Direct Variation
Two variables x and y show a __________________________
provided y = ax and a  0. The non-zero number a is called the
_______________ and y is said to vary directly with x.
Two variables, x and y, vary inversely if they are related as follows:
y
y  ax
Examples: Tell whether the equation represents a direct variation. If
so, identify the constant of variation.
a) y = 3x
d) x + y = 7
b) y = x + 3
c) y = - ½ x
e) 2x + y = 0
5
Inverse Variation
f)
a
,a  0
x
The constant a is the constant of variation, and y is said to vary
inversely with x.
Examples: Tell whether the equation represents an inverse variation.
If so, identify the constant of variation.
a) y 
3
x
b) xy = 10
d) y 
2
x
3
e) y =
c) y  3 x  4
y
1
x
Direct Variation Graphs
6
x
f)
y
7
x
The graph of a direct variation equation is a __________________
with a slope of ____ and a y – intercept of _______.
So the line passes through the __________.
Examples: Graph the direct variation equation.
a) y = ¾ x
b) y = -x
Graphs of Inverse Variation
The graph of an inverse variation model is a curve that DOES NOT
cross an axis.
EX:
How to determine if a set of data represents a direct variation.
How to determine if a set of data represents an inverse variation.
If a set of data represents a direct variation you should be able to write
a function rule in the form y = ax. The ratio _________ should always
Since inverse variation has the form
a
equal _____.
y
x
b)
x
y
a
.
x
A table has inverse
variation if xy gives the same answer for each ordered pair. The
product should always equal _________.
a  xy
Examples: Does the set of data represent a direct variation? If so
write the direct variation equation.
a)
x
y
y
Examples: Does the set of data represent an inverse variation? If so
write the inverse variation equation.
2
4
4
8
6
10
8
12
a)
x 1 2 3 4
5
y 48 24 16 12 9.6
2
-6
3
-9
4
-12
5
-15
b)
Writing direct variation equations given one set of data.
Use the set of data ( , ) to find your constant of variation______.
Write the direct variation equation.
a) x = 6, y = 9
b) x = 3, y = -12
x
y
1 2 3 4 5
2 4 12 16 20
Writing the inverse variation equations given one set of data.
Use the set of data ( , ) to find your constant of variation. Write the
inverse variation equation.
a) x = 9, y = 36
c) The amount of money m you make babysitting varies directly with
the number h of hours you babysit. If you make $24 babysitting for 2
hours, write the direct variation equation. How much would you make
if you babysat for 4.5 hours?
b) x = -21, y = 14
c) When setting up for a party, the hostess noticed the number of
bananas varied inversely as the number of watermelon on each table.
One table had 4 bananas and 5 watermelons. If another table had 7
bananas, how many watermelons would be needed to keep the same
proportions?
Homework:
6
Inverse and Direct Variation
Tell whether the equation represents direct variation, inverse variation or neither.
1. y = -4x
y
=4
x
4.
7. x = -8
For each problem find:
2. xy = 7
3. x = 7y
5. 3x + 4y = 8
6. 2xy = 20
8. 3 x 
12
y
a) the direct variation
9. y = x + 7
b) the inverse variation
10. If y = 21 when x = 10, find y when x = 4.
Direct:
Inverse:
11. If y = 17.5 when x = 12, find y when x = 8.
Direct:
Inverse:
12. To travel a fixed distance, the rate is inversely proportional to the time required. When the rate is
60 miles per hour, the time required is 4 hours. What time would be required for the same distance if
the rate were increased to 80 miles per hour?
13. Tell if the graph is direct, inverse, or neither?
x
y
1 2 3 4 5
3 6 9 12 15
__________________
x 1 2 3 4 5
y 2 4 9 16 25
__________________
x
y
1 2 3 4 6
24 12 8 6 4
__________________
14. Which equation represents an
inverse variation?
18. If m varies directly as p, and m = 5
when p = 7, what is the constant of
variation?
a
b
=
4
9
a
2
B
=
5
b
C 2a + 3 = 4b + 3
a
D =7
b
A
15. Which of these equations is a direct
variation?
F
G
H
J
y = -8
y = -8x
y = -8x + 1
y = -8x – 1
16. If y varies directly as x and the constant of
variation is -2, which equation represents this
relationship?
A y = -2x
2
B y= 
x
x
C y= 
2
D y = 2x
17. A farmer pumps water from an irrigation well to water
his field. The time it takes to water the field varies
inversely with the rate at which the pump operates. It
takes 20 hours to water the field when the pumping rate is
600 gallons per minute. If he adjusts the pump so that it
pumps at a rate of 400 gallons per minute, how long will
it take to water the field?
F 12.5 hours
G 15 hours
H 30 hours
J 40 hours
A 35
B 12
7
C
5
5
D
7
19.