{"id":169,"date":"2023-06-21T13:22:40","date_gmt":"2023-06-21T13:22:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-linear-functions\/"},"modified":"2023-06-21T13:22:40","modified_gmt":"2023-06-21T13:22:40","slug":"summary-linear-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-linear-functions\/","title":{"raw":"Summary: Characteristics of Linear Functions","rendered":"Summary: Characteristics of Linear Functions"},"content":{"raw":"\n\n<h2>Key Equations<\/h2>\n<section id=\"fs-id1165137784950\" class=\"key-equations\"><section id=\"fs-id1165137784950\" class=\"key-equations\">\n<table id=\"fs-id1165137784956\" summary=\"...\">\n<tbody>\n<tr>\n<td>slope-intercept form of a line<\/td>\n<td>[latex]y=mx+b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>slope<\/td>\n<td>[latex]m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>point-slope form of a line<\/td>\n<td>[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section><section id=\"fs-id1165135696154\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165137736447\">\n \t<li>The ordered pairs given by a linear function represent points on a line.<\/li>\n \t<li>Linear functions can be represented in words, function notation, tabular form and graphical form.<\/li>\n \t<li>The rate of change of a linear function is also known as the slope.<\/li>\n \t<li>An equation in slope-intercept form of a line includes the slope and the initial value of the function.<\/li>\n \t<li>The initial value, or <em>y<\/em>-intercept, is the output value when the input of a linear function is zero. It is the <em>y<\/em>-value of the point where the line crosses the <em>y<\/em>-axis.<\/li>\n \t<li>An increasing linear function results in a graph that slants upward from left to right and has a positive slope.<\/li>\n \t<li>A decreasing linear function results in a graph that slants downward from left to right and has a negative slope.<\/li>\n \t<li>A constant linear function results in a graph that is a horizontal line.<\/li>\n \t<li>Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant.<\/li>\n \t<li>The slope of a linear function can be calculated by dividing the difference between <em>y<\/em>-values by the difference in corresponding <em>x<\/em>-values of any two points on the line.<\/li>\n \t<li>The slope and initial value can be determined given a graph or any two points on the line.<\/li>\n \t<li>One form of a linear function is slope-intercept form.<\/li>\n \t<li>Point-slope form is useful for finding the equation of a linear function when given the slope of a line and one point.<\/li>\n \t<li>Point-slope form is also convenient for finding the equation of a linear function when given two points through which a line passes.<\/li>\n \t<li>The equation for a linear function can be written in slope-intercept form if the slope <em>m<\/em>&nbsp;and initial value <em>b&nbsp;<\/em>are known.<\/li>\n \t<li>A linear function can be used to solve real-world problems.<\/li>\n \t<li>A linear function can be written from tabular form.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137405111\" class=\"definition\">\n \t<dt><strong>decreasing linear function<\/strong><\/dt>\n \t<dd id=\"fs-id1165137405116\">a function with a negative slope: If [latex]m&lt;0, \\text{then }f\\left(x\\right)=mx+b[\/latex] is decreasing.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137863356\" class=\"definition\">\n \t<dt><strong>increasing linear function<\/strong><\/dt>\n \t<dd id=\"fs-id1165135188274\">a function with a positive slope: If [latex]m&gt;0, \\text{then }f\\left(x\\right)=mx+b[\/latex] is increasing.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n \t<dt><strong>linear function<\/strong><\/dt>\n \t<dd id=\"fs-id1165135429394\">a function with a constant rate of change that is a polynomial of degree 1 whosegraph is a straight line<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134389091\" class=\"definition\">\n \t<dt><strong>point-slope form<\/strong><\/dt>\n \t<dd id=\"fs-id1165134389097\">the equation of a linear function of the form [latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137635132\" class=\"definition\">\n \t<dt><strong>slope<\/strong><\/dt>\n \t<dd id=\"fs-id1165137635137\">the ratio of the change in output values to the change in input values; a measure of the steepness of a line<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137817449\" class=\"definition\">\n \t<dt><strong>slope-intercept form<\/strong><\/dt>\n \t<dd id=\"fs-id1165137817454\">the equation of a linear function of the form [latex]f\\left(x\\right)=mx+b[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135195656\" class=\"definition\">\n \t<dt><strong><em>y<\/em>-intercept<\/strong><\/dt>\n \t<dd id=\"fs-id1165137635107\">the value of a function when the input value is zero; also known as initial value<\/dd>\n<\/dl>\n<\/section><\/section>\n\n","rendered":"<h2>Key Equations<\/h2>\n<section id=\"fs-id1165137784950\" class=\"key-equations\">\n<section id=\"fs-id1165137784950\" class=\"key-equations\">\n<table id=\"fs-id1165137784956\" summary=\"...\">\n<tbody>\n<tr>\n<td>slope-intercept form of a line<\/td>\n<td>[latex]y=mx+b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>slope<\/td>\n<td>[latex]m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>point-slope form of a line<\/td>\n<td>[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165135696154\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165137736447\">\n<li>The ordered pairs given by a linear function represent points on a line.<\/li>\n<li>Linear functions can be represented in words, function notation, tabular form and graphical form.<\/li>\n<li>The rate of change of a linear function is also known as the slope.<\/li>\n<li>An equation in slope-intercept form of a line includes the slope and the initial value of the function.<\/li>\n<li>The initial value, or <em>y<\/em>-intercept, is the output value when the input of a linear function is zero. It is the <em>y<\/em>-value of the point where the line crosses the <em>y<\/em>-axis.<\/li>\n<li>An increasing linear function results in a graph that slants upward from left to right and has a positive slope.<\/li>\n<li>A decreasing linear function results in a graph that slants downward from left to right and has a negative slope.<\/li>\n<li>A constant linear function results in a graph that is a horizontal line.<\/li>\n<li>Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant.<\/li>\n<li>The slope of a linear function can be calculated by dividing the difference between <em>y<\/em>-values by the difference in corresponding <em>x<\/em>-values of any two points on the line.<\/li>\n<li>The slope and initial value can be determined given a graph or any two points on the line.<\/li>\n<li>One form of a linear function is slope-intercept form.<\/li>\n<li>Point-slope form is useful for finding the equation of a linear function when given the slope of a line and one point.<\/li>\n<li>Point-slope form is also convenient for finding the equation of a linear function when given two points through which a line passes.<\/li>\n<li>The equation for a linear function can be written in slope-intercept form if the slope <em>m<\/em>&nbsp;and initial value <em>b&nbsp;<\/em>are known.<\/li>\n<li>A linear function can be used to solve real-world problems.<\/li>\n<li>A linear function can be written from tabular form.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137405111\" class=\"definition\">\n<dt><strong>decreasing linear function<\/strong><\/dt>\n<dd id=\"fs-id1165137405116\">a function with a negative slope: If [latex]m<0, \\text{then }f\\left(x\\right)=mx+b[\/latex] is decreasing.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137863356\" class=\"definition\">\n<dt><strong>increasing linear function<\/strong><\/dt>\n<dd id=\"fs-id1165135188274\">a function with a positive slope: If [latex]m>0, \\text{then }f\\left(x\\right)=mx+b[\/latex] is increasing.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n<dt><strong>linear function<\/strong><\/dt>\n<dd id=\"fs-id1165135429394\">a function with a constant rate of change that is a polynomial of degree 1 whosegraph is a straight line<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134389091\" class=\"definition\">\n<dt><strong>point-slope form<\/strong><\/dt>\n<dd id=\"fs-id1165134389097\">the equation of a linear function of the form [latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137635132\" class=\"definition\">\n<dt><strong>slope<\/strong><\/dt>\n<dd id=\"fs-id1165137635137\">the ratio of the change in output values to the change in input values; a measure of the steepness of a line<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137817449\" class=\"definition\">\n<dt><strong>slope-intercept form<\/strong><\/dt>\n<dd id=\"fs-id1165137817454\">the equation of a linear function of the form [latex]f\\left(x\\right)=mx+b[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135195656\" class=\"definition\">\n<dt><strong><em>y<\/em>-intercept<\/strong><\/dt>\n<dd id=\"fs-id1165137635107\">the value of a function when the input value is zero; also known as initial value<\/dd>\n<\/dl>\n<\/section>\n<\/section>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-169\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"}]","CANDELA_OUTCOMES_GUID":"e6c60b3e-adfb-4cb9-96a5-3ab82dba3568","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-169","chapter","type-chapter","status-publish","hentry"],"part":160,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/169","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/169\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/160"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/169\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=169"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=169"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=169"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=169"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}