{"id":175,"date":"2023-06-21T13:22:40","date_gmt":"2023-06-21T13:22:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-graphs-of-linear-functions\/"},"modified":"2023-06-21T13:22:40","modified_gmt":"2023-06-21T13:22:40","slug":"summary-graphs-of-linear-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-graphs-of-linear-functions\/","title":{"raw":"Summary: Graphs of Linear Functions","rendered":"Summary: Graphs of Linear Functions"},"content":{"raw":"\n\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165134190780\">\n \t<li>Linear functions may be graphed by plotting points or by using the <em>y<\/em>-intercept and slope.<\/li>\n \t<li>Graphs of linear functions may be transformed by shifting the graph up, down, left, or right as well as using stretches, compressions, and reflections.<\/li>\n \t<li>The <em>y<\/em>-intercept and slope of a line may be used to write the equation of a line.<\/li>\n \t<li>The <em>x<\/em>-intercept is the point at which the graph of a linear function crosses the <em>x<\/em>-axis.<\/li>\n \t<li>Horizontal lines are written in the form, [latex]f(x)=b[\/latex].<\/li>\n \t<li>Vertical lines are written in the form, [latex]x=b[\/latex].<\/li>\n \t<li>Parallel lines have the same slope.<\/li>\n \t<li>Perpendicular lines have negative reciprocal slopes, assuming neither is vertical.<\/li>\n \t<li>A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the <em>x<\/em>- and <em>y<\/em>-values of the given point into the equation [latex]f\\left(x\\right)=mx+b[\/latex] and using the <em>b<\/em>&nbsp;that results. Similarly, point-slope form of an equation can also be used.<\/li>\n \t<li>A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope.<\/li>\n \t<li>The absolute value function is commonly used to measure distances between points.<\/li>\n \t<li>Applied problems, such as ranges of possible values, can also be solved using the absolute value function.<\/li>\n \t<li>The graph of the absolute value function resembles the letter V. It has a corner point at which the graph changes direction.<\/li>\n \t<li>In an absolute value equation, an unknown variable is the input of an absolute value function.<\/li>\n \t<li>If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable.<\/li>\n \t<li>An absolute value equation may have one solution, two solutions, or no solutions.<\/li>\n \t<li>An absolute value inequality is similar to an absolute value equation but takes the form [latex]|A|&lt;B,|A|\\le B,|A|&gt;B,\\text{ or }|A|\\ge B[\/latex]. It can be solved by determining the boundaries of the solution set and then testing which segments are in the set.<\/li>\n \t<li>Absolute value inequalities can also be solved graphically.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n \t<dt><strong>absolute value equation<\/strong><\/dt>\n \t<dd id=\"fs-id1165135429394\">an equation of the form [latex]|A|=B[\/latex], with [latex]B\\ge 0[\/latex]; it will have solutions when [latex]A=B[\/latex] or [latex]-A=B[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n \t<dt><strong>absolute value inequality<\/strong><\/dt>\n \t<dd id=\"fs-id1165135429394\">a relationship in the form [latex]|{ A }|&lt;{ B },|{ A }|\\le { B },|{ A }|&gt;{ B },\\text{or }|{ A }|\\ge{ B }[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n \t<dt><strong>horizontal line<\/strong><\/dt>\n \t<dd id=\"fs-id1165135429394\">a line defined by [latex]f\\left(x\\right)=b[\/latex] where <em>b<\/em> is a real number. The slope of a horizontal line is 0.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n \t<dt><strong>parallel lines<\/strong><\/dt>\n \t<dd id=\"fs-id1165135429394\">two or more lines with the same slope<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n \t<dt><strong>perpendicular lines<\/strong><\/dt>\n \t<dd id=\"fs-id1165135429394\">two lines that intersect at right angles and have slopes that are negative reciprocals of each other<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n \t<dt><strong>vertical line<\/strong><\/dt>\n \t<dd id=\"fs-id1165135429394\">a line defined by [latex]x=a[\/latex] where <em>a<\/em>&nbsp;is a real number. The slope of a vertical line is undefined.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n \t<dt><strong><em>x<\/em>-intercept<\/strong><\/dt>\n \t<dd id=\"fs-id1165135429394\">the point on the graph of a linear function when the output value is 0; the point at which the graph crosses the horizontal axis<\/dd>\n<\/dl>\n\n","rendered":"<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165134190780\">\n<li>Linear functions may be graphed by plotting points or by using the <em>y<\/em>-intercept and slope.<\/li>\n<li>Graphs of linear functions may be transformed by shifting the graph up, down, left, or right as well as using stretches, compressions, and reflections.<\/li>\n<li>The <em>y<\/em>-intercept and slope of a line may be used to write the equation of a line.<\/li>\n<li>The <em>x<\/em>-intercept is the point at which the graph of a linear function crosses the <em>x<\/em>-axis.<\/li>\n<li>Horizontal lines are written in the form, [latex]f(x)=b[\/latex].<\/li>\n<li>Vertical lines are written in the form, [latex]x=b[\/latex].<\/li>\n<li>Parallel lines have the same slope.<\/li>\n<li>Perpendicular lines have negative reciprocal slopes, assuming neither is vertical.<\/li>\n<li>A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the <em>x<\/em>&#8211; and <em>y<\/em>-values of the given point into the equation [latex]f\\left(x\\right)=mx+b[\/latex] and using the <em>b<\/em>&nbsp;that results. Similarly, point-slope form of an equation can also be used.<\/li>\n<li>A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope.<\/li>\n<li>The absolute value function is commonly used to measure distances between points.<\/li>\n<li>Applied problems, such as ranges of possible values, can also be solved using the absolute value function.<\/li>\n<li>The graph of the absolute value function resembles the letter V. It has a corner point at which the graph changes direction.<\/li>\n<li>In an absolute value equation, an unknown variable is the input of an absolute value function.<\/li>\n<li>If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable.<\/li>\n<li>An absolute value equation may have one solution, two solutions, or no solutions.<\/li>\n<li>An absolute value inequality is similar to an absolute value equation but takes the form [latex]|A|<B,|A|\\le B,|A|>B,\\text{ or }|A|\\ge B[\/latex]. It can be solved by determining the boundaries of the solution set and then testing which segments are in the set.<\/li>\n<li>Absolute value inequalities can also be solved graphically.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n<dt><strong>absolute value equation<\/strong><\/dt>\n<dd id=\"fs-id1165135429394\">an equation of the form [latex]|A|=B[\/latex], with [latex]B\\ge 0[\/latex]; it will have solutions when [latex]A=B[\/latex] or [latex]-A=B[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n<dt><strong>absolute value inequality<\/strong><\/dt>\n<dd id=\"fs-id1165135429394\">a relationship in the form [latex]|{ A }|<{ B },|{ A }|\\le { B },|{ A }|>{ B },\\text{or }|{ A }|\\ge{ B }[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n<dt><strong>horizontal line<\/strong><\/dt>\n<dd id=\"fs-id1165135429394\">a line defined by [latex]f\\left(x\\right)=b[\/latex] where <em>b<\/em> is a real number. The slope of a horizontal line is 0.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n<dt><strong>parallel lines<\/strong><\/dt>\n<dd id=\"fs-id1165135429394\">two or more lines with the same slope<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n<dt><strong>perpendicular lines<\/strong><\/dt>\n<dd id=\"fs-id1165135429394\">two lines that intersect at right angles and have slopes that are negative reciprocals of each other<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n<dt><strong>vertical line<\/strong><\/dt>\n<dd id=\"fs-id1165135429394\">a line defined by [latex]x=a[\/latex] where <em>a<\/em>&nbsp;is a real number. The slope of a vertical line is undefined.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135429388\" class=\"definition\">\n<dt><strong><em>x<\/em>-intercept<\/strong><\/dt>\n<dd id=\"fs-id1165135429394\">the point on the graph of a linear function when the output value is 0; the point at which the graph crosses the horizontal axis<\/dd>\n<\/dl>\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-175\">\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":15,"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":"37b3ba83-e2e0-4dc7-970e-28602a72abda","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-175","chapter","type-chapter","status-publish","hentry"],"part":160,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/175","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\/175\/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\/175\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=175"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=175"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=175"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=175"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}