{"id":8816,"date":"2017-02-01t11:28:52","date_gmt":"2017-02-01t19:28:52","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=8816"},"modified":"2019-03-17t18:33:02","modified_gmt":"2019-03-18t01:33:02","slug":"ap-calculus-10-step-guide-curve-sketching","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-10-step-guide-curve-sketching\/","title":{"rendered":"ap calculus: 10-step guide to curve sketching"},"content":{"rendered":"

what can calculus tell us about curve sketching? it turns out, quite a lot! in this article, you’ll see a list of the 10 key characteristics that describe a graph. while you may not be tested on your artistic ability to sketch a curve on the ap calculus exams, you will<\/em> be expected to determine these specific features of graphs.<\/p>\n

\"curve<\/p>\n

guide to curve sketching<\/h2>\n

the ten steps of curve sketching each require a specific tool. but some of the steps are closely related. in the list below, you’ll see some steps grouped if they are based on similar methods. <\/p>\n

    \nalgebra and pre-calculus<\/strong><\/p>\n
  1. domain and range<\/li>\n
  2. y<\/em>-intercept<\/li>\n
  3. x<\/em>-intercept(s)<\/li>\n
  4. symmetry<\/li>\n

    limits<\/strong><\/p>\n

  5. vertical asymptote(s)<\/li>\n
  6. horizontal and\/or oblique asymptote(s)<\/li>\n

    first derivative<\/strong><\/p>\n

  7. increase\/decrease<\/li>\n
  8. relative extrema<\/li>\n

    second derivative<\/strong><\/p>\n

  9. concavity<\/li>\n
  10. inflection points<\/li>\n<\/ol>\n

    some books outline these steps differently, sometimes combining items together. so it’s not uncommon to see “the eight steps for curve sketching,” etc.<\/p>\n

    let’s briefly review what each term means. more details can be found at ap calculus exam review: analysis of graphs<\/a>, for example.<\/p>\n

    step 1. determine the domain and range<\/h3>\n

    the domain<\/strong> of a function f<\/em>(x<\/em>) is the set of all input values (x<\/em>-values) for the function. <\/p>\n

    the range<\/strong> of a function f<\/em>(x<\/em>) is the set of all output values (y<\/em>-values) for the function.<\/p>\n

    methods for finding the domain and range vary from problem to problem. here is a good review<\/a>.<\/p>\n

    step 2. find the y<\/em>-intercept<\/h3>\n

    the y<\/em>-intercept<\/strong> of a function f<\/em>(x<\/em>) is the point where the graph crosses the y<\/em>-axis. <\/p>\n

    this is easy to find. simply plug in 0. the y<\/em>-intercept is: (0, f<\/em>(0<\/em>)).<\/p>\n

    step 3. find the x<\/em>-intercept(s)<\/h3>\n

    an x<\/em>-intercept<\/strong> of a function f<\/em>(x<\/em>) is any point where the graph crosses the x<\/em>-axis. <\/p>\n

    to find the x<\/em>-intercepts, solve f<\/em>(x<\/em>) = 0.<\/p>\n

    step 4. look for symmetry<\/h3>\n

    a graph can display various kinds of symmetry. three main symmetries are especially important: even<\/em>, odd<\/em>, and periodic<\/em> symmetry. <\/p>\n