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Rational Functions and their Graphs

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Definition Rational Function

A rational function f(x) is a function that can be written as

where P(x) and Q(x) are polynomial functions and Q(x) ≠ 0.

f(x) = P(x)Q(x)

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Examples of Rational Functions

In this graph, there is no value of x that makes the denominator 0. The graph is continuous because it has no jumps, breaks, or holes in it. It can be drawn with a pencil that never leaves the paper.

y = -2xx2 + 1

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In this graph, x cannot be 4 or -4 because then the denominator would equal 0.

y = 1x2 - 16

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In this graph, x cannot equal 1 or the denominator would equal 0.

y = (x+2)(x-1) x - 1

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Example of Point of Discontinuity

• Consider a function .

• This function is undefined for x = 2. But the simplified rational expression of this function, x + 3 which is obtained by canceling (x - 2) both in the numerator and the denominator is defined at x = 2. Thus we can say that the function f(x) has a point of discontinuity at x = 2.

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Example 1 Finding Points of Discontinuity

1x2 - 16

x2 - 1 x2 + 3

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Vertical Asymptotes

• An asymptote is a line that the curve approaches but does not cross. The equations of the vertical asymptotes can be found by finding the roots of q(x). Completely ignore the numerator when looking for vertical asymptotes, only the denominator matters.

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Example 1 Finding Points of Discontinuity

1x2 + 2x +1

-x + 1 x2 +1

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Example 2 Finding Vertical Asymptotes

x + 1(x – 2)(x – 3)

(x – 2) (x – 1) x - 2

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Example 2 Finding Vertical Asymptotes

(x – 3)(x + 4)(x – 3)(x – 3)(x+4)

x – 2 (x - 1)(x + 3)

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Horizontal Asymptotes

• The graph of a rational function has at most one HA.• The graph of a rational function has a HA at y=0 if the

degree of the denominator is greater than the degree of the numerator .

• If the degrees of the numerator and the denominator are =, then the graph has a HA at y = , a is the coefficient of the term of the highest degree in the numerator and b is the coefficient of the term of the highest degree in the denominator.

• If the degree of the numerator is greater than the degree of the denominator, then the graph has no HA

ab

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Example 1 Sketching Graphs of HA

y = x + 2(x+3)(x-4)

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Example 2 Sketching Graphs of HA

y = x + 3(x-1)(x-5)

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Example 5 Real World Connection

The CD-ROMs for a computer game can be manufactured for $.25 each. The development cost is $124,000. The first 100 discs are samples and will not be sold.

a. Write a function for the average cost of a salable disc. Graph the function.

b. What is the average cost if 2000 discs are produced? If 12,800 discs are produced?