Download The Reciprocal Function Family

The Reciprocal
Function Family
8-2
Vocabulary
Review
1. Place a ✓ in the box if the fractions are reciprocals. Place an ✗ in the box if they
are NOT reciprocals.
✓
7
8
8 and 7
✗
1
1
x and 2x
✓
1
3x and 3x
2. The reciprocal of a positive number is positive.
Yes / No
The reciprocal of a negative number is positive.
Yes / No
A negative number has no reciprocal.
Yes / No
The reciprocal of 0 is undefined.
Yes / No
reciprocal function (noun) rih SIP ruh kul FUNGK shun
Related Terms: reciprocal (noun), multiplicative inverse (noun), asymptote (noun),
improper fraction (noun)
Main Idea: A reciprocal function is any function that contains a fraction with the
independent variable in the denominator.
5
1
1
Examples: y 5 3x 1 1, f (x) 5 x 2 2 , g(x) 5 2x 2 8
x
Nonexamples: y 5 x 2 2, f (x) 5 2 , g(x) 5
x21
4x
5 , h(x) 5 8x 1 5
Use Your Vocabulary
Complete each equation with the given expressions to make a reciprocal function.
3. 5 and 3x
y5
6
3x
Chapter 8
4. 7 and x
5. (x 1 1) and 2
2
7
1 5
6. (x 2 8) and 1
f (x) 5
y5
x11
x
214
1
23
f (x) 5
x28
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Vocabulary Builder
Problem 1 Graphing an Inverse Variation Function
Got It? What is the graph of y 5 12
x ? Identify the x- and y-intercepts and the
asymptotes of the graph. Also, state the domain and range of the function.
7. Circle the graph that shows y 5 12
x.
y
y
y
8
8
8
4
4
4
x
8
4
O
4
x
8
8
4
O
4
x
8
8
4
O
4
4
4
8
8
8
4
8
8. Underline the correct word or words to complete the sentences about the graph.
The value of x cannot be 0, so there is no x- / y- intercept.
The value of y cannot be 0, so there is no x- / y- intercept.
The x-axis is the horizontal / vertical asymptote.
The y-axis is the horizontal / vertical asymptote.
9. The domain is the set of all real numbers except x 5
0 .
Key Concepts The Reciprocal Function Family
Parent function: y 5 1x , x 2 0
stretches by a factor of a.
shrinks by a factor of a.
reflects across the x-axis.
shifts horizontally
h units
and has a
shifts vertically
k units
฀฀฀฀฀฀฀฀฀฀฀
x h
215
Combined
y
k; x h.
vertical asymptote
at x h.
horizontal
asymptote at y k.
a
x
h
฀k,
where x h
2
2
The graph of y
1
฀฀฀฀฀
x
2
Translations occur based on the values of h and k in y
1
2
a
x
2
a
1,
When 0 ฀ a ฀ ฀1, the graph of y
a ฀0,
a
x , x 0.
2
2
Stretches and compressions occur based on the value of a in y
2
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
The range is the set of all real numbers except y 5 0 .
Lesson 8-2
Problem 3
Graphing a Translation
1
Got It? What is the graph of y 5 x 2
4 1 6? Identify the domain and range.
10. Find the equations of the asymptotes. Then sketch the asymptotes on the graph.
a
1
y xh k x4 6
The vertical asymptote is x 4 .
The horizontal asymptote is y 6 .
11. Graph the parent function, y 5 x1 , on the coordinate plane at
the right.
y
8
6
4
2
12. Translate the points (1, 1) and (21, 21) from the graph of y 5 1x .
Each point moves 4 units to the right and 6 units up.
1
13. Draw the branches of y 5 x 2
4 1 6 through the translated points.
14. Find the domain and range.
Domain: the set of all real numbers except x 5
Range: the set of all real numbers except y 5
6
Writing the Equation of a Transformation
Got It? This graph of a function is a translation of the graph of y 5 2x .
4
What is an equation for the function?
15. Identify the asymptotes: x 5
So h 5
x
1 and y 5 24
8 4
y5x2h1k
y5
2
8
1 24
x2 1
17. The simplified form of the equation is y 5
2
x21
24
.
Problem 5 Using a Reciprocal Function
Got It? The junior class is renting a laser tag facility with a capacity of 325 people.
The cost for the facility is $1200. The party must have 13 adult chaperones. If every
student who attends shares the facility cost equally, what function models the
cost per student C with respect to the number of students n who attend? What is
the domain of the function? How many students must attend to make the cost per
student no more than $7.50?
Chapter 8
O
1 and k 5 24 .
16. Substitute the values of h and k into the general form of a
translated reciprocal function.
a
x
2 4 6 8
216
4
8
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Problem 4
4
8 6 4 2 O
2
4
6
8
18. Complete the model.
C, cost per
is
student
Relate
n, the number of
students who attend
$1200 divided by
Capacity
Number of
chaperones
1200
Write
C
1200
or
n
C
n
19. The domain of this function is
#n#
1
20. Label the functions y 5 7.5 and y 5
312 .
1200
x in the graph at the right.
y 7.5
21. For the cost per student to be no more than $7.50,
160
Intersection
X160 Y7.5
students must attend.
y 1200
x
Lesson Check • Do you UNDERSTAND?
Write an equation of the reflection of the graph y 5 1x across the x-axis.
22. Assume y 5 ax is a reflection across the x-axis of y 5 1x . Circle the true statement.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
a must be a fraction between 0 and 1.
a must be negative.
23. Complete the equation with values of your choosing so the graph is a reflection
of y 5 x1 across the x-axis. Place a zero in any box you choose to leave blank.
y5
21
1
x2
The value of the numerator must be
negative. Sample values are given.
0
0
Math Success
Check off the vocabulary words that you understand.
reciprocal function
asymptote
translation
Rate how well you can graph translations of reciprocal functions.
Need to
review
0
2
4
6
8
Now I
get it!
10
217
Lesson 8-2