AP Calculus AB: 8.5.3 Graphing Functions with Asymptotes

Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero. A vertical asymptote to the graph of a function f is a line whose equation is x = a, where
, or .
• Identify horizontal asymptotes by taking the limit of the function as x approaches positive or negative infinity. A horizontal asymptote to the graph of a function f is a line whose equation is y = a, where , or .
• The behavior of a function can change from one side of a vertical asymptote to the other.

• When sketching the graph of a rational function, you should first look for asymptotes.

• Since this expression cannot be factored further and none of the factors cancel, set the denominator equal to 0. Solving for x indicates that the function has a vertical asymptote at x = 3.

• Then take the limit of the function as x approaches . For the sake of this limit, you can ignore the constants. Canceling produces 1 for the limit. Thus the function has a horizontal asymptote at y = 1.

• Next, you can find the first and second derivatives and determine the behavior of the function.

• The first derivative of the function never equals 0. It is undefined at x = 3, but that is not a critical point because the function is not defined there. When you make your sign chart, mark x = 3 as the vertical asymptote.

• The second derivative never equals 0 either. It is also
  undefined at x = 3, the vertical asymptote. Your sign chart should reflect this.

• Although the concavity changes at x = 3, the function is not defined there, so there is no inflection point.

• From the first derivative you can tell that the function is decreasing both to the left of the vertical asymptote and to its right. The second derivative indicates that the function is concave down on the left and concave up on the right.

The graph of f(x)=13xf(x) = \frac{1}{3}x is a straight line passing through the origin with slope 13\frac{1}{3}.

The graph of f(x)=xx−2f(x) = \frac{x}{x-2} has a vertical asymptote at x=2x=2 and a horizontal asymptote at y=1y=1, with the curve approaching but never touching these lines.

The graph of f(x)=x3−x=−23xf(x) = \frac{x}{3} - x = -\frac{2}{3}x is a straight line through the origin with slope −23-\frac{2}{3}.