{"id":9192,"date":"2017-02-24t12:04:54","date_gmt":"2017-02-24t20:04:54","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=9192"},"modified":"2017-02-24t12:04:54","modified_gmt":"2017-02-24t20:04:54","slug":"ap-calculus-review-functions-graphs","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-functions-graphs\/","title":{"rendered":"ap calculus review: functions and their graphs"},"content":{"rendered":"<p>at the heart of calculus is the concept of functions and their graphs.  all of the main areas of calculus, including <em>limits<\/em>, <em>derivatives<\/em>, and <em>integrals<\/em>, require a firm understanding of functions.  in this review we will explore what you need to know about functions and their graphs in order to succeed on the ap calculus exams.<\/p>\n<p>the main concepts concerning graphing and functions include:<\/p>\n<ul>\n<li>evaluating functions<\/li>\n<li>domain and range<\/li>\n<li>graphing (including intercepts and asymptotes)<\/li>\n<li>finding limits and determining continuity from the graph<\/li>\n<li>increase, decrease, and relative extrema<\/li>\n<li>concavity and inflection points<\/li>\n<\/ul>\n<h2>functions and their graphs<\/h2>\n<p>let&#8217;s start with the basics. <\/p>\n<p>by definition a <strong>function<\/strong> is just a rule that assigns to each input <em>x<\/em> a well-defined output <em>y<\/em>.  we use the notation <em>y<\/em> = <em>f<\/em>(<em>x<\/em>) to indicate that <em>x<\/em> is the input, <em>y<\/em> is the output, and <em>f<\/em> is the name of the function.<\/p>\n<p>in principle, the input and output could be anything: numbers, vectors, symbols, even other functions! typically in calculus the input and output are <a href=\"https:\/\/www.mathsisfun.com\/numbers\/real-numbers.html\" target=\"_blank\" rel=\"noopener noreferrer\">real numbers<\/a>.<\/p>\n<figure id=\"attachment_7411\" aria-describedby=\"caption-attachment-7411\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/06\/2000px-total_function.svg_-600x600.png\" alt=\"functions and their graphs -magoosh\" width=\"400\" height=\"400\" class=\"size-large wp-image-7411\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2016\/06\/2000px-total_function.svg_-600x600.png 600w, \/\/www.catharsisit.com\/hs\/files\/2016\/06\/2000px-total_function.svg_-300x300.png 300w, \/\/www.catharsisit.com\/hs\/files\/2016\/06\/2000px-total_function.svg_-150x150.png 150w, \/\/www.catharsisit.com\/hs\/files\/2016\/06\/2000px-total_function.svg_-768x768.png 768w, \/\/www.catharsisit.com\/hs\/files\/2016\/06\/2000px-total_function.svg_-1536x1536.png 1536w, \/\/www.catharsisit.com\/hs\/files\/2016\/06\/2000px-total_function.svg_.png 2000w\" sizes=\"(max-width: 400px) 100vw, 400px\" \/><figcaption id=\"caption-attachment-7411\" class=\"wp-caption-text\">a function f from x to y.  f(1) = d, f(2) = d, and f(3) = c.<\/figcaption><\/figure>\n<p>generally a function in calculus is defined by an expression that shows how to calculate the output from any given input.  for example, the squaring function <em>f<\/em>(<em>x<\/em>) = <em>x<\/em><sup>2<\/sup> has input <em>x<\/em> and output <em>y<\/em> = <em>x<\/em><sup>2<\/sup>.  <\/p>\n<h3>evaluating functions<\/h3>\n<p>evaluating a function on a given input is often called <em>plugging in<\/em>.  continuing with our example, <em>f<\/em>(<em>x<\/em>) = <em>x<\/em><sup>2<\/sup>, let&#8217;s see how to evaluate <em>f<\/em> at various inputs.<\/p>\n<ul>\n<li><em>f<\/em>(3) = 3<sup>2<\/sup> = 9.<\/li>\n<li><em>f<\/em>(-1\/5) = (-1\/5)<sup>2<\/sup> = 1\/25.<\/li>\n<li><em>f<\/em>(<em>c<\/em>) = <em>c<\/em><sup>2<\/sup>.<\/li>\n<li><em>f<\/em>(<em>c<\/em> + <em>h<\/em>) = (<em>c<\/em> + <em>h<\/em>)<sup>2<\/sup> = <em>c<\/em><sup>2<\/sup> + 2<em>ch<\/em> + <em>h<\/em><sup>2<\/sup>.<\/li>\n<li>if <em>g<\/em>(<em>x<\/em>) = <em>x<\/em> &#8211; 12, then <em>f<\/em>(<em>g<\/em>(<em>x<\/em>)) = <em>f<\/em>(<em>x<\/em> &#8211; 12) = <em>x<\/em><sup>2<\/sup> &#8211; 24<em>x<\/em> + 144.<\/li>\n<\/ul>\n<p>the last example in the list illustrates a <strong>composition<\/strong> of two functions, <em>f<\/em> and <em>g<\/em>.  understanding how compositions work helps you to make sense of the <a href=\"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/\">chain rule<\/a> for derivatives and the substitution rule for integrals.<\/p>\n<h3>domain<\/h3>\n<p>the <strong>domain<\/strong> of a function <em>f<\/em>(<em>x<\/em>) is the set of all input values (<em>x<\/em>-values) for the function.  <\/p>\n<p>unless further information is given, we look for the <strong>natural domain<\/strong>, which is the largest set of <em>x<\/em>-values that makes sense for the function.  <\/p>\n<p>for example, the domain of <em>f<\/em>(<em>x<\/em>) = <em>x<\/em><sup>2<\/sup> is the set of all real numbers, or (-&infin;, &infin;) in interval notation.  any real number at all can serve as input for this function.<\/p>\n<p>on the other hand, the domain of a rational function such as <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/rational_function.gif\" alt=\"example rational function\" width=\"161\" height=\"42\" class=\"alignnone size-full wp-image-9199\" \/> excludes any <em>x<\/em>-values for which the denominator evaluates to zero.  factoring the bottom in this example,<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/rational_function_factored1.gif\" alt=\"example rational function, factored denominator\" width=\"238\" height=\"45\" class=\"aligncenter size-full wp-image-9198\" \/><\/p>\n<p>we find that <em>x<\/em> = 5 and <em>x<\/em> = -2 must be excluded.  therefore the domain is: (-&infin;, -2) u (-2, 5) u (5, &infin;).<\/p>\n<h3>range<\/h3>\n<p>the <strong>range<\/strong> of a function is the set of all output values (<em>y<\/em>-values) for the function.<\/p>\n<p>finding the range is quite a bit more challenging than finding domain in general.  it helps to memorize the domains and ranges of simple functions first.  <\/p>\n<table id=\"tablepress-96\" class=\"tablepress tablepress-id-96 tablepress-responsive\">\n<thead>\n<tr class=\"row-1 odd\">\n<th class=\"column-1\">function<\/th>\n<th class=\"column-2\">domain<\/th>\n<th class=\"column-3\">range<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n<td class=\"column-1\"><em>x<\/em><sup>2<\/sup><\/td>\n<td class=\"column-2\">(-&infin;, &infin;)<\/td>\n<td class=\"column-3\">[0, &infin;)<\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n<td class=\"column-1\"><span style=\"font-size:larger\">&radic;<span style=\"text-decoration:overline\">&nbsp;<em>x<\/em><\/span><\/span><\/td>\n<td class=\"column-2\">[0, &infin;)<\/td>\n<td class=\"column-3\">[0, &infin;)<\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n<td class=\"column-1\">1\/<em>x<\/em><\/td>\n<td class=\"column-2\">(-&infin;, 0) u (0, &infin;)<\/td>\n<td class=\"column-3\">(-&infin;, 0) u (0, &infin;)<\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n<td class=\"column-1\">sin <em>x<\/em><\/td>\n<td class=\"column-2\">(-&infin;, &infin;)<\/td>\n<td class=\"column-3\">[-1, 1]<\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n<td class=\"column-1\">cos <em>x<\/em><\/td>\n<td class=\"column-2\">(-&infin;, &infin;)<\/td>\n<td class=\"column-3\">[-1, 1]<\/td>\n<\/tr>\n<tr class=\"row-7 odd\">\n<td class=\"column-1\">e<sup>x<\/sup><\/td>\n<td class=\"column-2\">(-&infin;, &infin;)<\/td>\n<td class=\"column-3\">(0, &infin;)<\/td>\n<\/tr>\n<tr class=\"row-8 even\">\n<td class=\"column-1\">ln <em>x<\/em><\/td>\n<td class=\"column-2\">(0, &infin;)<\/td>\n<td class=\"column-3\">(-&infin;, &infin;)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><!-- #tablepress-96 from cache --><\/p>\n<p>then more complicated functions may be analyzed using the properties of scaling and translations.<\/p>\n<p>for example, we can deduce that the range of <em>f<\/em>(<em>x<\/em>) = (<em>x<\/em> + 3)<sup>2<\/sup> &#8211; 7 is (-7, &infin;) because <em>f<\/em> is just a translation of the graph of <em>x<\/em><sup>2<\/sup> by 7 units down (and 3 units left, which doesn&#8217;t affect the range).<\/p>\n<p>here&#8217;s a good refresher for <a href=\"http:\/\/www.purplemath.com\/modules\/fcns2.htm\" target=\"_blank\" rel=\"noopener noreferrer\">domain and range<\/a>.<\/p>\n<h3>graphing<\/h3>\n<p>the <strong>graph<\/strong> of a function <em>f<\/em> is the set of all points (<em>x<\/em>, <em>y<\/em>) that satisfy <em>y<\/em> = <em>f<\/em>(<em>x<\/em>).  in other words, at each <em>x<\/em>-value, plug into the function to determine the corresponding <em>y<\/em>-value.<\/p>\n<p>for example, to graph <em>f<\/em>(<em>x<\/em>) = <em>x<\/em><sup>2<\/sup>, evaluate the function at a few points.  then plot each point (<em>x<\/em>, <em>x<\/em><sup>2<\/sup>) on the coordinate plane and connect the dots with a smooth curve.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/parabola.jpg\" alt=\"parabola graph with points labeled\" width=\"512\" height=\"590\" class=\"aligncenter size-full wp-image-9201\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/parabola.jpg 512w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/parabola-260x300.jpg 260w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/><\/p>\n<p>graphing a function can be time-consuming, but if you&#8217;re working on a section of the ap calculus exam that allows a calculator, then just use your calculator&#8217;s graphing feature.<\/p>\n<h2>intercepts and asymptotes<\/h2>\n<p>the intercepts and asymptotes are important features that help us to accurately graph a function.  in fact,  it&#8217;s also helpful to know how to locate the intercepts and asymptotes directly from a given graph.<\/p>\n<h3>the intercepts of a graph<\/h3>\n<p>the <strong><em>y<\/em>-intercept<\/strong> of a function <em>f<\/em>(<em>x<\/em>) is the point where the graph crosses the <em>y<\/em>-axis.  there can be only one.  simply plug in <em>x<\/em> = 0 to find it.  but if 0 is not in the domain of the function, then it has no <em>y<\/em>-intercept.<\/p>\n<p>an <strong><em>x<\/em>-intercept<\/strong> of a function <em>f<\/em>(<em>x<\/em>) is any point where the graph crosses the <em>x<\/em>-axis. in contrast to the <em>y<\/em>-intercept, there can be many <em>x<\/em>-intercepts for a given graph.  to find them, set <em>f<\/em>(<em>x<\/em>) = 0 and solve.  <\/p>\n<h3>the asymptotes of a graph<\/h3>\n<p>there are three types of asymptote.  <\/p>\n<ul>\n<li>a <strong>vertical asymptote<\/strong> for a function is a vertical line <em>x<\/em> = <em>k<\/em> showing where the function becomes unbounded.<\/li>\n<li>a <strong>horizontal asymptote<\/strong> for a function is a horizontal line that the graph of the function approaches as <em>x<\/em> approaches &infin; or -&infin;.<\/li>\n<li>an <strong>oblique asymptote<\/strong> for a function is a slanted line that the function approaches as <em>x<\/em> approaches &infin; or -&infin;.<\/li>\n<\/ul>\n<h3>example<\/h3>\n<p>below you can see an example graph with the intercepts and asymptotes labeled.<\/p>\n<figure id=\"attachment_9202\" aria-describedby=\"caption-attachment-9202\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/graph-600x560.jpg\" alt=\"graph showing intercepts and asymptotes\" width=\"600\" height=\"560\" class=\"size-large wp-image-9202\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/graph-600x560.jpg 600w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/graph-300x280.jpg 300w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/graph.jpg 639w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><figcaption id=\"caption-attachment-9202\" class=\"wp-caption-text\">this graph has a <em>y<\/em>-intercept at (0, 2), two <em>x<\/em>-intercepts at (-2, 0) and (4, 0), a vertical asymptote at <em>x<\/em> = 2, and an oblique asymptote at <em>y<\/em> = (1\/2)<em>x<\/em>.<\/figcaption><\/figure>\n<p>for more information and examples, check out these article about <a href=\"\/\/www.catharsisit.com\/hs\/ap\/find-horizontal-asymptotes\/\">horizontal asymptotes<\/a>, <a href=\"\/\/www.catharsisit.com\/hs\/ap\/find-vertical-asymptotes-function\/\">vertical asymptotes<\/a>, and <a href=\"\/\/www.catharsisit.com\/hs\/ap\/oblique-asymptotes\/\">oblique asymptotes<\/a>. <\/p>\n<h2>graphs, limits, and continuity<\/h2>\n<p>on the ap calculus exams you will sometimes need to find limits from a given graph.  <\/p>\n<p>for example, in the graph below, we can determine that <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/limit_statement1.gif\" alt=\"the limit of f as x approaches 2 is equal to 3.\" width=\"102\" height=\"25\" class=\"alignnone size-full wp-image-9204\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/limit_statement1.gif 102w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/limit_statement1-30x7.gif 30w\" sizes=\"(max-width: 102px) 100vw, 102px\" \/><\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/rational_function_limit_example1.png\" alt=\"rational function limit\" width=\"285\" height=\"283\" class=\"aligncenter size-full wp-image-8902\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/rational_function_limit_example1.png 285w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/rational_function_limit_example1-150x150.png 150w\" sizes=\"(max-width: 285px) 100vw, 285px\" \/><\/p>\n<p>you can also see that the above graph is continuous everywhere <em>except<\/em> at <em>x<\/em> = -1 (where there is a vertical asymptote) and <em>x<\/em> = 2 (where the limit value does not equal the function value).  thus, this function is continuous on (&infin;, -1) u (-1, 2) u (2, &infin;).<\/p>\n<p>for more information, check out <a href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-exam-review-limits-continuity\/\">ap exam review: limits and continuity<\/a>.<\/p>\n<h2>using derivatives to analyze functions and their graphs<\/h2>\n<p>part of the what makes calculus so useful is its ability to analyze the behavior of functions without having the graph as a guide.  using derivatives (and second derivatives), one can determine:<\/p>\n<ul>\n<li>on what interval(s) does the graph increase or decrease?<\/li>\n<li>where are the relative extrema (relative minimum and maximum points)?<\/li>\n<li>on what interval(s) is the graph concave up or down?<\/li>\n<li>where are the points of inflection?<\/li>\n<\/ul>\n<p>check out the <a href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-exam-review-analysis-graphs\/\">ap calculus exam review: analysis of graphs<\/a> for further explanation of these more advanced topics.<\/p>\n<h2>summary<\/h2>\n<p>there is a lot to know about functions and their graphs.  this short review just gives the basics.  now it&#8217;s up to you to explore further!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>functions and their graphs are the heart of calculus. click here to review what you need to know about functions and their graphs on the ap calculus exams.<\/p>\n","protected":false},"author":223,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24932],"class_list":["post-9192","post","type-post","status-publish","format-standard","hentry","category-ap","tag-ap-calculus"],"acf":[],"yoast_head":"<!-- this site is optimized with the yoast seo premium plugin v21.7 (yoast seo v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>ap calculus review: functions and their graphs - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"functions and their graphs are the heart of calculus. click here to review what 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