Name ________________________________________ Date __________________ Class__________________
LESSON
5-1
Reteach
Variation Functions
The variable y varies directly as the variable x if
k is called the constant of variation.
y  kx for some constant k.
To solve direct variation problems:
• Use the known x and y values in the problem to solve for k.
• Write the direct variation equation, substituting the value for k.
• Use the direct variation equation to solve for the missing variable.
If y varies directly as x, and y  52 when x  4, find y when x  6.
Step 1
Use y  52 when x  4.
y  kx
52  k · 4
13  k
Step 2
Write the direct
variation equation.
y  kx
y  13x
Step 3
Solve for y when x  6.
y  13x
y  13 · 6
y  78
The variable y varies jointly as the variables x and z if
y  kxz for some constant k.
Joint variation problems are solved
like direct variation problems.
If y varies jointly as x and z, and y  90 when x  36 and z  5, find y when x  40 and z  3.
Step 1
y  kxz
90  k · 36 · 5
90  180k
0.5  k
Step 2
Write the joint
variation equation.
y  kxz
y  0.5xz
Step 3
Solve for y when x  40
and z  3.
y  0.5xz
y  0.5 · 40 · 3
y  60
Solve each problem.
1. If y varies directly as x, and y  30 when x  20, find y when x  50.
a. Step 1:
b. Step 2:
c. Step 3:
y  kx
____________________
____________________
30  k · 20
____________________
____________________
____________________
2. If y varies jointly as x and z, and y  150 when x  2.5 and z  12, find y when
x  4 and z  6.5.
a. Step 1:
b. Step 2:
________________________
_________________________
c. Step 3:
________________________
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5-6
Holt McDougal Algebra 2
Name ________________________________________ Date __________________ Class__________________
LESSON
5-1
Reteach
Variation Functions (continued)
The variable y varies inversely as the variable x if
y
k
for some constant k.
x
If y varies inversely as x, and y  4 when x  30, find y when x  20.
Step 1
Use y  4 when x  30.
k
y
x
k
4
30
120  k
Step 2
Write the inverse
variation equation.
k
y
x
120
y
x
To graph the inverse variation function y 
x
y
x
y
10
12
10
12
20
6
20
6
30
4
30
4
40
3
40
3
Step 3
Solve for y when x  20.
120
y
x
120
y
20
y6
120
, make a table of values.
x
Because the function is undefined
for x  0, make separate tables for
negative and positive x-values.
Solve each problem.
3. If y varies inversely as x, and y  2 when x  9, find y when x  6. Then
graph the inverse variation function.
a. Step 1:
__________________________
b. Step 2:
__________________________
c. Step 3:
__________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
5-7
Holt McDougal Algebra 2
3. $3.00
Problem Solving
4. 15 h
5. 8 bushels
1. a. t =
6. 234 trillionths of a cm
b. k = 975
Reteach
c. It is the total number of student hours
that it takes to build a 7-foot sailboat.
1. a. k = 1.5
b. y = 1.5x
d. 81.25 h
c. y = 1.5x; y = 1.5 ⋅ 50; y = 75
2. 13 students
2. a. y = kxz; 150 = k ⋅ 2.5 ⋅ 12; 150 = 30k,
k=5
b. y = 5xz
b. y =
3. C
4. B
5. D
6. B
Reading Strategies
c. y = 5xz; y = 5 ⋅ 4 ⋅ 6.5; y = 130
3. a. y =
k
s
k
k
; 2 = ; k = 18
x
9
1. Direct
2. Inverse
3. Inverse
4. Direct
18
x
5. Direct
6. a. As her speed increases, the number
of miles she runs increases, so it is
direct variation.
18
18
;y =
;y =3
c. y =
x
6
b. d = ks
7. a. As the number of people at the party
increases, the number of slices per
person decreases, so it is inverse
variation.
b. p =
k
n
5-2 MULTIPLYING AND DIVIDING
RATIONAL EXPRESSIONS
Practice A
Challenge
1. F =
1. Possible answer: If the denominator is
0, then the expression is undefined
because division by 0 is impossible.
Gm1m2
r2
2. About 6.67 × 10−11Nm2per kg2
2. x = 3
kv 2
3. About 4.43 × 10 N 4. H =
r
−7
1
5. Reduce the resistance by a factor of .
3
6. Double the voltage.
k T
7. F =
L
8. It would increase by a factor of
2.
9. The length could be doubled or the
tension cut by a factor of 4.
3.
5
;x≠0
x2
4.
x+3
;x≠0
x2
5.
2
; x ≠ −3
3
6.
2x + 7
; x ≠ −3
x+3
7.
3x
y2
8.
2x 4
3y 2
9.
4
x−2
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A55
Holt McDougal Algebra 2