{"id":1706,"date":"2024-06-03T23:50:34","date_gmt":"2024-06-03T23:50:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1706"},"modified":"2025-01-18T23:10:52","modified_gmt":"2025-01-18T23:10:52","slug":"introduction-to-linear-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-linear-functions-fresh-take\/","title":{"raw":"Introduction to Linear Functions: Fresh Take","rendered":"Introduction to Linear Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Express a linear function using an equation, words, tables, and graphs<\/li>\r\n \t<li>Determine whether a linear function is increasing, decreasing, or constant<\/li>\r\n \t<li>Understand the slope of a line as the rate at which one quantity changes in relation to another<\/li>\r\n \t<li>Solve for the x-intercept in a linear function equation<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Linear Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nA <strong>linear function<\/strong> is characterized by a constant rate of change and can be represented as a polynomial of degree [latex]1[\/latex].\r\n\r\n<\/div>\r\n<h2>Representing Linear Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\n<strong>The Language of Linear Functions<\/strong>\r\n<ul>\r\n \t<li><strong>Word Form:<\/strong> Describing a linear function in words helps you understand the relationship between variables. For instance, if a train's distance from a station is determined by its speed and initial distance, you can say, \"The train's 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.\"<\/li>\r\n \t<li><strong>Rate of Change:<\/strong> This is another term for the slope. It tells you how fast one variable changes in relation to another.<\/li>\r\n<\/ul>\r\nUnderstanding the language used to describe linear functions can be a game-changer. For example, the term \"rate of change\" might sound complex, but it's just a fancy way of saying how one variable (like time) affects another (like distance). In real-world scenarios like the train example, this rate is constant, making it easier to predict future outcomes.\r\n\r\n<strong>Function Notation and Its Importance<\/strong>\r\n<ul>\r\n \t<li><strong>Slope-Intercept Form:<\/strong> This is the equation [latex]y=mx+b[\/latex], where [latex]m[\/latex] is the slope and [latex]b[\/latex] is the [latex]y[\/latex]-intercept. It's a standardized way to write linear functions.<\/li>\r\n \t<li><strong>Function Notation:<\/strong> [latex]f(x)=mx+b[\/latex] is another way to write the slope-intercept form, emphasizing that [latex]y[\/latex] is a function of [latex]x[\/latex].<\/li>\r\n<\/ul>\r\nFunction notation isn't just for mathematicians; it's a useful tool for anyone wanting to understand relationships between variables. For example, [latex]D(t)=83t+250[\/latex] tells us that the distance [latex]D[\/latex] is a function of time [latex]t[\/latex]. This notation makes it clear what the input and output variables are, which is crucial for understanding the function's behavior.\r\n\r\n<strong>Visualizing Linear Functions<\/strong>\r\n<ul>\r\n \t<li><strong>Tabular Form:<\/strong> A table can show how the output changes with each unit increase in the input.<\/li>\r\n \t<li><strong>Graphical Form:<\/strong> A graph provides a visual representation of a linear function. The slope and [latex]y[\/latex]-intercept can be easily identified.<\/li>\r\n<\/ul>\r\nGraphs offer a powerful way to visualize linear functions. For instance, the graph of [latex]D(t)=83t+250[\/latex] would be a straight line, allowing you to instantly see the rate of change and initial value. This can be particularly helpful in real-world applications like determining how far a train will be from a station at a given time.\r\n\r\n<\/div>\r\nWatch the following video for more on the slope intercept form.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ehgcfefh-IL3UCuXrUzE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/IL3UCuXrUzE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ehgcfefh-IL3UCuXrUzE\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844449&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ehgcfefh-IL3UCuXrUzE&vembed=0&video_id=IL3UCuXrUzE&video_target=tpm-plugin-ehgcfefh-IL3UCuXrUzE'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Slope-intercept+form+%7C+Algebra+I+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSlope-intercept form | Algebra I | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>Watch the following video for more on identifying linear functions from tables.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fdgcdfch-J3zpqUd5dMM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/J3zpqUd5dMM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fdgcdfch-J3zpqUd5dMM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=11328525&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fdgcdfch-J3zpqUd5dMM&vembed=0&video_id=J3zpqUd5dMM&video_target=tpm-plugin-fdgcdfch-J3zpqUd5dMM'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Identifying+Linear+Functions+From+Tables_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIdentifying Linear Functions From Tables\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>Watch the following video for more on identifying linear functions from graphs.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-adgfebab--VCwxDKOih0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/-VCwxDKOih0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-adgfebab--VCwxDKOih0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=11328526&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-adgfebab--VCwxDKOih0&vembed=0&video_id=-VCwxDKOih0&video_target=tpm-plugin-adgfebab--VCwxDKOih0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Identifying+Linear+Functions+From+Graphs_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIdentifying Linear Functions From Graphs\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Determine Whether a Linear Function is Increasing, Decreasing, or Constant<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\n<strong>Increasing Linear Function:<\/strong> If you're looking at a graph and see the line slanting upward from left to right, you're likely dealing with an increasing function.\r\n\r\n<strong>Decreasing Linear Function:<\/strong> A line that slants downward from left to right on a graph usually signifies a decreasing function.\r\n\r\n<strong>Constant Linear Function:<\/strong> A horizontal line on a graph is a dead giveaway for a constant function.\r\n\r\nGraphing is a powerful tool for understanding linear functions. Use online graphing calculators like Desmos to visualize these functions. You can even add sliders to manipulate the slope. For instance, the function [latex]f(x)=\u2212\\frac{2}{32}x\u2212\\frac{4}{3}[\/latex] can be graphed to visually represent its decreasing nature.\r\n\r\n<\/div>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]13521[\/ohm2_question]<\/section>Watch the following video for more on determining if a linear function is increasing or decreasing.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-habbbadd-AqDOZgNhnvA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/AqDOZgNhnvA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-habbbadd-AqDOZgNhnvA\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844467&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-habbbadd-AqDOZgNhnvA&vembed=0&video_id=AqDOZgNhnvA&video_target=tpm-plugin-habbbadd-AqDOZgNhnvA'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Determine+if+a+Linear+Function+is+Increasing+or+Decreasing_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine if a Linear Function is Increasing or Decreasing\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Calculate and Interpret Slope<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\n<strong>Slope<\/strong> is the measure of how steep a line is.\r\n\r\nIt's calculated as the <strong>change in output <\/strong>(rise) divided by the <strong>change in input <\/strong>(run).\r\n\r\nThe formula to calculate slope is [latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\Rightarrow \\dfrac{f(x_2)-f(x_1)}{x_2 - x_1}[\/latex]\r\n\r\nSlope isn't just a mathematical concept; it's a real-world indicator of change. Think of it as the speedometer of your car. A high absolute value of the slope means you're either accelerating fast or decelerating quickly\u2014essentially, you're going through a lot of change. A slope close to zero? You're cruising at a steady pace.\r\n\r\nQuick Tips:\r\n<ul>\r\n \t<li>Units for slope are always [latex]\\frac{\\text{units for the output}}{\\text{units for the input}}[\/latex]<\/li>\r\n \t<li>When you're calculating slope, don't forget about the units. They give context to your numbers. For instance, if you're looking at a graph that represents the speed of a car over time, the slope could be in \"miles per hour.\" This tells you how fast the car is going, which is much more informative than just a number.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">If [latex]f\\left(x\\right)[\/latex] is a linear function, and [latex]\\left(2,3\\right)[\/latex] and [latex]\\left(0,4\\right)[\/latex] are points on the line, find the slope. Is this function increasing or decreasing? [reveal-answer q=\"538866\"]Show Solution[\/reveal-answer] [hidden-answer a=\"538866\"]<center>[latex]m=\\frac{4 - 3}{0 - 2}=\\frac{1}{-2}=-\\frac{1}{2}[\/latex] ; decreasing because [latex]m&lt;0[\/latex]<\/center>[\/hidden-answer]<\/section><section class=\"textbox example\">The population of a small town increased from [latex]1,442[\/latex] to [latex]1,868[\/latex] between 2009 and 2012. Find the change in population per year if we assume the change was constant from 2009 to 2012. [reveal-answer q=\"706559\"]Show Solution[\/reveal-answer] [hidden-answer a=\"706559\"]<center>[latex]m=\\frac{1,868 - 1,442}{2,012 - 2,009}=\\frac{426}{3}=142\\text{ people per year}[\/latex]<\/center>[\/hidden-answer]<\/section>Watch the video below to see how to find the slope of a line passing through two points and then determine whether the line is increasing, decreasing or neither.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-heedcbfd-in3NTcx11I8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/in3NTcx11I8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-heedcbfd-in3NTcx11I8\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844468&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-heedcbfd-in3NTcx11I8&vembed=0&video_id=in3NTcx11I8&video_target=tpm-plugin-heedcbfd-in3NTcx11I8'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+the+Slope+Given+Two+Points+and+Describe+the+Line_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Slope Given Two Points and Describe the Line\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>Watch the next video to see an example of an application of slope in determining the increase in cost for producing solar panels given two data points.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aeacchde-4RbniDgEGE4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4RbniDgEGE4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aeacchde-4RbniDgEGE4\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844469&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-aeacchde-4RbniDgEGE4&vembed=0&video_id=4RbniDgEGE4&video_target=tpm-plugin-aeacchde-4RbniDgEGE4'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Slope+Application+Involving+Production+Costs_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Slope Application Involving Production Costs\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Finding the [latex]x[\/latex]-intercept of a Line<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nThe [latex]x[\/latex]-intercept is the point where a line crosses the [latex]x[\/latex]-axis. In mathematical terms, it's the value of [latex]x[\/latex] when [latex]f(x) = 0[\/latex]. For example, for the function [latex]f(x) = 3x - 6[\/latex], the [latex]x[\/latex]-intercept is found by setting [latex]f(x)[\/latex] equal to zero and solving for [latex]x[\/latex].\r\n\r\n<strong>Set the Function to Zero:<\/strong> To find the [latex]x[\/latex]-intercept, set [latex]f(x)=0[\/latex] and solve for [latex]x[\/latex].\r\n\r\n<strong>Check for Exceptions:<\/strong> Not all linear functions have [latex]x[\/latex]-intercepts. Functions of the form [latex]y=c[\/latex], where [latex]c[\/latex] is a nonzero real number, don't have an [latex]x[\/latex]-intercept.\r\n\r\n<\/div>\r\n<section class=\"textbox example\">Find the [latex]x[\/latex]-intercept of [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex]. [reveal-answer q=\"400055\"]Show Solution[\/reveal-answer] [hidden-answer a=\"400055\"] Set the function equal to zero to solve for [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}0=\\frac{1}{2}x - 3\\\\ 3=\\frac{1}{2}x\\\\ 6=x\\\\ x=6\\end{array}[\/latex]<\/p>\r\nThe graph crosses the [latex]x[\/latex]-axis at the point [latex](6, 0)[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nA graph of the function is shown below. We can see that the [latex]x[\/latex]-intercept is [latex](6, 0)[\/latex] as expected.\r\n\r\n<center><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184351\/CNX_Precalc_Figure_02_02_0132.jpg\" width=\"369\" height=\"378\" \/><\/center><center><strong><span style=\"font-size: 10pt;\">The graph of the linear function [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex]. \u00a0[\/hidden-answer]<\/span><\/strong><\/center><\/section>Watch the following video to see an example of finding the [latex]x[\/latex]-intercept.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dgcchhcf-MdXeVeVS42A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/MdXeVeVS42A?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dgcchhcf-MdXeVeVS42A\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=11328527&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dgcchhcf-MdXeVeVS42A&vembed=0&video_id=MdXeVeVS42A&video_target=tpm-plugin-dgcchhcf-MdXeVeVS42A'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Finding+x-intercepts+given+slope+intercept+form_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding x-intercepts given slope intercept form\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Express a linear function using an equation, words, tables, and graphs<\/li>\n<li>Determine whether a linear function is increasing, decreasing, or constant<\/li>\n<li>Understand the slope of a line as the rate at which one quantity changes in relation to another<\/li>\n<li>Solve for the x-intercept in a linear function equation<\/li>\n<\/ul>\n<\/section>\n<h2>Linear Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A <strong>linear function<\/strong> is characterized by a constant rate of change and can be represented as a polynomial of degree [latex]1[\/latex].<\/p>\n<\/div>\n<h2>Representing Linear Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>The Language of Linear Functions<\/strong><\/p>\n<ul>\n<li><strong>Word Form:<\/strong> Describing a linear function in words helps you understand the relationship between variables. For instance, if a train&#8217;s distance from a station is determined by its speed and initial distance, you can say, &#8220;The train&#8217;s 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.&#8221;<\/li>\n<li><strong>Rate of Change:<\/strong> This is another term for the slope. It tells you how fast one variable changes in relation to another.<\/li>\n<\/ul>\n<p>Understanding the language used to describe linear functions can be a game-changer. For example, the term &#8220;rate of change&#8221; might sound complex, but it&#8217;s just a fancy way of saying how one variable (like time) affects another (like distance). In real-world scenarios like the train example, this rate is constant, making it easier to predict future outcomes.<\/p>\n<p><strong>Function Notation and Its Importance<\/strong><\/p>\n<ul>\n<li><strong>Slope-Intercept Form:<\/strong> This is the equation [latex]y=mx+b[\/latex], where [latex]m[\/latex] is the slope and [latex]b[\/latex] is the [latex]y[\/latex]-intercept. It&#8217;s a standardized way to write linear functions.<\/li>\n<li><strong>Function Notation:<\/strong> [latex]f(x)=mx+b[\/latex] is another way to write the slope-intercept form, emphasizing that [latex]y[\/latex] is a function of [latex]x[\/latex].<\/li>\n<\/ul>\n<p>Function notation isn&#8217;t just for mathematicians; it&#8217;s a useful tool for anyone wanting to understand relationships between variables. For example, [latex]D(t)=83t+250[\/latex] tells us that the distance [latex]D[\/latex] is a function of time [latex]t[\/latex]. This notation makes it clear what the input and output variables are, which is crucial for understanding the function&#8217;s behavior.<\/p>\n<p><strong>Visualizing Linear Functions<\/strong><\/p>\n<ul>\n<li><strong>Tabular Form:<\/strong> A table can show how the output changes with each unit increase in the input.<\/li>\n<li><strong>Graphical Form:<\/strong> A graph provides a visual representation of a linear function. The slope and [latex]y[\/latex]-intercept can be easily identified.<\/li>\n<\/ul>\n<p>Graphs offer a powerful way to visualize linear functions. For instance, the graph of [latex]D(t)=83t+250[\/latex] would be a straight line, allowing you to instantly see the rate of change and initial value. This can be particularly helpful in real-world applications like determining how far a train will be from a station at a given time.<\/p>\n<\/div>\n<p>Watch the following video for more on the slope intercept form.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ehgcfefh-IL3UCuXrUzE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/IL3UCuXrUzE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ehgcfefh-IL3UCuXrUzE\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844449&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ehgcfefh-IL3UCuXrUzE&#38;vembed=0&#38;video_id=IL3UCuXrUzE&#38;video_target=tpm-plugin-ehgcfefh-IL3UCuXrUzE\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Slope-intercept+form+%7C+Algebra+I+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSlope-intercept form | Algebra I | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the following video for more on identifying linear functions from tables.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fdgcdfch-J3zpqUd5dMM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/J3zpqUd5dMM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fdgcdfch-J3zpqUd5dMM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=11328525&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fdgcdfch-J3zpqUd5dMM&#38;vembed=0&#38;video_id=J3zpqUd5dMM&#38;video_target=tpm-plugin-fdgcdfch-J3zpqUd5dMM\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Identifying+Linear+Functions+From+Tables_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIdentifying Linear Functions From Tables\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the following video for more on identifying linear functions from graphs.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-adgfebab--VCwxDKOih0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/-VCwxDKOih0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-adgfebab--VCwxDKOih0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=11328526&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-adgfebab--VCwxDKOih0&#38;vembed=0&#38;video_id=-VCwxDKOih0&#38;video_target=tpm-plugin-adgfebab--VCwxDKOih0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Identifying+Linear+Functions+From+Graphs_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIdentifying Linear Functions From Graphs\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Determine Whether a Linear Function is Increasing, Decreasing, or Constant<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Increasing Linear Function:<\/strong> If you&#8217;re looking at a graph and see the line slanting upward from left to right, you&#8217;re likely dealing with an increasing function.<\/p>\n<p><strong>Decreasing Linear Function:<\/strong> A line that slants downward from left to right on a graph usually signifies a decreasing function.<\/p>\n<p><strong>Constant Linear Function:<\/strong> A horizontal line on a graph is a dead giveaway for a constant function.<\/p>\n<p>Graphing is a powerful tool for understanding linear functions. Use online graphing calculators like Desmos to visualize these functions. You can even add sliders to manipulate the slope. For instance, the function [latex]f(x)=\u2212\\frac{2}{32}x\u2212\\frac{4}{3}[\/latex] can be graphed to visually represent its decreasing nature.<\/p>\n<\/div>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm13521\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13521&theme=lumen&iframe_resize_id=ohm13521&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Watch the following video for more on determining if a linear function is increasing or decreasing.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-habbbadd-AqDOZgNhnvA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/AqDOZgNhnvA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-habbbadd-AqDOZgNhnvA\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844467&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-habbbadd-AqDOZgNhnvA&#38;vembed=0&#38;video_id=AqDOZgNhnvA&#38;video_target=tpm-plugin-habbbadd-AqDOZgNhnvA\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Determine+if+a+Linear+Function+is+Increasing+or+Decreasing_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine if a Linear Function is Increasing or Decreasing\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Calculate and Interpret Slope<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Slope<\/strong> is the measure of how steep a line is.<\/p>\n<p>It&#8217;s calculated as the <strong>change in output <\/strong>(rise) divided by the <strong>change in input <\/strong>(run).<\/p>\n<p>The formula to calculate slope is [latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\Rightarrow \\dfrac{f(x_2)-f(x_1)}{x_2 - x_1}[\/latex]<\/p>\n<p>Slope isn&#8217;t just a mathematical concept; it&#8217;s a real-world indicator of change. Think of it as the speedometer of your car. A high absolute value of the slope means you&#8217;re either accelerating fast or decelerating quickly\u2014essentially, you&#8217;re going through a lot of change. A slope close to zero? You&#8217;re cruising at a steady pace.<\/p>\n<p>Quick Tips:<\/p>\n<ul>\n<li>Units for slope are always [latex]\\frac{\\text{units for the output}}{\\text{units for the input}}[\/latex]<\/li>\n<li>When you&#8217;re calculating slope, don&#8217;t forget about the units. They give context to your numbers. For instance, if you&#8217;re looking at a graph that represents the speed of a car over time, the slope could be in &#8220;miles per hour.&#8221; This tells you how fast the car is going, which is much more informative than just a number.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">If [latex]f\\left(x\\right)[\/latex] is a linear function, and [latex]\\left(2,3\\right)[\/latex] and [latex]\\left(0,4\\right)[\/latex] are points on the line, find the slope. Is this function increasing or decreasing? <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q538866\">Show Solution<\/button> <\/p>\n<div id=\"q538866\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]m=\\frac{4 - 3}{0 - 2}=\\frac{1}{-2}=-\\frac{1}{2}[\/latex] ; decreasing because [latex]m<0[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">The population of a small town increased from [latex]1,442[\/latex] to [latex]1,868[\/latex] between 2009 and 2012. Find the change in population per year if we assume the change was constant from 2009 to 2012. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q706559\">Show Solution<\/button> <\/p>\n<div id=\"q706559\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]m=\\frac{1,868 - 1,442}{2,012 - 2,009}=\\frac{426}{3}=142\\text{ people per year}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the video below to see how to find the slope of a line passing through two points and then determine whether the line is increasing, decreasing or neither.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-heedcbfd-in3NTcx11I8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/in3NTcx11I8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-heedcbfd-in3NTcx11I8\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844468&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-heedcbfd-in3NTcx11I8&#38;vembed=0&#38;video_id=in3NTcx11I8&#38;video_target=tpm-plugin-heedcbfd-in3NTcx11I8\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+the+Slope+Given+Two+Points+and+Describe+the+Line_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Slope Given Two Points and Describe the Line\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the next video to see an example of an application of slope in determining the increase in cost for producing solar panels given two data points.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aeacchde-4RbniDgEGE4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4RbniDgEGE4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aeacchde-4RbniDgEGE4\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844469&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-aeacchde-4RbniDgEGE4&#38;vembed=0&#38;video_id=4RbniDgEGE4&#38;video_target=tpm-plugin-aeacchde-4RbniDgEGE4\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Slope+Application+Involving+Production+Costs_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Slope Application Involving Production Costs\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Finding the [latex]x[\/latex]-intercept of a Line<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>The [latex]x[\/latex]-intercept is the point where a line crosses the [latex]x[\/latex]-axis. In mathematical terms, it&#8217;s the value of [latex]x[\/latex] when [latex]f(x) = 0[\/latex]. For example, for the function [latex]f(x) = 3x - 6[\/latex], the [latex]x[\/latex]-intercept is found by setting [latex]f(x)[\/latex] equal to zero and solving for [latex]x[\/latex].<\/p>\n<p><strong>Set the Function to Zero:<\/strong> To find the [latex]x[\/latex]-intercept, set [latex]f(x)=0[\/latex] and solve for [latex]x[\/latex].<\/p>\n<p><strong>Check for Exceptions:<\/strong> Not all linear functions have [latex]x[\/latex]-intercepts. Functions of the form [latex]y=c[\/latex], where [latex]c[\/latex] is a nonzero real number, don&#8217;t have an [latex]x[\/latex]-intercept.<\/p>\n<\/div>\n<section class=\"textbox example\">Find the [latex]x[\/latex]-intercept of [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex]. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q400055\">Show Solution<\/button> <\/p>\n<div id=\"q400055\" class=\"hidden-answer\" style=\"display: none\"> Set the function equal to zero to solve for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}0=\\frac{1}{2}x - 3\\\\ 3=\\frac{1}{2}x\\\\ 6=x\\\\ x=6\\end{array}[\/latex]<\/p>\n<p>The graph crosses the [latex]x[\/latex]-axis at the point [latex](6, 0)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>A graph of the function is shown below. We can see that the [latex]x[\/latex]-intercept is [latex](6, 0)[\/latex] as expected.<\/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\/21184351\/CNX_Precalc_Figure_02_02_0132.jpg\" width=\"369\" height=\"378\" alt=\"image\" \/><\/div>\n<div style=\"text-align: center;\"><strong><span style=\"font-size: 10pt;\">The graph of the linear function [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex]. \u00a0<\/div>\n<\/div>\n<p><\/span><\/strong><\/div>\n<\/section>\n<p>Watch the following video to see an example of finding the [latex]x[\/latex]-intercept.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dgcchhcf-MdXeVeVS42A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/MdXeVeVS42A?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dgcchhcf-MdXeVeVS42A\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=11328527&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dgcchhcf-MdXeVeVS42A&#38;vembed=0&#38;video_id=MdXeVeVS42A&#38;video_target=tpm-plugin-dgcchhcf-MdXeVeVS42A\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Finding+x-intercepts+given+slope+intercept+form_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding x-intercepts given slope intercept form\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":12,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Identifying Linear Functions From Tables \",\"author\":\"\",\"organization\":\"numberninja\",\"url\":\"https:\/\/www.youtube.com\/watch?v=J3zpqUd5dMM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Identifying Linear Functions From Graphs \",\"author\":\"\",\"organization\":\"numberninja\",\"url\":\"https:\/\/www.youtube.com\/watch?v=-VCwxDKOih0\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Determine if a Linear Function is Increasing or Decreasing 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