{"id":1286,"date":"2025-07-24T04:00:01","date_gmt":"2025-07-24T04:00:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1286"},"modified":"2026-03-18T02:28:16","modified_gmt":"2026-03-18T02:28:16","slug":"rates-of-change-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rates-of-change-fresh-take\/","title":{"raw":"Rates of Change: Fresh Take","rendered":"Rates of Change: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the average rate of change of a function over a specified interval.<\/li>\r\n \t<li>Use a graph to determine where a function is increasing, decreasing, or constant.<\/li>\r\n \t<li>Use a graph to locate local and absolute maxima and local minima.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Rates of Change<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Rate of change describes how an output quantity changes relative to the input quantity<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Units: output units per input units<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Average Rate of Change:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Calculated over an interval<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formula: [latex]\\frac{\\Delta y}{\\Delta x} = \\frac{f(x_2) - f(x_1)}{x_2 - x_1}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpretation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Positive rate: output increases as input increases<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Negative rate: output decreases as input increases<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Population growth<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Speed (distance per time)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Fuel efficiency (distance per volume)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Economic indicators (price changes over time)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Calculating Average Rate of Change:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the interval [latex][x_1, x_2][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate change in output: [latex]\\Delta y = f(x_2) - f(x_1)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate change in input: [latex]\\Delta x = x_2 - x_1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Divide: [latex]\\frac{\\Delta y}{\\Delta x}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpreting from Graphs:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Slope of secant line between two points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical change divided by horizontal change<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Working with Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Evaluate function at endpoints of interval<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply average rate of change formula<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Using the data in the table below, find the average rate of change between 2014 and 2019.<\/p>\r\n\r\n<div class=\"font-styrene relative\">\r\n<div class=\"bg-bg-100 text-text-500 flex items-center justify-center rounded-l-[inherit] w-14 bg-bg-200 border-border-200 border-r-[0.5px]\">\r\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2014<\/td>\r\n<td>2015<\/td>\r\n<td>2016<\/td>\r\n<td>2017<\/td>\r\n<td>2018<\/td>\r\n<td>2019<\/td>\r\n<td>2020<\/td>\r\n<td>2021<\/td>\r\n<td>2022<\/td>\r\n<td>2023<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>3.358<\/td>\r\n<td>2.429<\/td>\r\n<td>2.143<\/td>\r\n<td>2.415<\/td>\r\n<td>2.719<\/td>\r\n<td>2.604<\/td>\r\n<td>2.168<\/td>\r\n<td>3.008<\/td>\r\n<td>3.951<\/td>\r\n<td>3.519<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"padding min-w-0 flex-1 px-4 py-3\">\r\n<div class=\"break-words text-sm font-medium leading-tight\">[reveal-answer q=\"461891\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"461891\"]\r\n<p class=\"whitespace-pre-wrap break-words\">[latex] \\begin{align*} \\frac{\\Delta y}{\\Delta x} &amp;= \\frac{f(x_2) - f(x_1)}{x_2 - x_1} \\\\ &amp;= \\frac{$2.604 - $3.358}{2019 - 2014} \\\\ &amp;= \\frac{-$0.754}{5 \\text{ years}} \\\\ &amp;= -$0.1508 \\text{ per year} \\end{align*} [\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">The average rate of change is [latex]-$0.1508[\/latex] per year.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">This solution shows that between 2014 and 2019, the average price of gasoline decreased by about [latex]15.08[\/latex] cents per year. This negative rate of change indicates an overall downward trend in gasoline prices during this period, despite potential fluctuations within the timeframe.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the average rate of change of [latex]f\\left(x\\right)=x - 2\\sqrt{x}[\/latex] on the interval [latex]\\left[1,9\\right][\/latex].[reveal-answer q=\"609687\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"609687\"][latex]\\dfrac{1}{2}[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]1733[\/ohm_question]<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the average rate of change of [latex]f\\left(x\\right)={x}^{2}+2x - 8[\/latex] on the interval [latex]\\left[5,a\\right][\/latex].[reveal-answer q=\"312945\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"312945\"][latex]a+7[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]4083[\/ohm_question]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video provides another example of how to find the average rate of change between two points from a table of values.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hddacgab-iJ_0nPUUlOg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/iJ_0nPUUlOg?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hddacgab-iJ_0nPUUlOg\" 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=6405053&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hddacgab-iJ_0nPUUlOg&amp;vembed=0&amp;video_id=iJ_0nPUUlOg&amp;video_target=tpm-plugin-hddacgab-iJ_0nPUUlOg\" 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\/Ex+-+Find+the+Average+Rate+of+Change+From+a+Table+-+Temperatures_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Average Rate of Change From a Table - Temperatures\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video provides another example of finding the average rate of change of a function given a formula and an interval.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gchdhbcb-g93QEKJXeu4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/g93QEKJXeu4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gchdhbcb-g93QEKJXeu4\" 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=12844422&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gchdhbcb-g93QEKJXeu4&amp;vembed=0&amp;video_id=g93QEKJXeu4&amp;video_target=tpm-plugin-gchdhbcb-g93QEKJXeu4\" 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\/Ex+-+Find+the+Average+Rate+of+Change+Given+a+Function+Rule_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Average Rate of Change Given a Function Rule\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Behaviors of Functions<\/h2>\r\n<h3 data-type=\"title\">Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant<\/h3>\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\">Increasing and Decreasing Intervals:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Increasing: function values increase as input increases<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Decreasing: function values decrease as input increases<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Constant: function values remain the same as input increases<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Local Extrema:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Local Maximum: where function changes from increasing to decreasing<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Local Minimum: where function changes from decreasing to increasing<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Collectively called local extrema or relative extrema<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphical Interpretation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Increasing: graph slopes upward from left to right<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Decreasing: graph slopes downward from left to right<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Local maximum: highest point in a neighborhood<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Local minimum: lowest point in a neighborhood<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Mathematical Definitions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Increasing: [latex]f(b) &gt; f(a)[\/latex] for [latex]b &gt; a[\/latex] in an interval<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Decreasing: [latex]f(b) &lt; f(a)[\/latex] for [latex]b &gt; a[\/latex] in an interval<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Local maximum at [latex]x = b[\/latex]: [latex]f(x) \\leq f(b)[\/latex] for all [latex]x[\/latex] in some interval containing [latex]b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Local minimum at [latex]x = b[\/latex]: [latex]f(x) \\geq f(b)[\/latex] for all [latex]x[\/latex] in some interval containing [latex]b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identifying Increasing\/Decreasing Intervals:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Observe graph from left to right<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Note where slope changes from positive to negative or vice versa<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Locating Local Extrema:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Look for \"peaks\" (local maxima) and \"valleys\" (local minima)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Confirm by checking neighboring points<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using Technology:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Utilize graphing calculators or software to visualize functions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use built-in features to estimate extrema locations<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Analyzing Complex Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Break down the graph into smaller intervals<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Analyze behavior within each interval<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">For the function [latex]f[\/latex] whose graph is shown below, find all local maxima and minima.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194804\/CNX_Precalc_Figure_01_03_0112.jpg\" alt=\"Graph of a polynomial. The line curves down to x = negative 2 and up to x = 1.\" width=\"487\" height=\"368\" \/> Graph of a polynomial[\/caption]\r\n\r\n[reveal-answer q=\"523190\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"523190\"]Observe the graph of [latex]f[\/latex]. The graph attains a local maximum at [latex]x=1[\/latex] because it is the highest point in an open interval around [latex]x=1[\/latex]. The local maximum is the [latex]y[\/latex] -coordinate at [latex]x=1[\/latex], which is [latex]2[\/latex].The graph attains a local minimum at [latex]\\text{ }x=-1\\text{ }[\/latex] because it is the lowest point in an open interval around [latex]x=-1[\/latex]. The local minimum is the <em>y<\/em>-coordinate at [latex]x=-1[\/latex], which is [latex]-2[\/latex].[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]32571[\/ohm_question]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">This video further explains how to find where a function is increasing or decreasing.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-abedhfab-78b4HOMVcKM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/78b4HOMVcKM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-abedhfab-78b4HOMVcKM\" 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=12844423&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-abedhfab-78b4HOMVcKM&amp;vembed=0&amp;video_id=78b4HOMVcKM&amp;video_target=tpm-plugin-abedhfab-78b4HOMVcKM\" 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\/Determine+Where+a+Function+is+Increasing+and+Decreasing_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine Where a Function is Increasing and Decreasing\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h3>Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/h3>\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\">Constant Function: [latex]f(x) = c[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Behavior: Neither increasing nor decreasing<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph: Horizontal line<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identity Function: [latex]f(x) = x[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Behavior: Increasing on [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph: Straight line through origin<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quadratic Function: [latex]f(x) = x^2[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Increasing: [latex](0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Decreasing: [latex](-\\infty, 0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Minimum: At [latex]x = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph: Parabola opening upward<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cubic Function: [latex]f(x) = x^3[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Behavior: Increasing on [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph: S-shaped curve<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reciprocal Function: [latex]f(x) = \\frac{1}{x}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Decreasing: [latex](-\\infty, 0) \\cup (0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph: Hyperbola with vertical asymptote at x = 0<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reciprocal Squared: [latex]f(x) = \\frac{1}{x^2}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Increasing: [latex](-\\infty, 0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Decreasing: [latex](0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph: U-shaped curve with vertical asymptote at x = 0<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cube Root: [latex]f(x) = \\sqrt[3]{x}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Behavior: Increasing on [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph: S-shaped curve, less steep than cubic<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Square Root: [latex]f(x) = \\sqrt{x}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Increasing: [latex](0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex][0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph: Curve starting at origin, opening upward<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Absolute Value: [latex]f(x) = |x|[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Increasing: [latex](0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Decreasing: [latex](-\\infty, 0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Minimum: At [latex]x = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph: V-shaped graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<h3 data-type=\"title\">Use A Graph to Locate the Absolute Maximum and Absolute Minimum<\/h3>\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\">Absolute Extrema vs. Local Extrema:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Absolute: Highest\/lowest points over entire domain<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Local: Highest\/lowest points in a local region<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Absolute Maximum:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Highest point on the entire graph<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(c)[\/latex] is absolute max if [latex]f(c) \\geq f(x)[\/latex] for all [latex]x[\/latex] in domain<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Absolute Minimum:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Lowest point on the entire graph<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(d)[\/latex] is absolute min if [latex]f(d) \\leq f(x)[\/latex] for all [latex]x[\/latex] in domain<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Existence of Absolute Extrema:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Not all functions have absolute extrema<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = x^3[\/latex] has neither absolute max nor min<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Graphical Identification:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Observe entire graph within function's domain<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Locate highest and lowest points<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Comparing Extrema:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Compare all local maxima to find absolute maximum<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Compare all local minima to find absolute minimum<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Considering Domain:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Check domain boundaries for potential absolute extrema<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Be aware of asymptotic behavior<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiple Absolute Extrema:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Functions can have multiple absolute maxima or minima<\/li>\r\n \t<li class=\"whitespace-normal break-words\">These occur at same y-value but different x-values<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Consider the function:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] f(x) = \\begin{cases} x^2 &amp; \\text{if } x &lt; 0 \\\\ 16 - x^2 &amp; \\text{if } 0 \\leq x \\leq 4 \\\\ 0 &amp; \\text{if } x &gt; 4 \\end{cases} [\/latex]<\/p>\r\nFind the absolute maximum and minimum.\r\n\r\n[reveal-answer q=\"72361\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"72361\"]\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Absolute Maximum: [latex]16[\/latex] at [latex] x = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Absolute Minimum: [latex]0[\/latex] for all [latex]x &gt; 4[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Note: This function has a continuous absolute minimum over an interval<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the average rate of change of a function over a specified interval.<\/li>\n<li>Use a graph to determine where a function is increasing, decreasing, or constant.<\/li>\n<li>Use a graph to locate local and absolute maxima and local minima.<\/li>\n<\/ul>\n<\/section>\n<h2>Rates of Change<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Rate of change describes how an output quantity changes relative to the input quantity<\/li>\n<li class=\"whitespace-normal break-words\">Units: output units per input units<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Average Rate of Change:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Calculated over an interval<\/li>\n<li class=\"whitespace-normal break-words\">Formula: [latex]\\frac{\\Delta y}{\\Delta x} = \\frac{f(x_2) - f(x_1)}{x_2 - x_1}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Positive rate: output increases as input increases<\/li>\n<li class=\"whitespace-normal break-words\">Negative rate: output decreases as input increases<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Population growth<\/li>\n<li class=\"whitespace-normal break-words\">Speed (distance per time)<\/li>\n<li class=\"whitespace-normal break-words\">Fuel efficiency (distance per volume)<\/li>\n<li class=\"whitespace-normal break-words\">Economic indicators (price changes over time)<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Calculating Average Rate of Change:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the interval [latex][x_1, x_2][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate change in output: [latex]\\Delta y = f(x_2) - f(x_1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate change in input: [latex]\\Delta x = x_2 - x_1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Divide: [latex]\\frac{\\Delta y}{\\Delta x}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interpreting from Graphs:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Slope of secant line between two points<\/li>\n<li class=\"whitespace-normal break-words\">Vertical change divided by horizontal change<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Working with Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Evaluate function at endpoints of interval<\/li>\n<li class=\"whitespace-normal break-words\">Apply average rate of change formula<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Using the data in the table below, find the average rate of change between 2014 and 2019.<\/p>\n<div class=\"font-styrene relative\">\n<div class=\"bg-bg-100 text-text-500 flex items-center justify-center rounded-l-[inherit] w-14 bg-bg-200 border-border-200 border-r-[0.5px]\">\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2014<\/td>\n<td>2015<\/td>\n<td>2016<\/td>\n<td>2017<\/td>\n<td>2018<\/td>\n<td>2019<\/td>\n<td>2020<\/td>\n<td>2021<\/td>\n<td>2022<\/td>\n<td>2023<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>3.358<\/td>\n<td>2.429<\/td>\n<td>2.143<\/td>\n<td>2.415<\/td>\n<td>2.719<\/td>\n<td>2.604<\/td>\n<td>2.168<\/td>\n<td>3.008<\/td>\n<td>3.951<\/td>\n<td>3.519<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"padding min-w-0 flex-1 px-4 py-3\">\n<div class=\"break-words text-sm font-medium leading-tight\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q461891\">Show Answer<\/button><\/p>\n<div id=\"q461891\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\begin{align*} \\frac{\\Delta y}{\\Delta x} &= \\frac{f(x_2) - f(x_1)}{x_2 - x_1} \\\\ &= \\frac{$2.604 - $3.358}{2019 - 2014} \\\\ &= \\frac{-$0.754}{5 \\text{ years}} \\\\ &= -$0.1508 \\text{ per year} \\end{align*}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">The average rate of change is [latex]-$0.1508[\/latex] per year.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">This solution shows that between 2014 and 2019, the average price of gasoline decreased by about [latex]15.08[\/latex] cents per year. This negative rate of change indicates an overall downward trend in gasoline prices during this period, despite potential fluctuations within the timeframe.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the average rate of change of [latex]f\\left(x\\right)=x - 2\\sqrt{x}[\/latex] on the interval [latex]\\left[1,9\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q609687\">Show Solution<\/button><\/p>\n<div id=\"q609687\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{1}{2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm1733\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1733&theme=lumen&iframe_resize_id=ohm1733&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the average rate of change of [latex]f\\left(x\\right)={x}^{2}+2x - 8[\/latex] on the interval [latex]\\left[5,a\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q312945\">Show Solution<\/button><\/p>\n<div id=\"q312945\" class=\"hidden-answer\" style=\"display: none\">[latex]a+7[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm4083\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4083&theme=lumen&iframe_resize_id=ohm4083&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video provides another example of how to find the average rate of change between two points from a table of values.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hddacgab-iJ_0nPUUlOg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/iJ_0nPUUlOg?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hddacgab-iJ_0nPUUlOg\" 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=6405053&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hddacgab-iJ_0nPUUlOg&amp;vembed=0&amp;video_id=iJ_0nPUUlOg&amp;video_target=tpm-plugin-hddacgab-iJ_0nPUUlOg\" 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\/Ex+-+Find+the+Average+Rate+of+Change+From+a+Table+-+Temperatures_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Average Rate of Change From a Table &#8211; Temperatures\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video provides another example of finding the average rate of change of a function given a formula and an interval.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gchdhbcb-g93QEKJXeu4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/g93QEKJXeu4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gchdhbcb-g93QEKJXeu4\" 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=12844422&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gchdhbcb-g93QEKJXeu4&amp;vembed=0&amp;video_id=g93QEKJXeu4&amp;video_target=tpm-plugin-gchdhbcb-g93QEKJXeu4\" 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\/Ex+-+Find+the+Average+Rate+of+Change+Given+a+Function+Rule_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Average Rate of Change Given a Function Rule\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Behaviors of Functions<\/h2>\n<h3 data-type=\"title\">Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Increasing and Decreasing Intervals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Increasing: function values increase as input increases<\/li>\n<li class=\"whitespace-normal break-words\">Decreasing: function values decrease as input increases<\/li>\n<li class=\"whitespace-normal break-words\">Constant: function values remain the same as input increases<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Local Extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Local Maximum: where function changes from increasing to decreasing<\/li>\n<li class=\"whitespace-normal break-words\">Local Minimum: where function changes from decreasing to increasing<\/li>\n<li class=\"whitespace-normal break-words\">Collectively called local extrema or relative extrema<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphical Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Increasing: graph slopes upward from left to right<\/li>\n<li class=\"whitespace-normal break-words\">Decreasing: graph slopes downward from left to right<\/li>\n<li class=\"whitespace-normal break-words\">Local maximum: highest point in a neighborhood<\/li>\n<li class=\"whitespace-normal break-words\">Local minimum: lowest point in a neighborhood<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Mathematical Definitions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Increasing: [latex]f(b) > f(a)[\/latex] for [latex]b > a[\/latex] in an interval<\/li>\n<li class=\"whitespace-normal break-words\">Decreasing: [latex]f(b) < f(a)[\/latex] for [latex]b > a[\/latex] in an interval<\/li>\n<li class=\"whitespace-normal break-words\">Local maximum at [latex]x = b[\/latex]: [latex]f(x) \\leq f(b)[\/latex] for all [latex]x[\/latex] in some interval containing [latex]b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Local minimum at [latex]x = b[\/latex]: [latex]f(x) \\geq f(b)[\/latex] for all [latex]x[\/latex] in some interval containing [latex]b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identifying Increasing\/Decreasing Intervals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Observe graph from left to right<\/li>\n<li class=\"whitespace-normal break-words\">Note where slope changes from positive to negative or vice versa<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Locating Local Extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Look for &#8220;peaks&#8221; (local maxima) and &#8220;valleys&#8221; (local minima)<\/li>\n<li class=\"whitespace-normal break-words\">Confirm by checking neighboring points<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Using Technology:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Utilize graphing calculators or software to visualize functions<\/li>\n<li class=\"whitespace-normal break-words\">Use built-in features to estimate extrema locations<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Analyzing Complex Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Break down the graph into smaller intervals<\/li>\n<li class=\"whitespace-normal break-words\">Analyze behavior within each interval<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">For the function [latex]f[\/latex] whose graph is shown below, find all local maxima and minima.<\/p>\n<figure style=\"width: 487px\" 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\/18194804\/CNX_Precalc_Figure_01_03_0112.jpg\" alt=\"Graph of a polynomial. The line curves down to x = negative 2 and up to x = 1.\" width=\"487\" height=\"368\" \/><figcaption class=\"wp-caption-text\">Graph of a polynomial<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q523190\">Show Solution<\/button><\/p>\n<div id=\"q523190\" class=\"hidden-answer\" style=\"display: none\">Observe the graph of [latex]f[\/latex]. The graph attains a local maximum at [latex]x=1[\/latex] because it is the highest point in an open interval around [latex]x=1[\/latex]. The local maximum is the [latex]y[\/latex] -coordinate at [latex]x=1[\/latex], which is [latex]2[\/latex].The graph attains a local minimum at [latex]\\text{ }x=-1\\text{ }[\/latex] because it is the lowest point in an open interval around [latex]x=-1[\/latex]. The local minimum is the <em>y<\/em>-coordinate at [latex]x=-1[\/latex], which is [latex]-2[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm32571\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=32571&theme=lumen&iframe_resize_id=ohm32571&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">This video further explains how to find where a function is increasing or decreasing.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-abedhfab-78b4HOMVcKM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/78b4HOMVcKM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-abedhfab-78b4HOMVcKM\" 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=12844423&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-abedhfab-78b4HOMVcKM&amp;vembed=0&amp;video_id=78b4HOMVcKM&amp;video_target=tpm-plugin-abedhfab-78b4HOMVcKM\" 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\/Determine+Where+a+Function+is+Increasing+and+Decreasing_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine Where a Function is Increasing and Decreasing\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Constant Function: [latex]f(x) = c[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Behavior: Neither increasing nor decreasing<\/li>\n<li class=\"whitespace-normal break-words\">Graph: Horizontal line<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identity Function: [latex]f(x) = x[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Behavior: Increasing on [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Graph: Straight line through origin<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Quadratic Function: [latex]f(x) = x^2[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Increasing: [latex](0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Decreasing: [latex](-\\infty, 0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Minimum: At [latex]x = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Graph: Parabola opening upward<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Cubic Function: [latex]f(x) = x^3[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Behavior: Increasing on [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Graph: S-shaped curve<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reciprocal Function: [latex]f(x) = \\frac{1}{x}[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Decreasing: [latex](-\\infty, 0) \\cup (0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Graph: Hyperbola with vertical asymptote at x = 0<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reciprocal Squared: [latex]f(x) = \\frac{1}{x^2}[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Increasing: [latex](-\\infty, 0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Decreasing: [latex](0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Graph: U-shaped curve with vertical asymptote at x = 0<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Cube Root: [latex]f(x) = \\sqrt[3]{x}[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Behavior: Increasing on [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Graph: S-shaped curve, less steep than cubic<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Square Root: [latex]f(x) = \\sqrt{x}[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Increasing: [latex](0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Domain: [latex][0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Graph: Curve starting at origin, opening upward<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Absolute Value: [latex]f(x) = |x|[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Increasing: [latex](0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Decreasing: [latex](-\\infty, 0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Minimum: At [latex]x = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Graph: V-shaped graph<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<h3 data-type=\"title\">Use A Graph to Locate the Absolute Maximum and Absolute Minimum<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Absolute Extrema vs. Local Extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Absolute: Highest\/lowest points over entire domain<\/li>\n<li class=\"whitespace-normal break-words\">Local: Highest\/lowest points in a local region<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Absolute Maximum:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Highest point on the entire graph<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(c)[\/latex] is absolute max if [latex]f(c) \\geq f(x)[\/latex] for all [latex]x[\/latex] in domain<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Absolute Minimum:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Lowest point on the entire graph<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(d)[\/latex] is absolute min if [latex]f(d) \\leq f(x)[\/latex] for all [latex]x[\/latex] in domain<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Existence of Absolute Extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Not all functions have absolute extrema<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = x^3[\/latex] has neither absolute max nor min<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Graphical Identification:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Observe entire graph within function&#8217;s domain<\/li>\n<li class=\"whitespace-normal break-words\">Locate highest and lowest points<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Comparing Extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Compare all local maxima to find absolute maximum<\/li>\n<li class=\"whitespace-normal break-words\">Compare all local minima to find absolute minimum<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Considering Domain:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Check domain boundaries for potential absolute extrema<\/li>\n<li class=\"whitespace-normal break-words\">Be aware of asymptotic behavior<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiple Absolute Extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Functions can have multiple absolute maxima or minima<\/li>\n<li class=\"whitespace-normal break-words\">These occur at same y-value but different x-values<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Consider the function:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = \\begin{cases} x^2 & \\text{if } x < 0 \\\\ 16 - x^2 & \\text{if } 0 \\leq x \\leq 4 \\\\ 0 & \\text{if } x > 4 \\end{cases}[\/latex]<\/p>\n<p>Find the absolute maximum and minimum.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q72361\">Show Answer<\/button><\/p>\n<div id=\"q72361\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Absolute Maximum: [latex]16[\/latex] at [latex]x = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Absolute Minimum: [latex]0[\/latex] for all [latex]x > 4[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Note: This function has a continuous absolute minimum over an interval<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: Find the Average Rate of Change From a Table - Temperatures\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/iJ_0nPUUlOg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Find the Average Rate of Change Given a Function Rule\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/g93QEKJXeu4\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Determine Where a Function is Increasing and 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