{"id":1691,"date":"2024-06-03T21:47:08","date_gmt":"2024-06-03T21:47:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1691"},"modified":"2024-12-31T13:56:21","modified_gmt":"2024-12-31T13:56:21","slug":"introduction-to-linear-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-linear-functions-learn-it-2\/","title":{"raw":"Introduction to Linear Functions: Learn it 2","rendered":"Introduction to Linear Functions: Learn it 2"},"content":{"raw":"<h2>Representing Linear Functions<\/h2>\r\n<h3>Representing a Linear Function in Word Form<\/h3>\r\nLet\u2019s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.\r\n<ul>\r\n \t<li><em>The train\u2019s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at a constant speed.<\/em><\/li>\r\n<\/ul>\r\nThe speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable. The rate of change for this example is constant, which means that it is the same for each input value. As the time (input) increases by [latex]1[\/latex] second, the corresponding distance (output) increases by [latex]83[\/latex] meters. The train began moving at this constant speed at a distance of [latex]250[\/latex] meters from the station.\r\n<h3>Representing a Linear Function in Function Notation<\/h3>\r\nAnother approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as\u00a0<strong>slope-intercept form<\/strong> of a line, where [latex]x[\/latex] is the input value, [latex]m[\/latex] is the rate of change, and [latex]b[\/latex] is the initial value of the dependent variable.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>slope-intercept form<\/h3>\r\n<strong>Slope-intercept form,<\/strong> is a way to represent linear functions. It highlights the rate of change [latex]m[\/latex], the input value [latex]x[\/latex], and initial value of the dependent variable [latex]b[\/latex], making it foundational for understanding relationships between variables.\r\n\r\n&nbsp;\r\n\r\nSlope-intercept form is given below:\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}\\text{Equation form}\\hfill &amp; y=mx+b\\hfill \\\\ \\text{Function notation}\\hfill &amp; f\\left(x\\right)=mx+b\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/section>In the example of the train, we might use the notation [latex]D\\left(t\\right)[\/latex]\u00a0in which the total distance [latex]D[\/latex] is a function of the time [latex]t[\/latex].\u00a0The rate, [latex]m[\/latex],\u00a0is [latex]83[\/latex] meters per second. The initial value, [latex]b[\/latex], of the dependent variable is the original distance from the station, [latex]250[\/latex] meters. We can write a generalized equation to represent the motion of the train.\r\n<p style=\"text-align: center;\">[latex]D\\left(t\\right)=83t+250[\/latex]<\/p>\r\n\r\n<section class=\"textbox proTip\">In the example above, the function notation, [latex]y=f(x)[\/latex] is written, descriptively for the situation, as [latex]y = D(t)[\/latex].The function, [latex]D(t)=83t +250[\/latex], takes an input [latex]t[\/latex], multiplies it by [latex]83[\/latex], then adds [latex]250[\/latex]. The result is the output, [latex]D(t)[\/latex].We can choose input and output variables stylistically as desired. It is the form of the function that remains consistent from function to function.<\/section>\r\n<h3>Representing a Linear Function in Tabular Form<\/h3>\r\nA third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in the table below. From the table, we can see that the distance changes by [latex]83[\/latex] meters for every [latex]1[\/latex] second increase in time.\r\n\r\n<center><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18223049\/CNX_Precalc_Figure_02_01_0152.jpg\" alt=\"Table with the first row, labeled t, containing the seconds from 0 to 3, and with the second row, labeled D(t), containing the meters 250 to 499. The first row goes up by 1 second, and the second row goes up by 83 meters.\" width=\"487\" height=\"161\" \/><\/center><center><strong><span style=\"font-size: 10pt;\">Tabular representation of the function D showing selected input and output values.<\/span><\/strong><\/center>&nbsp;\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>representing a linear function in tabular form<\/h3>\r\nIn a table representing a linear function, each input-output pair forms a consistent pattern, exhibiting a constant rate of change between [latex]y[\/latex]-values. To identify the function as linear, ensure that the difference between consecutive [latex]y[\/latex]-values is the same when the [latex]x[\/latex]-values increase by a consistent amount.\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]13518[\/ohm2_question]<\/section>\r\n<h3>Representing a Linear Function in Graphical Form<\/h3>\r\nAnother way to represent linear functions is visually by using a graph. We can use the function relationship from above, [latex]D\\left(t\\right)=83t+250[\/latex], to draw a graph as seen below. Notice the graph is a line. When we plot a linear function, the graph is always a line.\r\n\r\n<section class=\"textbox proTip\">The y-intercept of the line in the graph below represents the\u00a0<strong>initial value\u00a0<\/strong>of the function.The term <strong>initial value<\/strong> is used in situations where it would be illogical to consider negative input. Even though it would be reasonable to graph the line [latex]y=83x + 250[\/latex] where [latex]x \\lt 0[\/latex], in this situation we don't consider negative input values of time.<\/section>The rate of change, which is always constant for linear functions, determines the slant or <strong>slope<\/strong> of the line. The point at which the input value is zero is the <strong>[latex]y[\/latex]-intercept<\/strong> of the line. We can see from the graph that the [latex]y[\/latex]-intercept in the train example we just saw is [latex]\\left(0,250\\right)[\/latex] and represents the distance of the train from the station when it began moving at a constant speed.\r\n\r\n<center><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18223051\/CNX_Precalc_Figure_02_01_0122.jpg\" alt=\"A graph of an increasing function with points at (-2, -4) and (0, 2).\" width=\"487\" height=\"289\" \/><\/center><center><strong><span style=\"font-size: 10pt;\">The graph of [latex]D\\left(t\\right)=83t+250[\/latex]. Graphs of linear functions are lines because the rate of change is constant.<\/span><\/strong><\/center>&nbsp;\r\n\r\nNotice the graph of the train example is restricted since the input value, time, must always be a nonnegative real number. However, this is not always the case for every linear function. Consider the graph of the line [latex]f\\left(x\\right)=2{x}_{}+1[\/latex].\u00a0Ask yourself what input values can be plugged into this function, that is, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled and then have one added to the product.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>representing a linear function in graphical form<\/h3>\r\nA <strong>linear function<\/strong> is a function whose graph is a line. Linear functions can be written in <strong>slope-intercept form<\/strong> of a line:\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=mx+b[\/latex]<\/p>\r\n&nbsp;\r\n\r\nwhere [latex]b[\/latex]\u00a0is the initial or starting value of the function (when input, [latex]x=0[\/latex]) and [latex]m[\/latex]\u00a0is the constant rate of change or <strong>slope<\/strong> of the function. The <strong><em>y<\/em>-intercept<\/strong> is at [latex]\\left(0,b\\right)[\/latex].\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\">The pressure, [latex]P[\/latex], in pounds per square inch (PSI) on a diver depends on depth below the water surface, [latex]d[\/latex], in feet.\r\n[latex]\\\\[\/latex]\r\nThis relationship may be modeled by the equation\r\n<center>[latex]P\\left(d\\right)=0.434d+14.696[\/latex]<\/center>\r\nRestate this function in words.[reveal-answer q=\"15899\"]Show Solution[\/reveal-answer] [hidden-answer a=\"15899\"]To restate the function in words, we need to describe each part of the equation. The pressure as a function of depth equals four hundred thirty-four thousandths times depth plus fourteen and six hundred ninety-six thousandths.\r\n[latex]\\\\[\/latex]<strong>Analysis of the Solution<\/strong>\r\n[latex]\\\\[\/latex]\r\nThe initial value, [latex]14.696[\/latex], is the pressure in PSI on the diver at a depth of [latex]0[\/latex] feet, which is the surface of the water. The rate of change, or slope, is [latex]0.434[\/latex] PSI per foot. This tells us that the pressure on the diver increases [latex]0.434[\/latex] PSI for each foot her depth increases.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]13519[\/ohm2_question]<\/section>","rendered":"<h2>Representing Linear Functions<\/h2>\n<h3>Representing a Linear Function in Word Form<\/h3>\n<p>Let\u2019s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.<\/p>\n<ul>\n<li><em>The train\u2019s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at a constant speed.<\/em><\/li>\n<\/ul>\n<p>The speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable. The rate of change for this example is constant, which means that it is the same for each input value. As the time (input) increases by [latex]1[\/latex] second, the corresponding distance (output) increases by [latex]83[\/latex] meters. The train began moving at this constant speed at a distance of [latex]250[\/latex] meters from the station.<\/p>\n<h3>Representing a Linear Function in Function Notation<\/h3>\n<p>Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as\u00a0<strong>slope-intercept form<\/strong> of a line, where [latex]x[\/latex] is the input value, [latex]m[\/latex] is the rate of change, and [latex]b[\/latex] is the initial value of the dependent variable.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>slope-intercept form<\/h3>\n<p><strong>Slope-intercept form,<\/strong> is a way to represent linear functions. It highlights the rate of change [latex]m[\/latex], the input value [latex]x[\/latex], and initial value of the dependent variable [latex]b[\/latex], making it foundational for understanding relationships between variables.<\/p>\n<p>&nbsp;<\/p>\n<p>Slope-intercept form is given below:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}\\text{Equation form}\\hfill & y=mx+b\\hfill \\\\ \\text{Function notation}\\hfill & f\\left(x\\right)=mx+b\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<p>In the example of the train, we might use the notation [latex]D\\left(t\\right)[\/latex]\u00a0in which the total distance [latex]D[\/latex] is a function of the time [latex]t[\/latex].\u00a0The rate, [latex]m[\/latex],\u00a0is [latex]83[\/latex] meters per second. The initial value, [latex]b[\/latex], of the dependent variable is the original distance from the station, [latex]250[\/latex] meters. We can write a generalized equation to represent the motion of the train.<\/p>\n<p style=\"text-align: center;\">[latex]D\\left(t\\right)=83t+250[\/latex]<\/p>\n<section class=\"textbox proTip\">In the example above, the function notation, [latex]y=f(x)[\/latex] is written, descriptively for the situation, as [latex]y = D(t)[\/latex].The function, [latex]D(t)=83t +250[\/latex], takes an input [latex]t[\/latex], multiplies it by [latex]83[\/latex], then adds [latex]250[\/latex]. The result is the output, [latex]D(t)[\/latex].We can choose input and output variables stylistically as desired. It is the form of the function that remains consistent from function to function.<\/section>\n<h3>Representing a Linear Function in Tabular Form<\/h3>\n<p>A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in the table below. From the table, we can see that the distance changes by [latex]83[\/latex] meters for every [latex]1[\/latex] second increase in time.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18223049\/CNX_Precalc_Figure_02_01_0152.jpg\" alt=\"Table with the first row, labeled t, containing the seconds from 0 to 3, and with the second row, labeled D(t), containing the meters 250 to 499. The first row goes up by 1 second, and the second row goes up by 83 meters.\" width=\"487\" height=\"161\" \/><\/div>\n<div style=\"text-align: center;\"><strong><span style=\"font-size: 10pt;\">Tabular representation of the function D showing selected input and output values.<\/span><\/strong><\/div>\n<p>&nbsp;<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>representing a linear function in tabular form<\/h3>\n<p>In a table representing a linear function, each input-output pair forms a consistent pattern, exhibiting a constant rate of change between [latex]y[\/latex]-values. To identify the function as linear, ensure that the difference between consecutive [latex]y[\/latex]-values is the same when the [latex]x[\/latex]-values increase by a consistent amount.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm13518\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13518&theme=lumen&iframe_resize_id=ohm13518&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Representing a Linear Function in Graphical Form<\/h3>\n<p>Another way to represent linear functions is visually by using a graph. We can use the function relationship from above, [latex]D\\left(t\\right)=83t+250[\/latex], to draw a graph as seen below. Notice the graph is a line. When we plot a linear function, the graph is always a line.<\/p>\n<section class=\"textbox proTip\">The y-intercept of the line in the graph below represents the\u00a0<strong>initial value\u00a0<\/strong>of the function.The term <strong>initial value<\/strong> is used in situations where it would be illogical to consider negative input. Even though it would be reasonable to graph the line [latex]y=83x + 250[\/latex] where [latex]x \\lt 0[\/latex], in this situation we don&#8217;t consider negative input values of time.<\/section>\n<p>The rate of change, which is always constant for linear functions, determines the slant or <strong>slope<\/strong> of the line. The point at which the input value is zero is the <strong>[latex]y[\/latex]-intercept<\/strong> of the line. We can see from the graph that the [latex]y[\/latex]-intercept in the train example we just saw is [latex]\\left(0,250\\right)[\/latex] and represents the distance of the train from the station when it began moving at a constant speed.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18223051\/CNX_Precalc_Figure_02_01_0122.jpg\" alt=\"A graph of an increasing function with points at (-2, -4) and (0, 2).\" width=\"487\" height=\"289\" \/><\/div>\n<div style=\"text-align: center;\"><strong><span style=\"font-size: 10pt;\">The graph of [latex]D\\left(t\\right)=83t+250[\/latex]. Graphs of linear functions are lines because the rate of change is constant.<\/span><\/strong><\/div>\n<p>&nbsp;<\/p>\n<p>Notice the graph of the train example is restricted since the input value, time, must always be a nonnegative real number. However, this is not always the case for every linear function. Consider the graph of the line [latex]f\\left(x\\right)=2{x}_{}+1[\/latex].\u00a0Ask yourself what input values can be plugged into this function, that is, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled and then have one added to the product.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>representing a linear function in graphical form<\/h3>\n<p>A <strong>linear function<\/strong> is a function whose graph is a line. Linear functions can be written in <strong>slope-intercept form<\/strong> of a line:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=mx+b[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>where [latex]b[\/latex]\u00a0is the initial or starting value of the function (when input, [latex]x=0[\/latex]) and [latex]m[\/latex]\u00a0is the constant rate of change or <strong>slope<\/strong> of the function. The <strong><em>y<\/em>-intercept<\/strong> is at [latex]\\left(0,b\\right)[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">The pressure, [latex]P[\/latex], in pounds per square inch (PSI) on a diver depends on depth below the water surface, [latex]d[\/latex], in feet.<br \/>\n[latex]\\\\[\/latex]<br \/>\nThis relationship may be modeled by the equation<\/p>\n<div style=\"text-align: center;\">[latex]P\\left(d\\right)=0.434d+14.696[\/latex]<\/div>\n<p>Restate this function in words.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q15899\">Show Solution<\/button> <\/p>\n<div id=\"q15899\" class=\"hidden-answer\" style=\"display: none\">To restate the function in words, we need to describe each part of the equation. The pressure as a function of depth equals four hundred thirty-four thousandths times depth plus fourteen and six hundred ninety-six thousandths.<br \/>\n[latex]\\\\[\/latex]<strong>Analysis of the Solution<\/strong><br \/>\n[latex]\\\\[\/latex]<br \/>\nThe initial value, [latex]14.696[\/latex], is the pressure in PSI on the diver at a depth of [latex]0[\/latex] feet, which is the surface of the water. The rate of change, or slope, is [latex]0.434[\/latex] PSI per foot. This tells us that the pressure on the diver increases [latex]0.434[\/latex] PSI for each foot her depth increases.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm13519\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13519&theme=lumen&iframe_resize_id=ohm13519&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":164,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1691"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1691\/revisions"}],"predecessor-version":[{"id":6822,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1691\/revisions\/6822"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/164"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1691\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1691"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1691"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1691"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1691"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}