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2.3 Polynomial and 2.3 Polynomial and Rational FunctionsRational Functions
• Identify a Polynomial Function• Identify a Rational Function• Find Vertical and Horizontal Asymptotes for
Rational Functions• Review finding x and y intercepts of graphs
DEFINITION:
The line x = a is a vertical asymptote if any of the following limit statements are true:
We will learn about limits in section 3.1
limx a
f x limx a
f x
limx a
f x .limx a
f x
•If c makes the denominator zero, but doesn’t make the numerator zero, then x = c is a vertical asymptote.
•If c makes both the denominator and the numerator zero, then there is a hole at x=c
Example 2
There are Vertical Asymptotes atx = 1 and x = -1.
There isn‘t a vertical asymptote at x = 0.Since 0 makes both the numerator and denominator equal zero, there is a hole where x = 0.
• 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
We will learn about limits in section 3.1.
limx
f x b limx
f x b.
The graph of a rational function may or may not cross a horizontal asymptote. Horizontal asymptotes are found by comparing the degree of the numerator to the degree of the denominator. 3 cases
Same: y = leading coefficient/leading coefficientBOB: y = 0 (bottom degree bigger)TUB: undefined-no H.A. (top degree bigger
Bob and tub are not in the textbook.
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)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) How much would 10 units cost? (c) Identify any intercepts & asymptotes. Graph the function to verify your answers.
C
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