Limits at Infinity and Asymptotes: Learn It 6

Drawing Graphs of Functions

Guidelines for Graphing a Function

We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let’s look at a general strategy to use when graphing any function.

How To: Draw the Graph of a Function

Given a function ff use the following steps to sketch a graph of f:

  • Step 1: Determine the domain of the function.
  • Step 2: Locate the x– and y-intercepts.
  • Step 3: Evaluate limxf(x) and limxf(x) to determine the end behavior.*
  • Step 4: Determine whether f has any vertical asymptotes.
  • Step 5: Calculate f. Find all critical points and determine the intervals where f is increasing and where f is decreasing. Determine whether f has any local extrema.
  • Step 6: Calculate f. Determine the intervals where f is concave up and where f is concave down. Use this information to determine whether f has any inflection points. The second derivative can also be used as an alternate means to determine or verify that f has a local extremum at a critical point.

*Note for Step 3: If either of these limits is a finite number L, then y=L is a horizontal asymptote. If either of these limits is or , determine whether f has an oblique asymptote. If f is a rational function such that f(x)=p(x)q(x), where the degree of the numerator is greater than the degree of the denominator, then f can be written as

f(x)=p(x)q(x)=g(x)+r(x)q(x),

where the degree of r(x) is less than the degree of q(x). The values of f(x) approach the values of g(x) as x±. If g(x) is a linear function, it is known as an oblique asymptote.

Now let’s use this strategy to graph several different functions. We start by graphing a polynomial function.

Sketch a graph of f(x)=(x1)2(x+2)

Sketch the graph of f(x)=x21x2

Sketch the graph of f(x)=x2x1

Sketch a graph of f(x)=(x1)23