UNIT 8.2 THE RECIPROCALUNIT 8.2 THE RECIPROCALFUNCTION FAMILYFUNCTION FAMILY

Each of the following equations is a translation of y = |x|. Describe each translation.

1. y = |x| + 2 2. y = |x + 2|

3. y = |x| – 3 4. y = |x – 3|

5. y = |x + 4| – 5 6. y = |x – 10| + 7

Solutions

1. y = |x| + 2 is y = |x| translated 2 units up.

2. y = |x + 2| is y = |x| translated 2 units left.

3. y = |x| – 3 is y = |x| translated 3 units down.

4. y = |x – 3| is y = |x| translated 3 units right.

5. y = |x + 4| – 5 is y = |x| translated 4 units left and 5 units down.

6. y = |x – 10| + 7 is y = |x| translated 10 units right and 7 units up.

Draw a graph of y = .0.5x

The graph has two parts. Each part is called a branch.

Graph the points.

The x-axis is the horizontal asymptote.The y-axis is the vertical asymptote.

Make a table of values that includes positive and negative values of x. Notice that x cannot be 0.

x –10 –5 –2 – – – 2 5 10

y –0.05 –0.1 –0.25 –1 –2 –5 5 2 1 0.25 0.1 0.05

12

14

1 10

1 10

14

12

Connect them with a smooth curve.

Compare the graphs of y = and y = shown below.0.25

x1x

The axes are the asymptotes for both graphs.

Both graphs are symmetric with respect to y = x and y = –x.

What points on the graphs are closest to the origin?

The points, (1, 1), (–1, –1), (0.5, 0.5), and (–0.5, –0.5) are closest to the origin.

The branches of y = are closer to the axis than are the branches of .0.25x

1x

Compare the graphs of y = and y = – shown below.2x

2x

Both graphs are symmetric with respect to y = x and y = –x.

Each is a 90° rotation of the other about the origin.

Graph the functions f = and f = 440.343w

Use the Intersection feature.

The wavelength is about 0.78 m/s.

The frequency f in hertz of a sound wave varies inversely with

its wavelength w. The function f = models the relationship

between f and w for a wave with a velocity of 343 m/s. Find the

wavelength of a sound wave with a frequency of 440 Hz.

343w

Sketch the graph of y = + 2. 1 x + 2

Step 2: Translate y = .

The graph y = includes (1, 1) and (–1, –1).

1x

1x

Step 1: Draw the asymptotes.

For y = + 2, b = –2 and c = 2.

The vertical asymptote is x = –2. The horizontal asymptote is y = 2.

1 x + 2

Translate these points 2 units to the left and 2 units up to (–1, 3) and (–3, 1). Draw the branches through these points.

5. Write an equation for the translation

of y = that has asymptotes at

x = 5 and y = –8.

1.Draw a graph of y = . 2. Compare the graphs of y = and y = .6x

12x

3. A group of college students rents a large two-story house. The amount of rent in dollars that each student pays per month is inversely proportional to the number of students. The rent r per month for one student is related to the number of students n by the equation r = . Find the monthly rent each student pays if there are 8 students in the house.

1200n

4.Sketch the asymptotes and

the graph of y = – – . 0. 5 x + 2

12

$150

13x

12x

Answers may vary. Sample: both graphs lie in quadrants I and III. Both have the axes as asymptotes. Both are symmetric with respect to y = x and y = –x.The graph of

y = can be obtained by stretching the graph of y = by a factor of 2.

6x

12x

y = – 8 13 x – 5

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