3-3:Rational 3-3:Rational FunctionsFunctions

3-3:Rational 3-3:Rational FunctionsFunctions

English CasbarroEnglish CasbarroUnit 3: Rational FunctionsUnit 3: Rational Functions

The first thing you are asked to do is substitute values into the original equation.What did you notice about f(–2) and f(3)?

You are next asked to reduce the fraction by factoring. Now, find f(–2) and f(3) again.What did you notice here?

The function has a removable discontinuity at x = –2 . This is because when you cancel the factor out, you can no longer see that there is a discontinuity there. This creates a hole in your graph at x = –2 .

The function has an asymptote at x = 3 because itIs factored and cancelled as much as possible. Thediscontinuity at x = 3 will remain in the graph.

Discontinuities

For the graph, , you must use the information you have been given:

1) Factor all polynomials2) Cancel out any factors that match

In order to graph this function, you must cancel out everything first. Then you can use the information on the following pages to graph the function.

Warm-up Review1. For a car moving with initial speed v0 and acceleration a, the distance d

that the car travels in time t is given by d = v0t + ½at2.a. Write a rational expression in terms of t for the average speed of the car

during a period of acceleration. Simplify the expression.b. During a race, a driver accelerates for 3s at a rate of 10 ft/s2 in order to

pass another car. The driver’s initial speed was 264 ft/s. What was the driver’s average speed during acceleration?

2. In an auto race, a car with an average speed of 200 mi/h takes an average of 31.5 s to complete one lap of the track.

a. Write an inverse variation function that gives the average speed s of a car in miles per hour as a function of the time t in seconds needed to complete one lap.

b. How many seconds does it take the car to complete one lap at an average speed of210 mi/h?

3. For which values of x is the expression undefined?

A. 0 and 1 B. 1 and 2 C. –1 and 2 D. –2 and 1

The parent function of rational functions is . The function

below should look familiar. The a is still the vertical stretch or compression.The k is still the vertical translation up or down. And the h is still the horizontalmovement, the opposite of the sign that you see.

How the graph is stretched or compressed

Moves the graph up or down k units (sign agrees with move)

Moves the graph left or right (sign is opposite the move)

Your standard function goes through the points (1,1) and ( –1, –1). You can use those points as “touchstones” for your moves.

Properties of the rational function

The graph is a hyperbolaThere is a vertical asymptote at x = h, and the domain will be , x ≠ hThere is a horizontal asymptote at y = k, and the range is , y ≠ k.

Turn in the following problems1. The average cost per DVD purchased from a movie club is a function of the number of DVDs the member buys. a. the function that describes the data in the table is described by What is the value of k? b. What is the total cost of buying 15 DVDs?

2. A student wrote the following description for the graph of

Explain the error. Write the correct description of the graph. The graph has vertical asymptotes at x = 1, and x = – 1, and a horizontal asymptote at y = 2. 3. The graph of which of the following functions has a hole?