{"id":396,"date":"2019-02-28T18:42:10","date_gmt":"2019-02-28T18:42:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&#038;p=396"},"modified":"2019-11-18T18:13:55","modified_gmt":"2019-11-18T18:13:55","slug":"1-4-concavity","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/1-4-concavity\/","title":{"raw":"1.4 Concavity","rendered":"1.4 Concavity"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Find intervals for concavity.<\/li>\r\n \t<li>Identify inflection points.<\/li>\r\n \t<li>Determine concavity's relationship to average rates of change.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAs part of exploring how functions change, it is interesting to explore the graphical behavior of functions.\u00a0 Concavity describes the shape of the function and how it is changing.\r\n\r\nConsider the graphs below that show the total sales, in thousands of dollars, for two companies over 4 weeks.\r\n\r\n<img class=\"aligncenter wp-image-2412 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14184936\/14CoABg1-1024x473.png\" alt=\"\" width=\"650\" height=\"300\" \/>As you can see, the sales for each company are increasing, but they are increasing in very different ways.\u00a0 Company A has a lot of sales immediately which could represent the release of a much anticipated product and then the increase in sales levels off. \u00a0Company B starts with slower sales perhaps representing an unknown product which then starts to sell more rapidly perhaps because of word of mouth.\u00a0 To describe the difference in behavior, we can investigate how the average rate of change varies over different intervals.\u00a0 Using tables of values, we can find the average rate of change between consecutive points.\u00a0 For example, in Company A, we can use the first pair of points to get the average rate of change [latex]\\frac{5-0}{1-0}=5[\/latex] and the second pair of points to get the average rate of change [latex]\\frac{7.1-5}{2-1}=2.1.[\/latex]\r\n\r\n<img class=\"wp-image-2406 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14180001\/14Fig1.png\" alt=\"\" width=\"619\" height=\"301\" \/>\r\n\r\n&nbsp;\r\n\r\nFrom the tables, we can see that the rate of change for company A is <em>decreasing<\/em>, while the rate of change for company B is <em>increasing<\/em>.\r\n\r\n<img class=\"aligncenter wp-image-2409 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14181537\/14CoABGraphic.png\" alt=\"\" width=\"665\" height=\"300\" \/>For an increasing function, when the rate of change is decreasing, as with Company A, we say the function is <strong>concave down<\/strong>.\u00a0 For an increasing function, when the rate of change is increasing, as with Company B, we say the function is <strong>concave up<\/strong>.\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<div>\r\n\r\nA function is <strong>concave up<\/strong> if the rate of change is increasing.\r\n\r\nA function is <strong>concave down<\/strong> if the rate of change is decreasing.\r\n\r\nA point where a function changes from concave up to concave down or vice versa is called an <strong>inflection point<\/strong>.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1: Describe the Concavity<\/h3>\r\n<div>\r\n\r\nAn object is thrown from the top of a building.\u00a0 The object\u2019s height in feet above ground after <em>t<\/em> seconds is given by the function $latex h(t)=144-16{{t}^{2}}$ for $latex 0\\le t\\le 3$.\u00a0 Describe the concavity of the graph.\r\n\r\n[reveal-answer q=\"962750\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"962750\"]\r\n\r\nSketching a graph of the function, we can see that the function is decreasing.\u00a0\u00a0<img class=\"aligncenter size-medium wp-image-2410\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14182534\/14Ex1G-212x300.png\" alt=\"\" width=\"212\" height=\"300\" \/>\r\n\r\nWe can calculate some rates of change to explore the behavior.\u00a0 For example, the interval [latex]t=0[\/latex] to [latex]t=1[\/latex] has an average rate of change of [latex]\\frac{128-144}{1-0}=-16.[\/latex] The remaining intervals are shown in the table below.<img class=\"aligncenter size-medium wp-image-2407\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14181232\/14Ex1-300x243.png\" alt=\"\" width=\"300\" height=\"243\" \/>\r\n\r\nNotice that the rates of change are becoming more negative, so the rates of change are <em>decreasing<\/em>.\u00a0 This means the function is concave down.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2: Concavity from a Table of Values<\/h3>\r\nThe value, <em>V<\/em>, of a car after <em>t<\/em> years is given in the table below. Is the value increasing or decreasing? Is the function concave up or concave down?\r\n<table>\r\n<tbody>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 34.6563px\"><em>t<\/em><\/td>\r\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">0<\/td>\r\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">2<\/td>\r\n<td class=\"border\" style=\"height: 11px;width: 54.6563px;text-align: center\">4<\/td>\r\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">6<\/td>\r\n<td class=\"border\" style=\"height: 11px;width: 57.6563px;text-align: center\">8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 34.6563px\"><em>V(t)<\/em><\/td>\r\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">28000<\/td>\r\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">24342<\/td>\r\n<td class=\"border\" style=\"height: 11px;width: 54.6563px;text-align: center\">21162<\/td>\r\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">18397<\/td>\r\n<td class=\"border\" style=\"height: 11px;width: 57.6563px;text-align: center\">15994<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"567041\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"567041\"]\r\n\r\nSince the values <em>V(t)<\/em>, \u00a0are getting smaller as we let [latex]t[\/latex] increase, we can determine that the value of the car is decreasing. We can compute rates of change to determine concavity.\r\n\r\n<img class=\"aligncenter wp-image-2408 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14181319\/14Ex2-1024x119.png\" alt=\"\" width=\"1024\" height=\"119\" \/>\r\n\r\nThese rate of change values are becoming less negative since they are moving to the right on a number line, so the rates of change are <em>increasing<\/em> meaning\u00a0this function is concave up.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It #1<\/h3>\r\n<div>\r\n\r\nIs the function described in the table below concave up or concave down?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 27.7188px\"><em>x<\/em><\/td>\r\n<td class=\"border\" style=\"width: 44.7188px;text-align: center\">0<\/td>\r\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">5<\/td>\r\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">10<\/td>\r\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">15<\/td>\r\n<td class=\"border\" style=\"width: 18.7188px;text-align: center\">20<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 27.7188px\"><em>g(x)<\/em><\/td>\r\n<td class=\"border\" style=\"width: 44.7188px;text-align: center\">10000<\/td>\r\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">9000<\/td>\r\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">7000<\/td>\r\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">4000<\/td>\r\n<td class=\"border\" style=\"width: 18.7188px;text-align: center\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"249682\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"249682\"]\r\n\r\nThe average rates of change are decreasing so the function is concave down.\r\n\r\n<img class=\"aligncenter wp-image-2411 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14183409\/14TryItAns-1024x124.png\" alt=\"\" width=\"1024\" height=\"124\" \/>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nGraphically, concave down functions bend downwards like a frown, and concave up function bend upwards like a smile.\r\n\r\n<img class=\"alignnone wp-image-1882 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08170332\/Screen-Shot-2019-04-08-at-1.03.09-PM-e1554743217160.png\" alt=\"\" width=\"477\" height=\"305\" \/>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3:\u00a0 Determine Intervals of Concavity from a Graph<\/h3>\r\n<div>\r\n\r\nFrom the graph shown, estimate the intervals on which the function is concave down and concave up.<img class=\"size-medium wp-image-1883 aligncenter\" style=\"font-size: 0.9em\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08171402\/Picture11-300x259.png\" alt=\"\" width=\"300\" height=\"259\" \/>[reveal-answer q=\"266317\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"266317\"]\r\n\r\nOn the far left, the graph is decreasing but concave up, since it is bending upwards.\u00a0 It begins increasing at <em>x<\/em> = -2, but it continues to bend upwards until about <em>x<\/em> = -1.\r\n\r\nFrom <em>x<\/em> = -1 the graph starts to bend downward, and continues to do so until about <em>x<\/em> = 2.\u00a0 The graph then begins curving upwards for the remainder of the graph shown.\r\n\r\nFrom this, we can estimate that the graph is concave up on the intervals $$(-\\infty ,-1)$$ and $$(2,\\infty )$$, and is concave down on the interval $$(-1,2)$$.\u00a0 The graph has inflection points at \u00a0<em>x<\/em> = -1 and <em>x<\/em> = 2.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It #2<\/h3>\r\nCreate a graph of $$f(x)={{x}^{3}}-6{{x}^{2}}-15x+20$$ and use it to estimate the intervals on which the function is concave up and concave down.\r\n\r\n[reveal-answer q=\"273303\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"273303\"]\r\n<div>\r\n\r\nLooking at the graph of\u00a0$$f(x)={{x}^{3}}-6{{x}^{2}}-15x+20$$, it appears the function is concave down on [latex]\\left(-\\infty,\\text{ }2\\right)[\/latex] and concave up on [latex]\\left(2,\\text{ }\\infty\\right).[\/latex]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><strong>Behaviors of the Toolkit Functions<\/strong><\/h2>\r\nWe will now return to our toolkit functions and discuss their graphical behavior.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">Function<\/td>\r\n<td class=\"border\">Increasing\/Decreasing<\/td>\r\n<td class=\"border\">Concavity<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Constant Function\r\n\r\n$latex f(x)=c$<\/td>\r\n<td class=\"border\">Neither increasing nor decreasing<\/td>\r\n<td class=\"border\">Neither concave up nor down<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Identity Function\r\n\r\n$latex f(x)=x$<\/td>\r\n<td class=\"border\">Increasing<\/td>\r\n<td class=\"border\">Neither concave up nor down<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Quadratic Function\r\n\r\n$latex f(x)={{x}^{2}}$<\/td>\r\n<td class=\"border\">Increasing on\u00a0$latex (0,\\infty )$\r\n\r\nDecreasing on\u00a0$latex (-\\infty ,0)$\r\n\r\nMinimum at <em>x<\/em> = 0<\/td>\r\n<td class=\"border\">Concave up\r\n\r\n$latex (-\\infty ,\\infty )$<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Cubic Function\r\n\r\n$latex f(x)={{x}^{3}}$<\/td>\r\n<td class=\"border\">Increasing<\/td>\r\n<td class=\"border\">Concave down on\u00a0$latex (-\\infty ,0)$\r\n\r\nConcave up on\u00a0$latex (0,\\infty )$\r\n\r\nInflection point at (0,0)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Reciprocal\r\n\r\n$latex f(x)=\\frac{1}{x}$<\/td>\r\n<td class=\"border\">Decreasing\u00a0$latex (-\\infty ,0)\\cup (0,\\infty )$<\/td>\r\n<td class=\"border\">Concave down on\u00a0$latex (-\\infty ,0)$\r\n\r\nConcave up on\u00a0$latex (0,\\infty )$\r\n\r\n&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Reciprocal squared\r\n\r\n$latex f(x)=\\frac{1}{{{x}^{2}}}$<\/td>\r\n<td class=\"border\">Increasing on\u00a0$latex (-\\infty ,0)$\r\n\r\nDecreasing on\u00a0$latex (0,\\infty )$<\/td>\r\n<td class=\"border\">Concave up on\u00a0$latex (-\\infty ,0)\\cup (0,\\infty )$<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Cube Root\r\n\r\n$latex f(x)=\\sqrt[3]{x}$<\/td>\r\n<td class=\"border\">Increasing<\/td>\r\n<td class=\"border\">Concave down on\u00a0$latex (0,\\infty )$\r\n\r\nConcave up on\u00a0$latex (-\\infty ,0)$\r\n\r\nInflection point at (0,0)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Square Root\r\n\r\n$latex f(x)=\\sqrt[{}]{x}$<\/td>\r\n<td class=\"border\">Increasing on\u00a0$latex (0,\\infty )$<\/td>\r\n<td class=\"border\">Concave down on\u00a0$latex (0,\\infty )$<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Absolute Value\r\n\r\n$latex f(x)=\\left| x \\right|$<\/td>\r\n<td class=\"border\">Increasing on\u00a0$latex (0,\\infty )$\r\n\r\nDecreasing on\u00a0$latex (-\\infty ,0)$<\/td>\r\n<td class=\"border\">Neither concave up or down\r\n\r\n&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul>\r\n \t<li>Concavity describes the shape of the curve.\u00a0 If the average rates are increasing on an interval then the function is concave up and if the average rates are decreasing on an interval then the function is concave down on the interval.<\/li>\r\n \t<li>A function has an inflection point when it switches from concave down to concave up or visa versa.<\/li>\r\n \t<li>Given a graph, intervals of concavity can be estimated by determining where the graph bends up versus where it bends down.<\/li>\r\n \t<li>Input values are used when describing intervals of concavity.\u00a0 Endpoint in the interval are not included so the notation uses parenthesis not square brackets.<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Find intervals for concavity.<\/li>\n<li>Identify inflection points.<\/li>\n<li>Determine concavity&#8217;s relationship to average rates of change.<\/li>\n<\/ul>\n<\/div>\n<p>As part of exploring how functions change, it is interesting to explore the graphical behavior of functions.\u00a0 Concavity describes the shape of the function and how it is changing.<\/p>\n<p>Consider the graphs below that show the total sales, in thousands of dollars, for two companies over 4 weeks.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2412\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14184936\/14CoABg1-1024x473.png\" alt=\"\" width=\"650\" height=\"300\" \/>As you can see, the sales for each company are increasing, but they are increasing in very different ways.\u00a0 Company A has a lot of sales immediately which could represent the release of a much anticipated product and then the increase in sales levels off. \u00a0Company B starts with slower sales perhaps representing an unknown product which then starts to sell more rapidly perhaps because of word of mouth.\u00a0 To describe the difference in behavior, we can investigate how the average rate of change varies over different intervals.\u00a0 Using tables of values, we can find the average rate of change between consecutive points.\u00a0 For example, in Company A, we can use the first pair of points to get the average rate of change [latex]\\frac{5-0}{1-0}=5[\/latex] and the second pair of points to get the average rate of change [latex]\\frac{7.1-5}{2-1}=2.1.[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2406 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14180001\/14Fig1.png\" alt=\"\" width=\"619\" height=\"301\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>From the tables, we can see that the rate of change for company A is <em>decreasing<\/em>, while the rate of change for company B is <em>increasing<\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2409\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14181537\/14CoABGraphic.png\" alt=\"\" width=\"665\" height=\"300\" \/>For an increasing function, when the rate of change is decreasing, as with Company A, we say the function is <strong>concave down<\/strong>.\u00a0 For an increasing function, when the rate of change is increasing, as with Company B, we say the function is <strong>concave up<\/strong>.<\/p>\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<div>\n<p>A function is <strong>concave up<\/strong> if the rate of change is increasing.<\/p>\n<p>A function is <strong>concave down<\/strong> if the rate of change is decreasing.<\/p>\n<p>A point where a function changes from concave up to concave down or vice versa is called an <strong>inflection point<\/strong>.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1: Describe the Concavity<\/h3>\n<div>\n<p>An object is thrown from the top of a building.\u00a0 The object\u2019s height in feet above ground after <em>t<\/em> seconds is given by the function [latex]h(t)=144-16{{t}^{2}}[\/latex] for [latex]0\\le t\\le 3[\/latex].\u00a0 Describe the concavity of the graph.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q962750\">Show Solution<\/span><\/p>\n<div id=\"q962750\" class=\"hidden-answer\" style=\"display: none\">\n<p>Sketching a graph of the function, we can see that the function is decreasing.\u00a0\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-2410\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14182534\/14Ex1G-212x300.png\" alt=\"\" width=\"212\" height=\"300\" \/><\/p>\n<p>We can calculate some rates of change to explore the behavior.\u00a0 For example, the interval [latex]t=0[\/latex] to [latex]t=1[\/latex] has an average rate of change of [latex]\\frac{128-144}{1-0}=-16.[\/latex] The remaining intervals are shown in the table below.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-2407\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14181232\/14Ex1-300x243.png\" alt=\"\" width=\"300\" height=\"243\" \/><\/p>\n<p>Notice that the rates of change are becoming more negative, so the rates of change are <em>decreasing<\/em>.\u00a0 This means the function is concave down.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2: Concavity from a Table of Values<\/h3>\n<p>The value, <em>V<\/em>, of a car after <em>t<\/em> years is given in the table below. Is the value increasing or decreasing? Is the function concave up or concave down?<\/p>\n<table>\n<tbody>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 34.6563px\"><em>t<\/em><\/td>\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">0<\/td>\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">2<\/td>\n<td class=\"border\" style=\"height: 11px;width: 54.6563px;text-align: center\">4<\/td>\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">6<\/td>\n<td class=\"border\" style=\"height: 11px;width: 57.6563px;text-align: center\">8<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 34.6563px\"><em>V(t)<\/em><\/td>\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">28000<\/td>\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">24342<\/td>\n<td class=\"border\" style=\"height: 11px;width: 54.6563px;text-align: center\">21162<\/td>\n<td class=\"border\" style=\"height: 11px;width: 56.6563px;text-align: center\">18397<\/td>\n<td class=\"border\" style=\"height: 11px;width: 57.6563px;text-align: center\">15994<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q567041\">Show Solution<\/span><\/p>\n<div id=\"q567041\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the values <em>V(t)<\/em>, \u00a0are getting smaller as we let [latex]t[\/latex] increase, we can determine that the value of the car is decreasing. We can compute rates of change to determine concavity.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2408 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14181319\/14Ex2-1024x119.png\" alt=\"\" width=\"1024\" height=\"119\" \/><\/p>\n<p>These rate of change values are becoming less negative since they are moving to the right on a number line, so the rates of change are <em>increasing<\/em> meaning\u00a0this function is concave up.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It #1<\/h3>\n<div>\n<p>Is the function described in the table below concave up or concave down?<\/p>\n<table>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 27.7188px\"><em>x<\/em><\/td>\n<td class=\"border\" style=\"width: 44.7188px;text-align: center\">0<\/td>\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">5<\/td>\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">10<\/td>\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">15<\/td>\n<td class=\"border\" style=\"width: 18.7188px;text-align: center\">20<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 27.7188px\"><em>g(x)<\/em><\/td>\n<td class=\"border\" style=\"width: 44.7188px;text-align: center\">10000<\/td>\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">9000<\/td>\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">7000<\/td>\n<td class=\"border\" style=\"width: 35.7188px;text-align: center\">4000<\/td>\n<td class=\"border\" style=\"width: 18.7188px;text-align: center\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q249682\">Show Solution<\/span><\/p>\n<div id=\"q249682\" class=\"hidden-answer\" style=\"display: none\">\n<p>The average rates of change are decreasing so the function is concave down.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2411 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/14183409\/14TryItAns-1024x124.png\" alt=\"\" width=\"1024\" height=\"124\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Graphically, concave down functions bend downwards like a frown, and concave up function bend upwards like a smile.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1882 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08170332\/Screen-Shot-2019-04-08-at-1.03.09-PM-e1554743217160.png\" alt=\"\" width=\"477\" height=\"305\" \/><\/p>\n<div class=\"textbox examples\">\n<h3>Example 3:\u00a0 Determine Intervals of Concavity from a Graph<\/h3>\n<div>\n<p>From the graph shown, estimate the intervals on which the function is concave down and concave up.<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1883 aligncenter\" style=\"font-size: 0.9em\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08171402\/Picture11-300x259.png\" alt=\"\" width=\"300\" height=\"259\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266317\">Show Solution<\/span><\/p>\n<div id=\"q266317\" class=\"hidden-answer\" style=\"display: none\">\n<p>On the far left, the graph is decreasing but concave up, since it is bending upwards.\u00a0 It begins increasing at <em>x<\/em> = -2, but it continues to bend upwards until about <em>x<\/em> = -1.<\/p>\n<p>From <em>x<\/em> = -1 the graph starts to bend downward, and continues to do so until about <em>x<\/em> = 2.\u00a0 The graph then begins curving upwards for the remainder of the graph shown.<\/p>\n<p>From this, we can estimate that the graph is concave up on the intervals $$(-\\infty ,-1)$$ and $$(2,\\infty )$$, and is concave down on the interval $$(-1,2)$$.\u00a0 The graph has inflection points at \u00a0<em>x<\/em> = -1 and <em>x<\/em> = 2.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It #2<\/h3>\n<p>Create a graph of $$f(x)={{x}^{3}}-6{{x}^{2}}-15x+20$$ and use it to estimate the intervals on which the function is concave up and concave down.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q273303\">Show Solution<\/span><\/p>\n<div id=\"q273303\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<p>Looking at the graph of\u00a0$$f(x)={{x}^{3}}-6{{x}^{2}}-15x+20$$, it appears the function is concave down on [latex]\\left(-\\infty,\\text{ }2\\right)[\/latex] and concave up on [latex]\\left(2,\\text{ }\\infty\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2><strong>Behaviors of the Toolkit Functions<\/strong><\/h2>\n<p>We will now return to our toolkit functions and discuss their graphical behavior.<\/p>\n<table>\n<tbody>\n<tr>\n<td class=\"border\">Function<\/td>\n<td class=\"border\">Increasing\/Decreasing<\/td>\n<td class=\"border\">Concavity<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Constant Function<\/p>\n<p>[latex]f(x)=c[\/latex]<\/td>\n<td class=\"border\">Neither increasing nor decreasing<\/td>\n<td class=\"border\">Neither concave up nor down<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Identity Function<\/p>\n<p>[latex]f(x)=x[\/latex]<\/td>\n<td class=\"border\">Increasing<\/td>\n<td class=\"border\">Neither concave up nor down<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Quadratic Function<\/p>\n<p>[latex]f(x)={{x}^{2}}[\/latex]<\/td>\n<td class=\"border\">Increasing on\u00a0[latex](0,\\infty )[\/latex]<\/p>\n<p>Decreasing on\u00a0[latex](-\\infty ,0)[\/latex]<\/p>\n<p>Minimum at <em>x<\/em> = 0<\/td>\n<td class=\"border\">Concave up<\/p>\n<p>[latex](-\\infty ,\\infty )[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Cubic Function<\/p>\n<p>[latex]f(x)={{x}^{3}}[\/latex]<\/td>\n<td class=\"border\">Increasing<\/td>\n<td class=\"border\">Concave down on\u00a0[latex](-\\infty ,0)[\/latex]<\/p>\n<p>Concave up on\u00a0[latex](0,\\infty )[\/latex]<\/p>\n<p>Inflection point at (0,0)<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Reciprocal<\/p>\n<p>[latex]f(x)=\\frac{1}{x}[\/latex]<\/td>\n<td class=\"border\">Decreasing\u00a0[latex](-\\infty ,0)\\cup (0,\\infty )[\/latex]<\/td>\n<td class=\"border\">Concave down on\u00a0[latex](-\\infty ,0)[\/latex]<\/p>\n<p>Concave up on\u00a0[latex](0,\\infty )[\/latex]<\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Reciprocal squared<\/p>\n<p>[latex]f(x)=\\frac{1}{{{x}^{2}}}[\/latex]<\/td>\n<td class=\"border\">Increasing on\u00a0[latex](-\\infty ,0)[\/latex]<\/p>\n<p>Decreasing on\u00a0[latex](0,\\infty )[\/latex]<\/td>\n<td class=\"border\">Concave up on\u00a0[latex](-\\infty ,0)\\cup (0,\\infty )[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Cube Root<\/p>\n<p>[latex]f(x)=\\sqrt[3]{x}[\/latex]<\/td>\n<td class=\"border\">Increasing<\/td>\n<td class=\"border\">Concave down on\u00a0[latex](0,\\infty )[\/latex]<\/p>\n<p>Concave up on\u00a0[latex](-\\infty ,0)[\/latex]<\/p>\n<p>Inflection point at (0,0)<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Square Root<\/p>\n<p>[latex]f(x)=\\sqrt[{}]{x}[\/latex]<\/td>\n<td class=\"border\">Increasing on\u00a0[latex](0,\\infty )[\/latex]<\/td>\n<td class=\"border\">Concave down on\u00a0[latex](0,\\infty )[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Absolute Value<\/p>\n<p>[latex]f(x)=\\left| x \\right|[\/latex]<\/td>\n<td class=\"border\">Increasing on\u00a0[latex](0,\\infty )[\/latex]<\/p>\n<p>Decreasing on\u00a0[latex](-\\infty ,0)[\/latex]<\/td>\n<td class=\"border\">Neither concave up or down<\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul>\n<li>Concavity describes the shape of the curve.\u00a0 If the average rates are increasing on an interval then the function is concave up and if the average rates are decreasing on an interval then the function is concave down on the interval.<\/li>\n<li>A function has an inflection point when it switches from concave down to concave up or visa versa.<\/li>\n<li>Given a graph, intervals of concavity can be estimated by determining where the graph bends up versus where it bends down.<\/li>\n<li>Input values are used when describing intervals of concavity.\u00a0 Endpoint in the interval are not included so the notation uses parenthesis not square brackets.<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n","protected":false},"author":158103,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-396","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/396","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/158103"}],"version-history":[{"count":22,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/396\/revisions"}],"predecessor-version":[{"id":3008,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/396\/revisions\/3008"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/396\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=396"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=396"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=396"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=396"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}