2.3 Polynomial and 2.3 Polynomial and Rational FunctionsRational 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:

limx a

f x limx a

f x

limx a

f x .limx a

f x

•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

f (x) x(x 2)

x(x 1)(x 1)

f (x) (x 2)

(x 1)(x 1)

• 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:

orlimx

f x b limx

f x b.

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 coefficientBOB: y = 0TUB: undefined (no H.A.)

f (x) 3x2 2x 4

2x2 x 1.

Determine the horizontal asymptote of the function given by

Example of holeExample of hole

Figure 45Figure 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)

3x-5

8y of intercepts theFind

5

8y

Suppose the average cost per unit in dollars, to produce x units of a product is given by

30

500

x

xC

)10(C )50(C )100(C(a) find

(b) Graph the function and identifyany intercepts and asymptotes

C