2.3 Polynomial and Rational Functions

Polynomial and rational functions are often used to express relationships in application problems.

DEFINITION: The line x = a is a vertical asymptote if any of the following limit statements are true:

If a makes the denominator zero, but doesn’t make the numerator zero, then x = a is a vertical asymptote. If a makes both the denominator and the numerator zero, then there is a hole at x=a

Example 2: Determine the vertical asymptotes of the function given by

Since x = 1 and x = –1 make the denominator 0, but don’t make the numerator 0, x = 1 and x = –1 are vertical asymptotes. x=0 is not a vertical asymptote since it makes both the numerator and denominator 0.

The line y = b is a horizontal asymptote if either or both of the following limit statements are true: or

The graph of a rational function may or may not cross a horizontal asymptote. Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator. Same: y = leading coefficient/leading coefficient BOB: y = 0 TUB: undefined (no H.A.)

Determine the horizontal asymptote of the function given by

Example of hole

Figure 45

Intercepts. The x-intercepts occur at values for which y = 0. For a fraction to = 0, the numerator must equal 0. Since 8 ≠ 0, there are no x-intercepts. To find the y-intercept, let x = 0. y-intercept (0, 8/5)

Suppose the average cost per unit in dollars, to produce x units of a product is given by (a) find (b) Graph the function and identify any intercepts and asymptotes