{"id":4256,"date":"2022-06-10T19:04:32","date_gmt":"2022-06-10T19:04:32","guid":{"rendered":"https:\/\/mdr.foobrdigital.com\/?p=4256"},"modified":"2022-06-10T19:04:32","modified_gmt":"2022-06-10T19:04:32","slug":"graphing-rational-functions","status":"publish","type":"post","link":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/2022\/06\/10\/graphing-rational-functions\/","title":{"rendered":"Graphing Rational Functions"},"content":{"rendered":"\n<p>Here are the steps for graphing a rational function:<\/p>\n\n\n\n<ol><li>Identify and draw the vertical asymptote using a dotted line.<\/li><li>Identify and draw the horizontal asymptote using a dotted line.<\/li><li>Plot the holes (if any)<\/li><li>Find\u00a0x-intercept\u00a0(by using y = 0) and\u00a0y-intercept\u00a0(by x = 0) and plot them.<\/li><li>Draw a table of two columns x and y and place the x-intercepts and vertical asymptotes in the table. Then take some random numbers in the x-column on either side of each of the x-intercepts and vertical asymptotes.<\/li><li>Compute the corresponding y-values by substituting each of them in the function.<\/li><li>Plot all points from the table and join them curves without touching the asymptotes.<\/li><\/ol>\n\n\n\n<p><strong>Example:<\/strong>&nbsp;Graph the rational function f(x) = (x<sup>2<\/sup>&nbsp;+ 5x + 6) \/ (x<sup>2<\/sup>&nbsp;+ x &#8211; 2).<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>We have already identified that its VA is x = 1, its HA is y = 1, and the hole is at (-2, -1\/3). We use dotted lines for asymptotes so that we can take care that the graph doesn&#8217;t touch those lines. Note that, the simplified form of the given function is, f(x) = (x + 3) \/ (x &#8211; 1). Now, we will find the intercepts.<\/p>\n\n\n\n<ul><li>For x-intercept, put y = 0. Then we get 0 = (x + 3) \/ (x &#8211; 1) \u21d2 x + 3 = 0 \u21d2 x = -3. So the x-intercept is at (-3, 0).<\/li><li>For y-intercept, put x = 0. Then we get y = (0 + 3) \/ (0 &#8211; 1) \u21d2 y = -3. So the y-intercept is at (0, -3).<\/li><\/ul>\n\n\n\n<p>We have the VA at x = 1 and x-intercept is at x = -3. Let us construct a table now with these two values in the column of x and some random numbers on either side of each of these numbers -3 and 1.<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th scope=\"col\">x<\/th><th scope=\"col\">y<\/th><\/tr><\/thead><tbody><tr><td>-5<\/td><td>y = (-5 + 3) \/ (-5 &#8211; 1) = 0.33<\/td><\/tr><tr><td>-4<\/td><td>y = (-4 + 3) \/ (-4 &#8211; 1) = 0.2<\/td><\/tr><tr><td><strong>-3<\/strong><\/td><td><strong>0 (x-int)<\/strong><\/td><\/tr><tr><td>-2<\/td><td>y = (-2 + 3) \/ (-2 &#8211; 1) = -0.33<\/td><\/tr><tr><td>0<\/td><td>-3 (y-int)<\/td><\/tr><tr><td><strong>1<\/strong><\/td><td><strong>VA<\/strong><\/td><\/tr><tr><td>2<\/td><td>y = (2 + 3) \/ (2 &#8211; 1) = 5<\/td><\/tr><tr><td>3<\/td><td>y = (3 + 3) \/ (3 &#8211; 1) = 3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Let us plot all these points on the graph along with all asymptotes, hole, and intercepts.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/mdr.foobrdigital.com\/wp-content\/uploads\/2022\/06\/rational-function-graph-1642146935.png\" alt=\"\" class=\"wp-image-4257\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Here are the steps for graphing a rational function: Identify and draw the vertical asymptote using a dotted line. Identify and draw the horizontal asymptote using a dotted line. Plot the holes (if any) Find\u00a0x-intercept\u00a0(by using y = 0) and\u00a0y-intercept\u00a0(by x = 0) and plot them. Draw a table of two columns x and y [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[715],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/4256"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=4256"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/4256\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=4256"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=4256"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=4256"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}