Rational Functions

Graphing Rational Functions and Finding Asymptotes

Rational functions are functions of the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0. Understanding their graphs involves identifying key features such as asymptotes and intercepts.

• Vertical Asymptotes: Occur at values of x where the denominator Q(x) = 0 and the numerator P(x) ≠ 0.

• Horizontal Asymptotes: Determined by the degrees of P(x) and Q(x). If degrees are equal, the asymptote is y = leading coefficient of P(x) divided by leading coefficient of Q(x).

• Intercepts: The y-intercept is found by evaluating f(0). The x-intercept(s) are found by solving P(x) = 0.

Example: Graphing

• Vertical Asymptote: Set denominator to zero:

• Horizontal Asymptote: Degrees of numerator and denominator are both 1. Asymptote is (since leading coefficients are both 1).

• x-intercept: Set numerator to zero:

• y-intercept:

To graph the function:

1. Draw the vertical asymptote at (dashed line).

2. Draw the horizontal asymptote at (dashed line).

3. Plot the intercepts: and .

4. Choose additional points on either side of the vertical asymptote to sketch the curve.

Summary Table:

Additional info: When graphing rational functions, always check for holes (removable discontinuities) by factoring numerator and denominator and canceling common factors. In this example, there are no holes.