{"id":1303,"date":"2025-07-24T04:12:15","date_gmt":"2025-07-24T04:12:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1303"},"modified":"2026-03-18T03:36:54","modified_gmt":"2026-03-18T03:36:54","slug":"linear-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-functions-fresh-take\/","title":{"raw":"Linear Models: Fresh Take","rendered":"Linear Models: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Build linear models from verbal descriptions.<\/li>\r\n \t<li>Draw and interpret scatter plots.<\/li>\r\n \t<li>Find the line of best fit.<\/li>\r\n \t<li>Use a linear regression model to make predictions.<\/li>\r\n<\/ul>\r\n<\/section><section>\r\n<h2>Building Linear Models<\/h2>\r\n<\/section><section>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\n<strong>Modeling Linear Functions Problem-Solving Strategy<\/strong>\r\n<ol>\r\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>\r\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>\r\n \t<li>Determine what we are trying to find, identify, solve, or interpret.<\/li>\r\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>\r\n \t<li>When needed, write a formula for the function.<\/li>\r\n \t<li>Solve or evaluate the function using the formula.<\/li>\r\n \t<li>Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.<\/li>\r\n \t<li>Clearly convey your result using appropriate units, and answer in full sentences when necessary.<\/li>\r\n<\/ol>\r\n<strong>Helpful tips:<\/strong>\r\n<ul>\r\n \t<li>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.\"<\/li>\r\n \t<li>When given two points, use them to find your slope.<\/li>\r\n \t<li>Diagrams are not just doodles; they're visual aids. Use them to map out the problem and see the relationships between variables.<\/li>\r\n<\/ul>\r\n<\/div>\r\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.\r\n<ol>\r\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>\r\n \t<li>Find and interpret the [latex]y[\/latex]-intercept.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"218698\"]Show Solution[\/reveal-answer] [hidden-answer a=\"218698\"]\r\n<ol>\r\n \t<li>[latex]C\\left(x\\right)=0.25x+25,000[\/latex]<\/li>\r\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>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cgadehaa-xKH1Evwu150\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/xKH1Evwu150?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cgadehaa-xKH1Evwu150\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844471&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cgadehaa-xKH1Evwu150&amp;vembed=0&amp;video_id=xKH1Evwu150&amp;video_target=tpm-plugin-cgadehaa-xKH1Evwu150\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Linear+equation+word+problem+%7C+Linear+equations+%7C+Algebra+I+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLinear equation word problem | Linear equations | Algebra I | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><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.\r\n<ol>\r\n \t<li>Predict the population in 2014.<\/li>\r\n \t<li>Identify the year in which the population will reach [latex]54,000[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"171054\"]Show Solution[\/reveal-answer] [hidden-answer a=\"171054\"]\r\n<ol>\r\n \t<li>[latex]41,100[\/latex]<\/li>\r\n \t<li>2020<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bagcedef-BR7MJpvUL0Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/BR7MJpvUL0Q?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bagcedef-BR7MJpvUL0Q\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780696&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bagcedef-BR7MJpvUL0Q&amp;vembed=0&amp;video_id=BR7MJpvUL0Q&amp;video_target=tpm-plugin-bagcedef-BR7MJpvUL0Q\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Linear+Modeling_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLinear Modeling\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><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><\/section><section>\r\n<h2>Drawing and Interpreting Scatterplots<\/h2>\r\n<\/section>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Scatterplots visually represent relationships between two variables.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each point on a scatterplot represents a pair of values [latex](x, y)[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The pattern of points can indicate the type and strength of a relationship.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Linear relationships in scatterplots suggest a constant rate of change.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Not all scatterplots show clear relationships; some may show no pattern at all.<strong>\u00a0<\/strong><\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Finding the Line of Best Fit<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">The line of best fit represents the overall trend in a scatterplot.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">It minimizes the overall distance between itself and all data points.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The line can be estimated visually or calculated mathematically.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slope of the line indicates the rate of change between variables.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The line of best fit is used for making predictions within the data range.<strong>\u00a0<\/strong><\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Understanding Interpolation and Extrapolation<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Interpolation predicts values within the range of observed data.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Extrapolation estimates values outside the range of observed data.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpolation is generally more reliable than extrapolation.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Both methods use the line of best fit or other trend models.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Understanding data limits is crucial for accurate predictions.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Model breakdown can occur, especially with extrapolation.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">According to the data from the table in the cricket-chirp example, what temperature can we predict if we counted 20 chirps in 15 seconds?\r\n<table summary=\"Two rows and ten columns. The first row is labeled, 'chirps'. The second row is labeled is labeled, 'Temp'. Reading the remaining rows as ordered pairs (i.e., (chirps, Temp), we have the following values: (44, 80.5), (35, 70.5), (20.4, 57), (33, 66), (31, 68), (35, 72), (18.5, 52), (37, 73.5) and (26, 53).\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>Chirps<\/strong><\/td>\r\n<td>44<\/td>\r\n<td>35<\/td>\r\n<td>20.4<\/td>\r\n<td>33<\/td>\r\n<td>31<\/td>\r\n<td>35<\/td>\r\n<td>18.5<\/td>\r\n<td>37<\/td>\r\n<td>26<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Temperature<\/strong><\/td>\r\n<td>80.5<\/td>\r\n<td>70.5<\/td>\r\n<td>57<\/td>\r\n<td>66<\/td>\r\n<td>68<\/td>\r\n<td>72<\/td>\r\n<td>52<\/td>\r\n<td>73.5<\/td>\r\n<td>53<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"\" align=\"aligncenter\" width=\"381\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014341\/CNX_Precalc_Figure_02_04_0042.jpg\" alt=\"Scatter plot, showing the line of best fit and where interpolation and extrapolation occurs. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"381\" height=\"336\" \/> Scatterplot with extrapolation and interpolation labeled[\/caption]\r\n\r\n[reveal-answer q=\"271439\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"271439\"]\r\n\r\n[latex]54^\\circ \\text{F}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fchchhde-c4_MJg_c49k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/c4_MJg_c49k?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fchchhde-c4_MJg_c49k\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780697&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fchchhde-c4_MJg_c49k&amp;vembed=0&amp;video_id=c4_MJg_c49k&amp;video_target=tpm-plugin-fchchhde-c4_MJg_c49k\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/What+is+Interpolation+and+Extrapolation%3F_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is Interpolation and Extrapolation?\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Distinguishing Between Linear and Nonlinear Models<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Data relationships can be linear or nonlinear.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The correlation coefficient (r) measures the strength and direction of linear relationships.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">r ranges from -1 to 1, with values closer to \u00b11 indicating stronger linear relationships.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Correlation does not imply causation.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Visual inspection of scatterplots is crucial alongside numerical measures.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Nonlinear relationships require different modeling approaches.<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-lg font-bold\">Interpreting Correlation<\/p>\r\n\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Strong Positive ([latex]0.7 &lt; r \\leq 1[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to increase strongly.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Moderate Positive ([latex]0.3 &lt; r \\leq 0.7[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to increase moderately.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Weak Positive ([latex]0 &lt; r \\leq 0.3[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to increase weakly.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>No Linear Correlation ([latex]r \\approx 0[\/latex])<\/strong>: No clear linear trend between [latex]x[\/latex] and [latex]y[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Weak Negative ([latex]-0.3 \\leq r &lt; 0[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to decrease weakly.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Moderate Negative ([latex]-0.7 \\leq r &lt; -0.3[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to decrease moderately.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Strong Negative ([latex]-1 \\leq r &lt; -0.7[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to decrease strongly.<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Build linear models from verbal descriptions.<\/li>\n<li>Draw and interpret scatter plots.<\/li>\n<li>Find the line of best fit.<\/li>\n<li>Use a linear regression model to make predictions.<\/li>\n<\/ul>\n<\/section>\n<section>\n<h2>Building Linear Models<\/h2>\n<\/section>\n<section>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/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><strong>Helpful tips:<\/strong><\/p>\n<ul>\n<li>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;<\/li>\n<li>When given two points, use them to find your slope.<\/li>\n<li>Diagrams are not just doodles; they&#8217;re visual aids. Use them to map out the problem and see the relationships between variables.<\/li>\n<\/ul>\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 watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cgadehaa-xKH1Evwu150\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/xKH1Evwu150?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cgadehaa-xKH1Evwu150\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844471&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cgadehaa-xKH1Evwu150&amp;vembed=0&amp;video_id=xKH1Evwu150&amp;video_target=tpm-plugin-cgadehaa-xKH1Evwu150\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Linear+equation+word+problem+%7C+Linear+equations+%7C+Algebra+I+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLinear equation word problem | Linear equations | Algebra I | Khan Academy\u201d here (opens in new window).<\/a><\/p>\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 watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bagcedef-BR7MJpvUL0Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/BR7MJpvUL0Q?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bagcedef-BR7MJpvUL0Q\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780696&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bagcedef-BR7MJpvUL0Q&amp;vembed=0&amp;video_id=BR7MJpvUL0Q&amp;video_target=tpm-plugin-bagcedef-BR7MJpvUL0Q\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Linear+Modeling_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLinear Modeling\u201d here (opens in new window).<\/a><\/p>\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<\/section>\n<section>\n<h2>Drawing and Interpreting Scatterplots<\/h2>\n<\/section>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Scatterplots visually represent relationships between two variables.<\/li>\n<li class=\"whitespace-normal break-words\">Each point on a scatterplot represents a pair of values [latex](x, y)[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">The pattern of points can indicate the type and strength of a relationship.<\/li>\n<li class=\"whitespace-normal break-words\">Linear relationships in scatterplots suggest a constant rate of change.<\/li>\n<li class=\"whitespace-normal break-words\">Not all scatterplots show clear relationships; some may show no pattern at all.<strong>\u00a0<\/strong><\/li>\n<\/ul>\n<\/div>\n<h2>Finding the Line of Best Fit<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">The line of best fit represents the overall trend in a scatterplot.<\/li>\n<li class=\"whitespace-normal break-words\">It minimizes the overall distance between itself and all data points.<\/li>\n<li class=\"whitespace-normal break-words\">The line can be estimated visually or calculated mathematically.<\/li>\n<li class=\"whitespace-normal break-words\">Slope of the line indicates the rate of change between variables.<\/li>\n<li class=\"whitespace-normal break-words\">The line of best fit is used for making predictions within the data range.<strong>\u00a0<\/strong><\/li>\n<\/ul>\n<\/div>\n<h2>Understanding Interpolation and Extrapolation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Interpolation predicts values within the range of observed data.<\/li>\n<li class=\"whitespace-normal break-words\">Extrapolation estimates values outside the range of observed data.<\/li>\n<li class=\"whitespace-normal break-words\">Interpolation is generally more reliable than extrapolation.<\/li>\n<li class=\"whitespace-normal break-words\">Both methods use the line of best fit or other trend models.<\/li>\n<li class=\"whitespace-normal break-words\">Understanding data limits is crucial for accurate predictions.<\/li>\n<li class=\"whitespace-normal break-words\">Model breakdown can occur, especially with extrapolation.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">According to the data from the table in the cricket-chirp example, what temperature can we predict if we counted 20 chirps in 15 seconds?<\/p>\n<table summary=\"Two rows and ten columns. The first row is labeled, 'chirps'. The second row is labeled is labeled, 'Temp'. Reading the remaining rows as ordered pairs (i.e., (chirps, Temp), we have the following values: (44, 80.5), (35, 70.5), (20.4, 57), (33, 66), (31, 68), (35, 72), (18.5, 52), (37, 73.5) and (26, 53).\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>Chirps<\/strong><\/td>\n<td>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong>Temperature<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure style=\"width: 381px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014341\/CNX_Precalc_Figure_02_04_0042.jpg\" alt=\"Scatter plot, showing the line of best fit and where interpolation and extrapolation occurs. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"381\" height=\"336\" \/><figcaption class=\"wp-caption-text\">Scatterplot with extrapolation and interpolation labeled<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q271439\">Show Solution<\/button><\/p>\n<div id=\"q271439\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]54^\\circ \\text{F}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fchchhde-c4_MJg_c49k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/c4_MJg_c49k?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fchchhde-c4_MJg_c49k\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780697&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fchchhde-c4_MJg_c49k&amp;vembed=0&amp;video_id=c4_MJg_c49k&amp;video_target=tpm-plugin-fchchhde-c4_MJg_c49k\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/What+is+Interpolation+and+Extrapolation%3F_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is Interpolation and Extrapolation?\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Distinguishing Between Linear and Nonlinear Models<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Data relationships can be linear or nonlinear.<\/li>\n<li class=\"whitespace-normal break-words\">The correlation coefficient (r) measures the strength and direction of linear relationships.<\/li>\n<li class=\"whitespace-normal break-words\">r ranges from -1 to 1, with values closer to \u00b11 indicating stronger linear relationships.<\/li>\n<li class=\"whitespace-normal break-words\">Correlation does not imply causation.<\/li>\n<li class=\"whitespace-normal break-words\">Visual inspection of scatterplots is crucial alongside numerical measures.<\/li>\n<li class=\"whitespace-normal break-words\">Nonlinear relationships require different modeling approaches.<\/li>\n<\/ul>\n<p class=\"font-600 text-lg font-bold\">Interpreting Correlation<\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Strong Positive ([latex]0.7 < r \\leq 1[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to increase strongly.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Moderate Positive ([latex]0.3 < r \\leq 0.7[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to increase moderately.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Weak Positive ([latex]0 < r \\leq 0.3[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to increase weakly.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>No Linear Correlation ([latex]r \\approx 0[\/latex])<\/strong>: No clear linear trend between [latex]x[\/latex] and [latex]y[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Weak Negative ([latex]-0.3 \\leq r < 0[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to decrease weakly.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Moderate Negative ([latex]-0.7 \\leq r < -0.3[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to decrease moderately.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Strong Negative ([latex]-1 \\leq r < -0.7[\/latex])<\/strong>: As [latex]x[\/latex] increases, [latex]y[\/latex] tends to decrease strongly.<\/li>\n<\/ul>\n<\/div>\n","protected":false},"author":67,"menu_order":29,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Linear equation word problem | Linear equations | Algebra I | Khan Academy\",\"author\":\"\",\"organization\":\"Khan Academy\",\"url\":\"https:\/\/youtu.be\/xKH1Evwu150\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Linear Modeling\",\"author\":\"\",\"organization\":\"LawrenceAcademy Math\",\"url\":\"https:\/\/youtu.be\/BR7MJpvUL0Q\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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