the location of the vertex\u2014the \\(x\\)-coordinate\u2014all we need to do is substitute into the function to find the \\(y\\)-coordinate. For example, consider the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=-2(x+4)(x-2)\\)<\/span>. The \\(x\\)-intercepts are at <span style=\"font-style:normal; font-size:90%\">\\(-4\\)<\/span> and <span style=\"font-style:normal; font-size:90%\">\\(+2\\)<\/span>, and the vertex is located at <span style=\"font-style:normal; font-size:90%\">\\(\\frac{-4+2}{2}=-1\\)<\/span> (simply take the \u201caverage\u201d of the \\(x\\)-intercepts). And we\u2019re going to plug that into our original equation, so we have: <span style=\"font-style:normal; font-size:90%\">\\(f(-1)=-2(-1+4)(-1-2)=-2(3)(-3)=18\\)<\/span>. Since \\(a\\) is negative, the range of all real numbers is less than or equal to 18.<\/p>\n<hr>\n<h2><span id=\"Review\" class=\"m-toc-anchor\"><\/span>Review<\/h2>\n<p>Ok, let\u2019s do a quick review before we go.<\/p>\n<ul>\n<li>Domain is the set of input values, while range is the set of output values.<\/li>\n<li>To determine the domain and range of any function on a graph, the general idea is to assume that they are both real numbers, then look for places where no values exist.<\/li>\n<li>And finally, when looking at things algebraically, we have three forms of quadratic equations: standard form, vertex form, and factored form.<\/li>\n<\/ul>\n<p>Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"FAQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Frequently_Asked_Questions\" class=\"m-toc-anchor\"><\/span>Frequently Asked Questions<\/h2>\n<div class=\"faq-list\">\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is the range of a quadratic function?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>The range of a quadratic function is a list of all the possible \\(y\\)-values of a quadratic function.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">How do you find the domain of a quadratic function?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>The domain of a quadratic function is always \\((-\u221e,\u221e)\\) because quadratic functions always extend forever in either direction along the \\(x\\)-axis.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What makes the function quadratic?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>A function is considered quadratic if it has a degree of 2. In other words, a function is quadratic if it has an \\(x^2\\)-term.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">Can a quadratic function have a range of \\((-\u221e,\u221e)\\)?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>No, a quadratic function cannot have a range of \\((-\u221e,\u221e)\\). In order for this to happen, the parabola would have to be on its side, like this:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Quadratic-Function-Infinity-Example.svg\" alt=\"A graph of an upward-opening parabola with arrows at both ends, drawn on an x-y coordinate plane.\" width=\"253.3\" height=\"299.2\" class=\"aligncenter size-full wp-image-286837\"  role=\"img\" \/><\/p>\n<p>However, if this were to happen, it would not be considered a function because the graph fails the vertical line test.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">How do you find the range of a quadratic function?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>To find the range of a standard quadratic function in the form \\(f(x)=ax^2+bx+c\\), find the vertex of the parabola and determine if the parabola opens up or down.<\/p>\n<p>To find the vertex of a quadratic in this form, use the formula \\(x=-\\frac{b}{2a}\\). This will give you the \\(x\\)-coordinate of the vertex, and the \\(y\\)-coordinate can be found by plugging this \\(x\\)-value into the original equation and solving for \\(f(x)\\) (or \\(y\\)).<\/p>\n<p>Next, determine if the parabola opens up or down. If the value of \\(a\\) is positive, then the parabola opens up and the range is \\((y\\)-coordinate of the vertex, \\(\u221e)\\). If the value of \\(a\\) is negative, then the parabola opens down and the range is \\((-\u221e, y\\)-coordinate of the vertex\\()\\).<\/p>\n<div class=\"lightbulb-example-2\"><span class=\"lightbulb-icon\">\ud83d\udca1<\/span><span class=\"faq-example-question\">Example: What is the range of \\(y=2x^2+4x-5\\)?<\/span><\/p>\n<hr style=\"padding: 0; margin-top: -0.2em; margin-bottom: 1.2em\">First, find the \\(x\\)-coordinate of the vertex: <\/p>\n<p style=\"text-align: center\">\\(x=-\\dfrac{b}{2a}\\)\\(\\:=-\\dfrac{4}{2(2)}=-1\\)<\/p>\n<p> Then, find the \\(y\\)-coordinate of the vertex: <\/p>\n<p style=\"text-align: center\">\\(y=2x^2+4x-5\\)\\(\\:=2(-1)^2+4(-1)-5\\)\\(\\:=-7\\)<\/p>\n<p> Next, determine if the parabola opens up or down. In this case, it opens up because \\(a=2\\) and 2 is positive. <\/p>\n<p style=\"margin-bottom: 0em\">Therefore, the range of \\(y=2x^2+4x-5\\) is \\((-7,\u221e)\\).<\/p>\n<\/div>\n<\/p><\/div>\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"spoiler\" id=\"factsheet-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Quadratic_Function_Domain_and_Range_PDF\" class=\"m-toc-anchor\"><\/span>Quadratic Function Domain and Range PDF<\/h2>\n<div>\n\t\t\t\t\t<img width=\"1361\" height=\"1768\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/image_2022-01-21_110207.png\" class=\"attachment-full size-full\" alt=\"Infographic titled &quot;Domain and Range of Quadratics&quot; explaining the standard form, vertex form, and factored form of quadratic functions, including their domains, ranges, and example calculations.\" decoding=\"async\" loading=\"lazy\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/image_2022-01-21_110207.png 1361w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/image_2022-01-21_110207-231x300.png 231w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/image_2022-01-21_110207-788x1024.png 788w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/image_2022-01-21_110207-768x998.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/image_2022-01-21_110207-1182x1536.png 1182w\" sizes=\"auto, (max-width: 1361px) 100vw, 1361px\" \/><\/p>\n<div class=\"sub_categories\">\n\t\t\t\t\t\t<a href=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/01\/Domain-and-Range-of-Quadratic-Functions-Fact-Sheet.pdf\"><span id=\"Your_Quadratic_Functions_PDF_Download\" class=\"m-toc-anchor\"><\/span>Your Quadratic Functions PDF Download<\/a>\n\t\t\t\t\t<\/div>\n<\/p>\n<\/div>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Domain_and_Range_of_Quadratic_Functions_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Domain and Range of Quadratic Functions 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 \/>\nFind the domain and range of the equation \\(f(x)=3x^2+6x-2\\).<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">Domain: all real numbers | Range: all real numbers<\/div><div class=\"PQ\"  id=\"PQ-1-2\">Domain: all real numbers \u2265\u22121 | Range: all real numbers \u22655<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-3\">Domain: all real numbers | Range: all real numbers \u2265\u22125<\/div><div class=\"PQ\"  id=\"PQ-1-4\">Domain: all real numbers | Range: all real numbers \u2265\u22121<\/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>The domain of this function is all real numbers because there is no limit on the values that can be plugged in for \\(x\\). However, there are limits to the output values.<\/p>\n<p>To find the possible output values, or the range, two things must be known:<\/p>\n<ol>\n<li>If the graph opens up or down<\/li>\n<li>What the \\(y\\)-value of the vertex is<\/li>\n<\/ol>\n<p>This equation is in standard form, and \\(a\\) is positive, which indicates that the graph opens up. This means the range will be greater than or equal to some value. Here\u2019s how to find that value:<\/p>\n<p>First, evaluate \\(f(\\frac{-b}{2a})\\).<\/p>\n<p style=\"text-align: center; line-height: 50px\">\n\\(f(\\frac{-6}{2(3)})\\)<br \/>\n\\(f(\\frac{-6}{6})\\)<br \/>\n\\(f(-1)\\)\n<\/p>\n<p>Now, plug this back into the original equation. <\/p>\n<p style=\"text-align: center; line-height: 45px\">\n\\(f(x)=3x^2+6x-2\\)<br \/>\n\\(f(-1)=3(-1)^2+6(-1)-2\\)<br \/>\n\\(f(-1)=3-6-2\\)<br \/>\n\\(f(-1)=-5\\)\n<\/p>\n<p>Since \\(a\\) is positive, we know that the range is all real numbers \u2265\u22125.<\/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 \/>\nFind the domain and range of the equation \\(f(x)=-3x^2+6x-3\\).<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-2-1\">Domain: all real numbers | Range: all real numbers \u22640<\/div><div class=\"PQ\"  id=\"PQ-2-2\">Domain: all real numbers \u22650 | Range: all real numbers<\/div><div class=\"PQ\"  id=\"PQ-2-3\">Domain: all real numbers >3 | Range: all real numbers <6<\/div><div class=\"PQ\"  id=\"PQ-2-4\">Domain: all real numbers | Range: all real numbers<\/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>The domain of this function is all real numbers because there is no limit on the values that can be plugged in for \\(x\\). However, there are limits to the output values.<\/p>\n<p>To find the possible output values, or the range, two things must be known:<\/p>\n<ol>\n<li>If the graph opens up or down<\/li>\n<li>What the \\(y\\)-value of the vertex is<\/li>\n<\/ol>\n<p>This equation is in standard form, and \\(a\\) is negative, which indicates that the graph opens down. This means the range will be less than or equal to some value. <\/p>\n<p>Here\u2019s how to find that value:<\/p>\n<p>First, evaluate \\(f(\\frac{-b}{2a})\\).<\/p>\n<p style=\"text-align: center; line-height: 50px\">\n\\(f(\\frac{-6}{2(-3)})\\)<br \/>\n\\(f(\\frac{-6}{-6})\\)<br \/>\n\\(f(1)\\)\n<\/p>\n<p>Now, plug this back into the original equation.<\/p>\n<p style=\"text-align:center; line-height: 45px\">\n\\(f(x)=-3x^2+6x-3\\)<br \/>\n\\(f(1)=-3(1)^2+6(1)-3\\)<br \/>\n\\(f(1)=-3+6-3\\)<br \/>\n\\(f(1)=0\\)\n<\/p>\n<p>Since \\(a\\) is negative, the range is all real numbers \u22640.<\/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 \/>\nFind the domain and range of the equation \\(f(x)=2(x+3)^2-8\\).<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">Domain: all real numbers | Range: all real numbers<\/div><div class=\"PQ\"  id=\"PQ-3-2\">Domain: all real numbers \u22658 | Range: all real numbers \u22643<\/div><div class=\"PQ\"  id=\"PQ-3-3\">Domain: all real numbers | Range: all real numbers \u2264\u22128<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-4\">Domain: all real numbers | Range: all real numbers \u2265\u22128<\/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>This equation is in vertex form: \\(f(x)=a(x-h)^2+k\\).<\/p>\n<p>The domain, or values for \\(x\\), can be any real number, but the range does have restrictions. Not all \\(y\\)-values will appear on the graph for this equation.<\/p>\n<p>To find the range, first find the vertex, which is located at \\((h,k)\\). Referring back to the original equation shows that \\((h,k)\\) would be \\((-3,-8)\\).<\/p>\n<p>Since \\(a\\) is positive, the range is all real numbers greater than or equal to \u22128.<\/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 \/>\nFind the domain and range of the equation \\(f(x)=4(x+7)^2+5\\).<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">Domain: all real numbers \u22655 | Range: all real numbers<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-2\">Domain: all real numbers | Range: all real numbers \u22655<\/div><div class=\"PQ\"  id=\"PQ-4-3\">Domain: no solution | Range: all real numbers<\/div><div class=\"PQ\"  id=\"PQ-4-4\">Domain: all real numbers | Range: all real numbers \u2265\u22125<\/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>This equation is in vertex form. The domain, or values for \\(x\\), can be any real number, but the range does have restrictions. Not all \\(y\\)-values will appear on the graph for this equation.<\/p>\n<p>To find the range, first find the vertex, which is located at \\((h,k)\\). Referring back to the original equation shows that \\((h,k)\\) would be \\((-7,5)\\).<\/p>\n<p>Since \\(a\\) is positive, the range is all real numbers \u22655.<\/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 \/>\nFind the domain and range of the equation \\(y=-4(x+4)(x-6)\\).<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">Domain: all real numbers \u2264100 | Range: all real numbers<\/div><div class=\"PQ\"  id=\"PQ-5-2\">Domain: all real numbers \u226440 | Range: all real numbers<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-3\">Domain: all real numbers | Range: all real numbers \u2264100<\/div><div class=\"PQ\"  id=\"PQ-5-4\">Domain: all real numbers >100 | Range: all real numbers<\/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>This equation is in a factored form. The domain is all real numbers because there is no restriction for the value of \\(x\\), or the input.<\/p>\n<p>Because the equation is in a factored form, the \\(x\\)-intercepts can be identified easily. The equation \\(f(x)=-4(x+4)(x-6)\\) shows that the \\(x\\)-intercepts are at -4 and 6. These two values can be averaged in order to find the vertex, or \\(y\\)-intercept.<\/p>\n<p>Average \u22124 and 6 to get \\(\\frac{-4+6}{2}\\), which simplifies to 1. Now, take this value and plug it into the original equation:<\/p>\n<p style=\"text-align: center\">\\(f(1)=-4(1+4)(1-6)\\)<\/p>\n<p>This simplifies to 100. In the original equation, \\(a\\) was negative, so the range is all real numbers \u2264100.<\/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":114833,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-4326","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-domain-and-range-videos","7":"page_category-math-advertising-group","8":"page_category-video-pages-for-study-course-sidebar-ad","9":"page_type-video","10":"content_type-fact-sheet","11":"content_type-practice-questions","12":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4326","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=4326"}],"version-history":[{"count":7,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4326\/revisions"}],"predecessor-version":[{"id":198728,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4326\/revisions\/198728"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/114833"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=4326"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}