{"id":8816,"date":"2017-02-01T11:28:52","date_gmt":"2017-02-01T19:28:52","guid":{"rendered":"https:\/\/magoosh.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":"https:\/\/magoosh.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