{"id":220,"date":"2024-10-18T21:13:11","date_gmt":"2024-10-18T21:13:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/more-on-linear-functions-fresh-take\/"},"modified":"2024-10-18T21:25:29","modified_gmt":"2024-10-18T21:25:29","slug":"more-on-linear-functions-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/more-on-linear-functions-fresh-take\/","title":{"raw":"More on Linear Functions: Fresh Take","rendered":"More on Linear Functions: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Create and interpret equations of linear functions<\/li>\n\t<li>Use linear functions to model and draw conclusions from real-world problems<\/li>\n\t<li>Plot the graphs of linear equations<\/li>\n<\/ul>\n<\/section>\n<h2>Point-Slope Form<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>You've been using slope-intercept form, [latex]y=mx+b[\/latex], to describe linear functions. But did you know there's another superhero in town? Meet <strong>Point-Slope Form<\/strong>, [latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex], where [latex]m[\/latex] is the slope and [latex]({x}_{1},{y}_{1})[\/latex] is a point on the line. This form is a game-changer when you know a specific point and the slope but don't have the [latex]y[\/latex]-intercept.<\/p>\n<p>You can convert between point-slope and slope-intercept forms using basic algebra.<\/p>\n<p>To convert to slope-intercept form, distribute any constants and isolate [latex]y[\/latex]<\/p>\n<p>Both forms describe the same line, so choose the one that makes your life easier!<\/p>\n<\/div>\n<section class=\"textbox example\">Write the equation of a line in point-slope form with a slope of [latex]-2[\/latex] that passes through the point [latex]\\left(-2,\\text{ }2\\right)[\/latex]. Then rewrite the equation in the slope-intercept form. [reveal-answer q=\"624568\"]Show Solution[\/reveal-answer] [hidden-answer a=\"624568\"]<center>Point-slope form: [latex]y - 2=-2\\left(x+2\\right)[\/latex]<\/center><center>Slope-intercept form: [latex]y=-2x - 2[\/latex]<\/center>[\/hidden-answer]<\/section>\n<p>Watch the following video for a worked example of finding point-slope form and converting it to slope-intercept form.<\/p>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328528&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=FntpEHhLHvw&amp;video_target=tpm-plugin-rffzfoe7-FntpEHhLHvw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Writing+an+equation+using+point+slope+form+given+a+point+and+slope.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting an equation using point slope form given a point and slope\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Writing and Interpreting Equations of Linear Functions: Two Points, Graphs, and Real-World Applications<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>You've learned that two points can define a line. But did you know that this is like having two pieces of a puzzle that reveal the entire picture? When you have two points, you can find the slope and then use point-slope form to find the equation of the line.<\/p>\n<p>Sometimes, all you have is a graph. No worries! You can still find the equation of the line. Just pick two points on the graph and use them to find the slope. Then, use the slope and one point to write the equation in point-slope form.<\/p>\n<strong>Quick Tips<\/strong>\n<ul>\n\t<li><strong>Choose Wisely<\/strong>: Pick points that are easy to read on the graph.<\/li>\n\t<li><strong>Y-Intercept<\/strong>: Look for where the line crosses the [latex]y[\/latex]-axis; this is your [latex]y[\/latex]-intercept [latex]b[\/latex].<\/li>\n<\/ul>\n<p>Linear functions are not just theoretical constructs; they're practical tools. For example, if you're starting a business, you can use a linear function to model your costs. The slope might represent the cost per item, and the y-intercept could be your fixed costs like rent.<\/p>\n<\/div>\n<section class=\"textbox example\">Write the point-slope form of an equation of a line that passes through the points [latex]\\left(-1,3\\right)[\/latex] and [latex]\\left(0,0\\right)[\/latex]. Then rewrite the equation in slope-intercept form. [reveal-answer q=\"85979\"]Show Solution[\/reveal-answer] [hidden-answer a=\"85979\"]<center>Point-slope form: [latex]y - 0=-3\\left(x - 0\\right)[\/latex]<\/center><center>Slope-intercept form: [latex]y=-3x[\/latex]<\/center>[\/hidden-answer]<\/section>\n<section class=\"textbox example\">Match each equation of a linear function with one of the lines in the graph below.\n\n<ol>\n\t<li>[latex]f\\left(x\\right)=2x+3[\/latex]<\/li>\n\t<li>[latex]g\\left(x\\right)=2x - 3[\/latex]<\/li>\n\t<li>[latex]h\\left(x\\right)=-2x+3[\/latex]<\/li>\n\t<li>[latex]j\\left(x\\right)=\\frac{1}{2}x+3[\/latex]<\/li>\n<\/ol>\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184343\/CNX_Precalc_Figure_02_02_0112.jpg\" alt=\"Graph of three lines, line 1) passes through (0,3) and (-2, -1), line 2) passes through (0,3) and (-6,0), line 3) passes through (0,-3) and (2,1)\" width=\"393\" height=\"305\"><\/center>[reveal-answer q=\"659573\"]Show Solution[\/reveal-answer] [hidden-answer a=\"659573\"] Analyze the information for each function.\n\n<ol>\n\t<li>This function has a slope of [latex]2[\/latex] and a [latex]y[\/latex]-intercept of [latex]3[\/latex]. It must pass through the point [latex](0, 3)[\/latex] and slant upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function [latex]g[\/latex]&nbsp;has the same slope, but a different [latex]y[\/latex]-intercept. Lines [latex]I[\/latex] and [latex]III[\/latex] have the same slant because they have the same slope. Line [latex]III[\/latex] does not pass through [latex](0, 3)[\/latex] so [latex]f[\/latex]&nbsp;must be represented by line [latex]I[\/latex].<\/li>\n\t<li>This function also has a slope of [latex]2[\/latex], but a [latex]y[\/latex]-intercept of [latex]\u20133[\/latex]. It must pass through the point [latex](0, \u20133)[\/latex] and slant upward from left to right. It must be represented by line [latex]III[\/latex].<\/li>\n\t<li>This function has a slope of [latex]\u20132[\/latex] and a [latex]y[\/latex]-intercept of [latex]3[\/latex]. This is the only function listed with a negative slope, so it must be represented by line [latex]IV[\/latex] because it slants downward from left to right.<\/li>\n\t<li>This function has a slope of [latex]\\frac{1}{2}[\/latex] and a [latex]y[\/latex]-intercept of [latex]3[\/latex]. It must pass through the point [latex](0, 3)[\/latex] and slant upward from left to right. Lines [latex]I[\/latex] and [latex]II[\/latex] pass through [latex](0, 3)[\/latex], but the slope of [latex]j[\/latex] is less than the slope of [latex]f[\/latex]&nbsp;so the line for [latex]j[\/latex]&nbsp;must be flatter. This function is represented by Line [latex]II[\/latex].<\/li>\n<\/ol>\n<p>Now we can re-label the lines.<\/p>\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184346\/CNX_Precalc_Figure_02_02_0122.jpg\" width=\"489\" height=\"374\"><\/center>[\/hidden-answer]<\/section>\n<section class=\"textbox example\">If [latex]f(x)[\/latex] is a linear function, with [latex]f(2)=\u201311[\/latex], and [latex]f(4)=\u221225[\/latex], write an equation for the function in slope-intercept form. [reveal-answer q=\"659513\"]Show Solution[\/reveal-answer] [hidden-answer a=\"659513\"]\n\n<p>We can write the given points using coordinates.<\/p>\n<p style=\"text-align: center;\">[latex] f(2) = -11 \\rightarrow (2, -11) [\/latex]<\/p>\n<center>[latex] f(4) = -25 \\rightarrow (4, -25) [\/latex]<p><\/p><\/center>\n<p>We can then use the points to calculate the slope.<\/p>\n<p style=\"text-align: center;\">[latex] \\begin{array}{ccl} m &amp; = &amp; \\frac{-25 + 11}{4 - 2} \\\\ &amp; = &amp; \\frac{-14}{2} \\\\ &amp; = &amp; -7 \\end{array} [\/latex]<\/p>\n<p>Substitute the slope and the coordinates of one of the points into the point-slope form.<\/p>\n<p style=\"text-align: center;\">[latex] \\begin{array}{ccl} y + 11 &amp; = &amp; -7(x - 2) \\\\ y + 11 &amp; = &amp; -7x + 14 \\end{array} [\/latex]<\/p>\n<p>Finally, we can write the equation in the slope-intercept form.<\/p>\n<p style=\"text-align: center;\">[latex] y = -7x + 3 [\/latex]<\/p>\n\n[\/hidden-answer]<\/section>\n<p>Watch the following video for more on writing point-slope and slope-intercept form from two points.<\/p>\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/LtpXvUCrgrM?si=BDxIdeG1r4FWq9Op\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Point-slope+and+slope-intercept+form+from+two+points+_+Algebra+I+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPoint-slope and slope-intercept form from two points | Algebra I | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the following video for more on writing the equation of a line from a graph.<\/p>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328529&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=JaX_dIDUYBg&amp;video_target=tpm-plugin-y5bucx9j-JaX_dIDUYBg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+To+Find+The+Equation+of+a+Line+From+a+Graph+%7C+Algebra.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Find The Equation of a Line From a Graph | Algebra\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Building Linear Models<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<strong>Modeling Linear Functions Problem-Solving Strategy<\/strong>\n<ol>\n\t<li>Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.<\/li>\n\t<li>Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.<\/li>\n\t<li>Determine what we are trying to find, identify, solve, or interpret.<\/li>\n\t<li>Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.<\/li>\n\t<li>When needed, write a formula for the function.<\/li>\n\t<li>Solve or evaluate the function using the formula.<\/li>\n\t<li>Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.<\/li>\n\t<li>Clearly convey your result using appropriate units, and answer in full sentences when necessary.<\/li>\n<\/ol>\n<p>You've probably heard the phrase \"starting point\" a lot, right? The [latex]y[\/latex]-intercept is your starting point, and the slope guides you from there. Always remember, slope is your \"rate of change,\" and the [latex]y[\/latex]-intercept is your \"initial value.\"<\/p>\n<p>When given two points, use them to find your slope.<\/p>\n<p>Diagrams are not just doodles; they're visual aids. Use them to map out the problem and see the relationships between variables.<\/p>\n<\/div>\n<section class=\"textbox example\">A company sells doughnuts. They incur a fixed cost of [latex]$25,000[\/latex] for rent, insurance, and other expenses. It costs [latex]$0.25[\/latex] to produce each doughnut.\n\n<ol>\n\t<li>Write a linear model to represent the cost [latex]C[\/latex] of the company as a function of [latex]x[\/latex], the number of doughnuts produced.<\/li>\n\t<li>Find and interpret the [latex]y[\/latex]-intercept.<\/li>\n<\/ol>\n\n[reveal-answer q=\"218698\"]Show Solution[\/reveal-answer] [hidden-answer a=\"218698\"]\n\n<ol>\n\t<li>[latex]C\\left(x\\right)=0.25x+25,000[\/latex]<\/li>\n\t<li>The [latex]y[\/latex]-intercept is [latex](0, 25,000)[\/latex]. If the company does not produce a single doughnut, they still incur a cost of [latex]$25,000[\/latex].<\/li>\n<\/ol>\n\n[\/hidden-answer]<\/section>\n<section class=\"textbox example\">A city\u2019s population has been growing linearly. In 2008, the population was [latex]28,200[\/latex]. By 2012, the population was [latex]36,800[\/latex]. Assume this trend continues.\n\n<ol>\n\t<li>Predict the population in 2014.<\/li>\n\t<li>Identify the year in which the population will reach [latex]54,000[\/latex].<\/li>\n<\/ol>\n\n[reveal-answer q=\"171054\"]Show Solution[\/reveal-answer] [hidden-answer a=\"171054\"]\n\n<ol>\n\t<li>[latex]41,100[\/latex]<\/li>\n\t<li>2020<\/li>\n<\/ol>\n\n[\/hidden-answer]<\/section>\n<section class=\"textbox example\">There is a straight road leading from the town of Timpson to Ashburn [latex]60[\/latex] miles east and [latex]12[\/latex] miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located [latex]22[\/latex] miles directly east of the town of Timpson, how far is the road junction from Timpson? [reveal-answer q=\"788968\"]Show Solution[\/reveal-answer] [hidden-answer a=\"788968\"] [latex]\\approx 21.57[\/latex] miles[\/hidden-answer]<\/section>\n<h2>Graphing Linear Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p><strong>Graphing a Function by Plotting Points&nbsp;<\/strong><\/p>\n<p><strong>Plotting Points:<\/strong> When graphing a function by plotting points, always choose at least two input values to get a minimum of two points on the graph. This helps you draw a more accurate line.<\/p>\n<p>Plotting points is often the first method taught for graphing linear functions. But did you know that choosing three points can serve as a self-check? If all three don't fall on the same line, you've likely made an error. So, next time you're plotting points, go for that extra one; it's like a built-in error detector!<\/p>\n<p><strong>Graphing a Linear Function Using y-intercept and Slope<\/strong><\/p>\n<p>Think of the [latex]y[\/latex]-intercept and slope as the DNA of your graph. They uniquely identify how your graph will look. The [latex]y[\/latex]-intercept tells you where the graph starts on the [latex]y[\/latex]-axis, and the slope tells you how steep the line is. Knowing these two can save you time and effort in plotting multiple points.<\/p>\n<p>Do all linear functions have [latex]y[\/latex]-intercepts? Yes, they do, except for vertical lines, which aren't functions. So, the next time you're dealing with a linear function, you can be pretty confident that you'll find a [latex]y[\/latex]-intercept.<\/p>\n<p><strong>Graphing a Linear Function Using Transformations<\/strong><\/p>\n<p>You've learned that linear functions can be graphed using the slope-intercept form [latex]y=mx+b[\/latex]. But did you know that you can also use transformations to graph these functions? Transformations like vertical stretches, compressions, and shifts can give you a new perspective on how linear functions behave.<\/p>\n<p><strong>Vertical Stretch\/Compression:<\/strong> The coefficient [latex]f(x)=mx[\/latex] acts as a vertical stretch or compression. If [latex]m&gt;1[\/latex], the graph stretches; if [latex]0&lt;m&lt;1[\/latex], it compresses.<\/p>\n<p><strong>Vertical Shift:<\/strong> The constant [latex]b[\/latex] in [latex]f(x)=mx+b[\/latex] moves the graph up or down. Positive [latex]b[\/latex] values shift the graph upwards, while negative [latex]b[\/latex] values shift it downwards.<\/p>\n<p>When graphing using transformations, the order in which you apply them is crucial. For instance, if you have [latex]f(x)= \\frac{1}{2}x\u22123[\/latex], you should first apply the vertical compression by a factor of [latex]\\frac{1}{2}[\/latex] and then shift the graph down by [latex]3[\/latex] units. This sequence aligns with the order of operations in mathematics, ensuring that you get an accurate graph.<\/p>\n<\/div>\n<section class=\"textbox example\">Graph [latex]f\\left(x\\right)=-\\frac{3}{4}x+6[\/latex] by plotting points. [reveal-answer q=\"156351\"]Show Solution[\/reveal-answer] [hidden-answer a=\"156351\"]<center><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15184550\/CNX_Precalc_Figure_02_02_0022.jpg\"><img class=\"aligncenter size-full wp-image-2803\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15184550\/CNX_Precalc_Figure_02_02_0022.jpg\" alt=\"cnx_precalc_figure_02_02_0022\" width=\"487\" height=\"316\"><\/a><\/center>[\/hidden-answer]<\/section>\n<section class=\"textbox example\">Graph [latex]f\\left(x\\right)=4+2x[\/latex], using transformations. [reveal-answer q=\"350962\"]Show Solution[\/reveal-answer] [hidden-answer a=\"350962\"]<center><\/center><center><img class=\"aligncenter size-full wp-image-2804\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15185737\/CNX_Precalc_Figure_02_02_0092.jpg\" alt=\"cnx_precalc_figure_02_02_0092\" width=\"510\" height=\"520\"><\/center>[\/hidden-answer]<\/section>\n<p>Watch the following video for more on graphing linear functions using the slope and the [latex]y[\/latex]-intercept.<\/p>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328530&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=NAblGVxxJZo&amp;video_target=tpm-plugin-k15gsaw3-NAblGVxxJZo\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Graphing+Linear+Equations+-+Best+Explanation.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing Linear Equations - Best Explanation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the following video for more on graphing linear functions with transformations.<\/p>\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/h9zn_ODlgbM?si=fKJtgjor-qNPYbXI\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Graph+a+Linear+Function+as+a+Transformation+of+f(x)%3Dx.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a Linear Function as a Transformation of f(x)=x\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Create and interpret equations of linear functions<\/li>\n<li>Use linear functions to model and draw conclusions from real-world problems<\/li>\n<li>Plot the graphs of linear equations<\/li>\n<\/ul>\n<\/section>\n<h2>Point-Slope Form<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>You&#8217;ve been using slope-intercept form, [latex]y=mx+b[\/latex], to describe linear functions. But did you know there&#8217;s another superhero in town? Meet <strong>Point-Slope Form<\/strong>, [latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex], where [latex]m[\/latex] is the slope and [latex]({x}_{1},{y}_{1})[\/latex] is a point on the line. This form is a game-changer when you know a specific point and the slope but don&#8217;t have the [latex]y[\/latex]-intercept.<\/p>\n<p>You can convert between point-slope and slope-intercept forms using basic algebra.<\/p>\n<p>To convert to slope-intercept form, distribute any constants and isolate [latex]y[\/latex]<\/p>\n<p>Both forms describe the same line, so choose the one that makes your life easier!<\/p>\n<\/div>\n<section class=\"textbox example\">Write the equation of a line in point-slope form with a slope of [latex]-2[\/latex] that passes through the point [latex]\\left(-2,\\text{ }2\\right)[\/latex]. Then rewrite the equation in the slope-intercept form. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q624568\">Show Solution<\/button> <\/p>\n<div id=\"q624568\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">Point-slope form: [latex]y - 2=-2\\left(x+2\\right)[\/latex]<\/div>\n<div style=\"text-align: center;\">Slope-intercept form: [latex]y=-2x - 2[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for a worked example of finding point-slope form and converting it to slope-intercept form.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328528&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=FntpEHhLHvw&amp;video_target=tpm-plugin-rffzfoe7-FntpEHhLHvw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Writing+an+equation+using+point+slope+form+given+a+point+and+slope.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting an equation using point slope form given a point and slope\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Writing and Interpreting Equations of Linear Functions: Two Points, Graphs, and Real-World Applications<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p>You&#8217;ve learned that two points can define a line. But did you know that this is like having two pieces of a puzzle that reveal the entire picture? When you have two points, you can find the slope and then use point-slope form to find the equation of the line.<\/p>\n<p>Sometimes, all you have is a graph. No worries! You can still find the equation of the line. Just pick two points on the graph and use them to find the slope. Then, use the slope and one point to write the equation in point-slope form.<\/p>\n<p><strong>Quick Tips<\/strong><\/p>\n<ul>\n<li><strong>Choose Wisely<\/strong>: Pick points that are easy to read on the graph.<\/li>\n<li><strong>Y-Intercept<\/strong>: Look for where the line crosses the [latex]y[\/latex]-axis; this is your [latex]y[\/latex]-intercept [latex]b[\/latex].<\/li>\n<\/ul>\n<p>Linear functions are not just theoretical constructs; they&#8217;re practical tools. For example, if you&#8217;re starting a business, you can use a linear function to model your costs. The slope might represent the cost per item, and the y-intercept could be your fixed costs like rent.<\/p>\n<\/div>\n<section class=\"textbox example\">Write the point-slope form of an equation of a line that passes through the points [latex]\\left(-1,3\\right)[\/latex] and [latex]\\left(0,0\\right)[\/latex]. Then rewrite the equation in slope-intercept form. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q85979\">Show Solution<\/button> <\/p>\n<div id=\"q85979\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">Point-slope form: [latex]y - 0=-3\\left(x - 0\\right)[\/latex]<\/div>\n<div style=\"text-align: center;\">Slope-intercept form: [latex]y=-3x[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Match each equation of a linear function with one of the lines in the graph below.<\/p>\n<ol>\n<li>[latex]f\\left(x\\right)=2x+3[\/latex]<\/li>\n<li>[latex]g\\left(x\\right)=2x - 3[\/latex]<\/li>\n<li>[latex]h\\left(x\\right)=-2x+3[\/latex]<\/li>\n<li>[latex]j\\left(x\\right)=\\frac{1}{2}x+3[\/latex]<\/li>\n<\/ol>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184343\/CNX_Precalc_Figure_02_02_0112.jpg\" alt=\"Graph of three lines, line 1) passes through (0,3) and (-2, -1), line 2) passes through (0,3) and (-6,0), line 3) passes through (0,-3) and (2,1)\" width=\"393\" height=\"305\" \/><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q659573\">Show Solution<\/button> <\/p>\n<div id=\"q659573\" class=\"hidden-answer\" style=\"display: none\"> Analyze the information for each function.<\/p>\n<ol>\n<li>This function has a slope of [latex]2[\/latex] and a [latex]y[\/latex]-intercept of [latex]3[\/latex]. It must pass through the point [latex](0, 3)[\/latex] and slant upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function [latex]g[\/latex]&nbsp;has the same slope, but a different [latex]y[\/latex]-intercept. Lines [latex]I[\/latex] and [latex]III[\/latex] have the same slant because they have the same slope. Line [latex]III[\/latex] does not pass through [latex](0, 3)[\/latex] so [latex]f[\/latex]&nbsp;must be represented by line [latex]I[\/latex].<\/li>\n<li>This function also has a slope of [latex]2[\/latex], but a [latex]y[\/latex]-intercept of [latex]\u20133[\/latex]. It must pass through the point [latex](0, \u20133)[\/latex] and slant upward from left to right. It must be represented by line [latex]III[\/latex].<\/li>\n<li>This function has a slope of [latex]\u20132[\/latex] and a [latex]y[\/latex]-intercept of [latex]3[\/latex]. This is the only function listed with a negative slope, so it must be represented by line [latex]IV[\/latex] because it slants downward from left to right.<\/li>\n<li>This function has a slope of [latex]\\frac{1}{2}[\/latex] and a [latex]y[\/latex]-intercept of [latex]3[\/latex]. It must pass through the point [latex](0, 3)[\/latex] and slant upward from left to right. Lines [latex]I[\/latex] and [latex]II[\/latex] pass through [latex](0, 3)[\/latex], but the slope of [latex]j[\/latex] is less than the slope of [latex]f[\/latex]&nbsp;so the line for [latex]j[\/latex]&nbsp;must be flatter. This function is represented by Line [latex]II[\/latex].<\/li>\n<\/ol>\n<p>Now we can re-label the lines.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184346\/CNX_Precalc_Figure_02_02_0122.jpg\" width=\"489\" height=\"374\" alt=\"image\" \/><\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">If [latex]f(x)[\/latex] is a linear function, with [latex]f(2)=\u201311[\/latex], and [latex]f(4)=\u221225[\/latex], write an equation for the function in slope-intercept form. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q659513\">Show Solution<\/button> <\/p>\n<div id=\"q659513\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can write the given points using coordinates.<\/p>\n<p style=\"text-align: center;\">[latex]f(2) = -11 \\rightarrow (2, -11)[\/latex]<\/p>\n<div style=\"text-align: center;\">[latex]f(4) = -25 \\rightarrow (4, -25)[\/latex]<\/p>\n<\/div>\n<p>We can then use the points to calculate the slope.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccl} m & = & \\frac{-25 + 11}{4 - 2} \\\\ & = & \\frac{-14}{2} \\\\ & = & -7 \\end{array}[\/latex]<\/p>\n<p>Substitute the slope and the coordinates of one of the points into the point-slope form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccl} y + 11 & = & -7(x - 2) \\\\ y + 11 & = & -7x + 14 \\end{array}[\/latex]<\/p>\n<p>Finally, we can write the equation in the slope-intercept form.<\/p>\n<p style=\"text-align: center;\">[latex]y = -7x + 3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more on writing point-slope and slope-intercept form from two points.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/LtpXvUCrgrM?si=BDxIdeG1r4FWq9Op\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Point-slope+and+slope-intercept+form+from+two+points+_+Algebra+I+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPoint-slope and slope-intercept form from two points | Algebra I | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the following video for more on writing the equation of a line from a graph.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328529&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=JaX_dIDUYBg&amp;video_target=tpm-plugin-y5bucx9j-JaX_dIDUYBg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+To+Find+The+Equation+of+a+Line+From+a+Graph+%7C+Algebra.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Find The Equation of a Line From a Graph | Algebra\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Building Linear Models<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p><strong>Modeling Linear Functions Problem-Solving Strategy<\/strong><\/p>\n<ol>\n<li>Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.<\/li>\n<li>Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.<\/li>\n<li>Determine what we are trying to find, identify, solve, or interpret.<\/li>\n<li>Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.<\/li>\n<li>When needed, write a formula for the function.<\/li>\n<li>Solve or evaluate the function using the formula.<\/li>\n<li>Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.<\/li>\n<li>Clearly convey your result using appropriate units, and answer in full sentences when necessary.<\/li>\n<\/ol>\n<p>You&#8217;ve probably heard the phrase &#8220;starting point&#8221; a lot, right? The [latex]y[\/latex]-intercept is your starting point, and the slope guides you from there. Always remember, slope is your &#8220;rate of change,&#8221; and the [latex]y[\/latex]-intercept is your &#8220;initial value.&#8221;<\/p>\n<p>When given two points, use them to find your slope.<\/p>\n<p>Diagrams are not just doodles; they&#8217;re visual aids. Use them to map out the problem and see the relationships between variables.<\/p>\n<\/div>\n<section class=\"textbox example\">A company sells doughnuts. They incur a fixed cost of [latex]$25,000[\/latex] for rent, insurance, and other expenses. It costs [latex]$0.25[\/latex] to produce each doughnut.<\/p>\n<ol>\n<li>Write a linear model to represent the cost [latex]C[\/latex] of the company as a function of [latex]x[\/latex], the number of doughnuts produced.<\/li>\n<li>Find and interpret the [latex]y[\/latex]-intercept.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q218698\">Show Solution<\/button> <\/p>\n<div id=\"q218698\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]C\\left(x\\right)=0.25x+25,000[\/latex]<\/li>\n<li>The [latex]y[\/latex]-intercept is [latex](0, 25,000)[\/latex]. If the company does not produce a single doughnut, they still incur a cost of [latex]$25,000[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">A city\u2019s population has been growing linearly. In 2008, the population was [latex]28,200[\/latex]. By 2012, the population was [latex]36,800[\/latex]. Assume this trend continues.<\/p>\n<ol>\n<li>Predict the population in 2014.<\/li>\n<li>Identify the year in which the population will reach [latex]54,000[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q171054\">Show Solution<\/button> <\/p>\n<div id=\"q171054\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]41,100[\/latex]<\/li>\n<li>2020<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">There is a straight road leading from the town of Timpson to Ashburn [latex]60[\/latex] miles east and [latex]12[\/latex] miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located [latex]22[\/latex] miles directly east of the town of Timpson, how far is the road junction from Timpson? <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q788968\">Show Solution<\/button> <\/p>\n<div id=\"q788968\" class=\"hidden-answer\" style=\"display: none\"> [latex]\\approx 21.57[\/latex] miles<\/div>\n<\/div>\n<\/section>\n<h2>Graphing Linear Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<p><strong>Graphing a Function by Plotting Points&nbsp;<\/strong><\/p>\n<p><strong>Plotting Points:<\/strong> When graphing a function by plotting points, always choose at least two input values to get a minimum of two points on the graph. This helps you draw a more accurate line.<\/p>\n<p>Plotting points is often the first method taught for graphing linear functions. But did you know that choosing three points can serve as a self-check? If all three don&#8217;t fall on the same line, you&#8217;ve likely made an error. So, next time you&#8217;re plotting points, go for that extra one; it&#8217;s like a built-in error detector!<\/p>\n<p><strong>Graphing a Linear Function Using y-intercept and Slope<\/strong><\/p>\n<p>Think of the [latex]y[\/latex]-intercept and slope as the DNA of your graph. They uniquely identify how your graph will look. The [latex]y[\/latex]-intercept tells you where the graph starts on the [latex]y[\/latex]-axis, and the slope tells you how steep the line is. Knowing these two can save you time and effort in plotting multiple points.<\/p>\n<p>Do all linear functions have [latex]y[\/latex]-intercepts? Yes, they do, except for vertical lines, which aren&#8217;t functions. So, the next time you&#8217;re dealing with a linear function, you can be pretty confident that you&#8217;ll find a [latex]y[\/latex]-intercept.<\/p>\n<p><strong>Graphing a Linear Function Using Transformations<\/strong><\/p>\n<p>You&#8217;ve learned that linear functions can be graphed using the slope-intercept form [latex]y=mx+b[\/latex]. But did you know that you can also use transformations to graph these functions? Transformations like vertical stretches, compressions, and shifts can give you a new perspective on how linear functions behave.<\/p>\n<p><strong>Vertical Stretch\/Compression:<\/strong> The coefficient [latex]f(x)=mx[\/latex] acts as a vertical stretch or compression. If [latex]m>1[\/latex], the graph stretches; if [latex]0<m<1[\/latex], it compresses.<\/p>\n<p><strong>Vertical Shift:<\/strong> The constant [latex]b[\/latex] in [latex]f(x)=mx+b[\/latex] moves the graph up or down. Positive [latex]b[\/latex] values shift the graph upwards, while negative [latex]b[\/latex] values shift it downwards.<\/p>\n<p>When graphing using transformations, the order in which you apply them is crucial. For instance, if you have [latex]f(x)= \\frac{1}{2}x\u22123[\/latex], you should first apply the vertical compression by a factor of [latex]\\frac{1}{2}[\/latex] and then shift the graph down by [latex]3[\/latex] units. This sequence aligns with the order of operations in mathematics, ensuring that you get an accurate graph.<\/p>\n<\/div>\n<section class=\"textbox example\">Graph [latex]f\\left(x\\right)=-\\frac{3}{4}x+6[\/latex] by plotting points. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q156351\">Show Solution<\/button> <\/p>\n<div id=\"q156351\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15184550\/CNX_Precalc_Figure_02_02_0022.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2803\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15184550\/CNX_Precalc_Figure_02_02_0022.jpg\" alt=\"cnx_precalc_figure_02_02_0022\" width=\"487\" height=\"316\" \/><\/a><\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Graph [latex]f\\left(x\\right)=4+2x[\/latex], using transformations. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q350962\">Show Solution<\/button> <\/p>\n<div id=\"q350962\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\"><\/div>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2804\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15185737\/CNX_Precalc_Figure_02_02_0092.jpg\" alt=\"cnx_precalc_figure_02_02_0092\" width=\"510\" height=\"520\" \/><\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more on graphing linear functions using the slope and the [latex]y[\/latex]-intercept.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328530&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=NAblGVxxJZo&amp;video_target=tpm-plugin-k15gsaw3-NAblGVxxJZo\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Graphing+Linear+Equations+-+Best+Explanation.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing Linear Equations &#8211; Best Explanation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the following video for more on graphing linear functions with transformations.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/h9zn_ODlgbM?si=fKJtgjor-qNPYbXI\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Graph+a+Linear+Function+as+a+Transformation+of+f(x)%3Dx.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a Linear Function as a Transformation of f(x)=x\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":6,"menu_order":29,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Writing an equation using point slope form given a point and slope\",\"author\":\"Brian McLogan\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/FntpEHhLHvw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Point-slope and slope-intercept form from two points | Algebra I | Khan Academy\",\"author\":\"Khan Academy\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/LtpXvUCrgrM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"How To Find The Equation of a Line From a Graph | Algebra\",\"author\":\"The Organic Chemistry Tutor\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/JaX_dIDUYBg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Graphing Linear Equations - Best Explanation\",\"author\":\"BetterThanYourProf\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/NAblGVxxJZo\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Graph a Linear Function as a Transformation of f(x)=x\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/h9zn_ODlgbM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"module-header":"","content_attributions":[{"type":"copyrighted_video","description":"Writing an equation using point slope form given a point and slope","author":"Brian McLogan","organization":"","url":"https:\/\/youtu.be\/FntpEHhLHvw","project":"","license":"arr","license_terms":""},{"type":"copyrighted_video","description":"Point-slope and slope-intercept form from two points | Algebra I | Khan Academy","author":"Khan Academy","organization":"","url":"https:\/\/youtu.be\/LtpXvUCrgrM","project":"","license":"arr","license_terms":""},{"type":"copyrighted_video","description":"How To Find The Equation of a Line From a Graph | Algebra","author":"The Organic Chemistry Tutor","organization":"","url":"https:\/\/youtu.be\/JaX_dIDUYBg","project":"","license":"arr","license_terms":""},{"type":"copyrighted_video","description":"Graphing Linear Equations - Best Explanation","author":"BetterThanYourProf","organization":"","url":"https:\/\/youtu.be\/NAblGVxxJZo","project":"","license":"arr","license_terms":""},{"type":"copyrighted_video","description":"Graph a Linear Function as a Transformation of f(x)=x","author":"James Sousa (Mathispower4u.com)","organization":"","url":"https:\/\/youtu.be\/h9zn_ODlgbM","project":"","license":"arr","license_terms":""}],"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\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/220"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/220\/revisions"}],"predecessor-version":[{"id":308,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/220\/revisions\/308"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/220\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/media?parent=220"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapter-type?post=220"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/contributor?post=220"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/license?post=220"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}