8-3 Rational Functions and Their Graphs

Quick Review

The rational function f ( x ) = P ( x ) Q ( x ) f , open x close , equals . fraction p , open x close , over q , open x close end fraction  has a point of discontinuity for each real zero of Q(x).

If P(x) and Q(x) have

• no common factors, then f(x) has a vertical asymptote when Q(x) = 0.
• a common real zero a, then there is a hole or a vertical asymptote at x = a . x equals eh .
• degree of P ( x ) <  degree  p open x close less than , degree  of Q(x), then the graph of f(x) has a horizontal asymptote at y = 0. y equals 0.
• degree of P(x) = degree of Q(x), then there is a horizontal asymptote at y = a b , y equals , eh over b , comma  where a and b are the coefficients of the terms of greatest degree in P(x) and Q(x), respectively.
• degree of P ( x ) >  degree  p open x close greater than , degree  of Q(x), then there is no horizontal asymptote.

Example

Find any points of discontinuity for the graph of the rational function y = 2.5 x + 7 . y equals . fraction 2.5 , over x plus 7 end fraction . .  Describe any vertical or horizontal asymptotes and any holes.

There is a vertical asymptote at x = − 7 x equals negative 7  and a horizontal asymptote at y = 0. y equals 0.

Exercises

Find any points of discontinuity for each rational function. Sketch the graph. Describe any vertical or horizontal asymptotes and any holes.

19. y = x − 1 ( x + 2 ) ( x − 1 ) y equals . fraction x minus 1 , over open , x plus 2 , close . open , x minus 1 , close end fraction
20. y = x 3 − 1 x 2 − 1 y equals . fraction x cubed , minus 1 , over x squared , minus 1 end fraction
21. y = 2 x 2 + 3 x 2 + 2 y equals . fraction 2 , x squared , plus 3 , over x squared , plus 2 end fraction
22. The start-up cost of a company is $150,000. It costs $.17 to manufacture each headset. Graph the function that represents the average cost of a headset. How many must be manufactured to result in a cost of less than $5 per headset?

8-4 Rational Expressions

Quick Review

A rational expression is in simplest form when its numerator and denominator are polynomials that have no common factors.

Example

Simplify the rational expression. State any restrictions on the variable.

Image Long Description

Exercises

Simplify each rational expression. State any restrictions on the variable.

23. x 2 + 10 x + 25 x 2 + 9 x + 20 fraction x squared , plus 10 x plus 25 , over x squared , plus 9 x plus 20 end fraction
24. x 2 − 2 x − 24 x 2 + 7 x + 12 ⋅ x 2 − 1 x − 6 fraction x squared , minus 2 x minus 24 , over x squared , plus 7 x plus 12 end fraction . dot . fraction x squared , minus 1 , over x minus 6 end fraction
25. 4 x 2 − 2 x x 2 + 5 x + 4 ÷ 2 x x 2 + 2 x + 1 fraction 4 , x squared , minus 2 x , over x squared , plus 5 x plus 4 end fraction . divides . fraction 2 x , over x squared , plus 2 x plus 1 end fraction
26. What is the ratio of the volume of a sphere to its surface area?

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