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Introduction to Second Derivatives And Beyond - At A Glance:

There are three types of points to find on the function, and the great thing is that you already know how to find all of them.

  • Intercepts: To find the y-intercept of a function, plug x = 0 into the function and see what we find. To find the x-intercepts, also known as roots, we set the function equal to zero and solve. Sometimes we'll also find vertical asymptotes in this step (find where f is undefined).
      
  • Critical points: To find the critical points, set '(x) = 0 (or undefined) and solve.
      
  • Inflection points: To find the inflection points, set "(x) = 0 (or undefined), solve, and check each solution to see if it's a real inflection point.

In summary, we're finding where f, ', and " are zero or undefined. These will mostly be dots, but there may be asymptotes or holes where f is undefined.

Here's the only thing you need to do that we didn't do earlier: after finding the x-value of a CP or IP, plug that x-value back into the original function f to find the corresponding y-value. In order to graph a point, we need to know both coordinates.

Example 1

Find and graph all critical points of the function

f (x) = 3x2 + 4x + 7


Exercise 1

Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.

f (x) = ex

Exercise 2

Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.

f (x) = ( x - 2 )3

Exercise 3

Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.

f (x) = (1-x)ex

Exercise 4

Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.

f (x) = x2ex - 4ex

Exercise 5

Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.

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