{"id":7985,"date":"2016-10-23t11:25:20","date_gmt":"2016-10-23t18:25:20","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=7985"},"modified":"2016-12-12t18:12:59","modified_gmt":"2016-12-13t02:12:59","slug":"ap-calculus-difference-graph-function","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-difference-graph-function\/","title":{"rendered":"ap calculus: the difference between a graph and a function?"},"content":{"rendered":"<p>did you know that a graph of a function <i>f<\/i> is not the same as the function itself?<\/p>\n<p>it might seem like there is clearly a difference, but sometimes it\u2019s hard to articulate into words.<\/p>\n<p>we have spoken about the definition of a function. simply put, it\u2019s a rule that transforms one real number into another real number. a graph is a geometric representation of that rule.<\/p>\n<h2>a graph is a set.<\/h2>\n<p>if this is true, then according to the definition of a set, a graph is an unordered collection of objects. for this lesson, you need to know a little more about sets: the <b>cartesian product<\/b> of two sets <i>a<\/i> and <i>b<\/i> is again a set, denoted \u00a0<i>a<\/i>\u00a0x\u00a0<i>b\u00a0and read \u201ca cross b.\u201d it is the set of all elements of the form <i>(a, b)<\/i> with\u00a0<a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/11\/wtdbagaaf_img1.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-133\" alt=\"wtdbagaaf_img1\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/11\/wtdbagaaf_img1.png\" width=\"87\" height=\"21\" \/><\/a><\/i><\/p>\n<p>the cartesian product is named after the famous french philosopher <a href=\"http:\/\/en.wikipedia.org\/wiki\/ren%c3%a9_descartes\">ren\u00e9 descartes<\/a>:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/11\/wtdbagaaf_img2.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-134\" alt=\"wtdbagaaf_img2\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/11\/wtdbagaaf_img2.png\" width=\"245\" height=\"300\" \/><\/a><\/p>\n<h2>the points in a graph<\/h2>\n<p>the objects in the graph of a function are points\u00a0<a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/11\/wtdbagaaf_img3.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-135\" alt=\"wtdbagaaf_img3\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/11\/wtdbagaaf_img3.png\" width=\"105\" height=\"20\" \/><\/a>\u00a0ordered pairs of real numbers in the cartesian product of the set of real numbers with itself. \u00a0we call these points <b>cartesian coordinates.<\/b><\/p>\n<p>we represent these points geometrically in what is known as the <b>cartesian plane,<\/b> or simply the <b>plane<\/b>:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/11\/wtdbagaaf_img4.png\"><img decoding=\"async\" class=\"alignnone  wp-image-136\" alt=\"difference between a graph and a function\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/11\/wtdbagaaf_img4.png\" width=\"504\" height=\"467\" \/><\/a><\/p>\n<p>the center of the plane, the point (0, 0) is called the <b>origin<\/b>, the horizontal axis is called the <b>x-axis<\/b> while the vertical axis is known as the <b>y-axis.<\/b> the two axes naturally divide the plane into <b>quadrants<\/b>. if a point <i>(x, y)<\/i> has both <i>x, y<\/i> positive, we say that <i>(x, y)<\/i> lies in the first quadrant; if (<i>x, y) <\/i>has <i>x<\/i> negative and <i>y<\/i> positive, then <i>(x, y)<\/i> lies in the second quadrant, etc.<\/p>\n<h2>the graph of a function<\/h2>\n<p>given a function <i>f <\/i>whose domain is the set of real numbers and whose codomain is the set of real numbers, we say that the graph is the set of all points in the set\u00a0<a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/11\/wtdbagaaf_img5.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-137\" alt=\"ap calculus\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/11\/wtdbagaaf_img5.png\" width=\"48\" height=\"18\" \/><\/a>\u00a0of the form <i>(x, f(x))<\/i> where <i>x <\/i>is a point in the domain of <i>f<\/i>.<\/p>\n<p>therefore the graph is a set that is unique for a given function, which geometrically represents the function. we can often use the graph of a function in order to deduce properties of said function.<\/p>\n<h2>conclusion<\/h2>\n<p>the graph of a function and a function are closely related but not the same. therefore when you are explaining a solution to a problem, make sure that you use \u201cthe function\u201d and \u201cthe graph of the function\u201d in the right places, depending on which you really mean.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>did you know that a graph of a function f is not the same as the function itself? it might seem like there is clearly a difference, but sometimes it\u2019s hard to articulate into words. we have spoken about the definition of a function. simply put, it\u2019s a rule that transforms one real number into [&hellip;]<\/p>\n","protected":false},"author":48,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24929],"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: the difference between a graph and a function? - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"here we define and express the difference between a graph and a function in simple terms! although closely related, they are not the same!\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-difference-graph-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_us\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"ap calculus: the difference between a graph and a function?\" \/>\n<meta property=\"og:description\" content=\"here we define and express the difference between a graph and a function in simple terms! although closely related, they are not the same!\" \/>\n<meta property=\"og:url\" content=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-difference-graph-function\/\" \/>\n<meta property=\"og:site_name\" content=\"magoosh blog | high school\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/magooshsat\/\" \/>\n<meta property=\"article:published_time\" content=\"2016-10-23t18:25:20+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2016-12-13t02:12:59+00:00\" \/>\n<meta property=\"og:image\" content=\"\/\/www.catharsisit.com\/ap\/files\/2013\/11\/wtdbagaaf_img1.png\" \/>\n<meta name=\"author\" content=\"christopher wirick\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@magooshsat_act\" \/>\n<meta name=\"twitter:site\" content=\"@magooshsat_act\" \/>\n<meta name=\"twitter:label1\" content=\"written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"christopher wirick\" \/>\n\t<meta name=\"twitter:label2\" content=\"est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"article\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-difference-graph-function\/#article\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-difference-graph-function\/\"},\"author\":{\"name\":\"christopher wirick\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/5efe20458a8ff2240fb51dd6812c21c6\"},\"headline\":\"ap calculus: the difference between a graph and a function?\",\"datepublished\":\"2016-10-23t18:25:20+00:00\",\"datemodified\":\"2016-12-13t02:12:59+00:00\",\"mainentityofpage\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-difference-graph-function\/\"},\"wordcount\":433,\"commentcount\":0,\"publisher\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#organization\"},\"keywords\":[\"ap calculus\"],\"articlesection\":[\"ap\"],\"inlanguage\":\"en-us\",\"potentialaction\":[{\"@type\":\"commentaction\",\"name\":\"comment\",\"target\":[\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-difference-graph-function\/#respond\"]}]},{\"@type\":\"webpage\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-difference-graph-function\/\",\"url\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-difference-graph-function\/\",\"name\":\"ap calculus: the difference between a graph and a function? 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