4.3
Direct and Partial Variation
Try These
The rate of change for a number pattern is 8. What could the pattern be?
e.g., 10, 18, 26, 34, 42, 50,…
y
6
Jenny drew these graphs. How can she check that her graphs
match the equations?
4
y 3x 4
2
0
–2
–4
y 3x
1
4 x
2
Use the graphs. How much does y increase when x increases
by 1? What is the slope of each graph?
When x increases by 1, y increases by
–2
Slope of y 5 3x is
y-intercept
the value of
the dependent
variable when
the independent
variable is 0;
sometimes called
the initial value
2
3
3
.
. Slope of y 5 3x 1 4 is
3
.
How are the graphs the same?
e.g., Both represent linear relations. Both are increasing. Both
show continuous data. Both have the same slope.
3
Reflecting
What part of each
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equation shows
the slope? What
part shows the
y-intercept?
How are the graphs different?
They cross the y-axis at different points. For y 5 3x, the
y-intercept is 0 . For y 5 3x 1 4, the y-intercept is
.
4
Example
Two marinas charge different amounts for boat repairs. How can you
write an equation to describe the cost of repairs at each marina?
• Lakeside Marina charges $50/h.
• Bayview Marina charges a fixed fee of $30, plus $40/h.
Solution
A. Complete a table of values for each relation.
Lakeside Marina
Bayview Marina
Time (h), x
0
1
2
3
4
Time (h), x
0
1
2
3
4
Cost ($), y
0
50
100
150
200
Cost ($), y
30
70
110
150
190
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B. Graph both relations from Part A on the grid.
Cost of Boat Repairs
200
C. What is the rate of change for each linear relation?
$100
, or $50/h
2 h
$110 2 $30
5
2 h20 h
5
Rate of changeBayview
5
160
Cost ($)
$100 2 $0
Rate of changeLakeside 5
2 h20 h
120
80
40
0
$80
, or $40/h
2 h
50
SlopeBayview 5
40
The
is the hourly rate that the marina charges for
y-intercept
is the fixed fee.
F. Which relation is a direct variation? Which is a partial
variation?
The relation for Lakeside Marina is a direct variation
because the total cost equals the number of hours times the
cost per hour.
The relation for Bayview Marina is a partial variation because
the total cost equals the number of hours times the cost per
hour, plus a fixed fee.AW12
0176519637
G. Each equation describes
situation atC04-F23-AW12.ai
one of the marinas.
Figurethe
Number
Company
MPS
• The variable y is the
total cost.
Technical
• The variable x is the number of hours needed to repair
Pass
1st pass
a boat.
Approved
Complete the equations.
Lakeside Marina: y 5
Bayview Marina: y 5
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Not Approved
50
40
x x1
30
4
Bayview
Marina
Reflecting
E. What do the slope and y-intercept represent in each situation?
The slope
repairs.
2
3
Time (h)
Lakeside
Marina
D. What is the slope of each linear relation?
SlopeLakeside 5
1
For the cost of
repairs at Bayview
Marina, how are
the rate of change
and the slope the
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same? How are
they different?
direct variation
a linear relation in
which one variable
is a multiple of the
other variable;
y 5 mx
partial variation
a linear relation
in which one
variable is equal
to a multiple of
the other variable,
plus a constant;
y 5 mx 1 b
Reflecting
How can you tell if
the linear relation
is a direct variation
or a partial
variation with each
of these?
· table of values
· graph
· equation
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Practice
1. Circle the direct variations. Underline the partial variations.
Hint
You can write any
linear equation
in the form
y 5 mx 1 b, where
m is the slope and
b is the y-intercept.
a) y 5 7x
c) y 5 5x 2 11
b) y 5 10x 1 4
d) y 5 26x
2. a) Record the slope and y-intercept for each linear relation.
y 5 5x
y 5 212x 1 9
Slope 5 5 Slope 5 212 y-intercept 5 0
y-intercept 5 9
b) Is each equation in Part a) a direct or partial variation?
The equation y 5 5x is a
direct variation.
The equation y 5 212x 1 9 is a partial variation.
3. Each graph matches one of these situations. Record the letter
for the correct situation under each graph. Label the axes and
write a title. Record whether the relation is a direct variation or
partial variation.
• Situation A: Jim works at a fast food restaurant. He earns
$11.50/h. He is paid for each full hour that he works.
• Situation B: Rhonda has 10 L of gas in the gas tank of her
car. She fills up the gas tank at a gas station. The pump
flows at a rate of 20 L/min.
• Situation C: Jung has $500 in his savings account. He
plans to withdraw $50 per week. He will not make any
deposits.
How did you
decide which
graph matches
each situation in
Question 3?
Earnings ($)
___________________
Volume of fuel (L)
___________________
100
80
60
40
20
0
1
2 3 4
Time (min)
_____________
5
partial variation B 100
Savings Account Balance
________________________
Wages Earned
________________________
Filling a Gas Tank
________________________
100
Account balance ($)
___________________
Reflecting
80
60
40
20
0
2
4
6
8 10
Time (h)
_____________
direct variation A 500
400
300
200
100
0
2
4
6
8 10
Time (wk)
_____________
partial variation C
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4. Complete the chart for each graph in Question 3.
Graph
Situation A
Situation B
Situation C
Independent
variable, x
Dependent variable, y
time
earnings
time
week
number
Slope
y-intercept
11.5
20
y 5 11.5x
y 5 20x 1 10
0
10
volume of fuel
account
250
balance
Equation
500
y 5 250x 1 500
5. Sasha is a welder in Fort McMurray. The expansion of a metal
welding rod varies directly with the change in temperature. For
every 50 8C increase in temperature, the rod gets 1 mm longer.
a) What are the independent and dependent variables?
Hint
The phrase “varies
directly with”
indicates a direct
variation.
The independent variable, x, is the temperature .
The dependent variable, y, is
the length of the rod .
b) In a graph for Part a), what would the slope be?
Each time the temperature increases 50 8C, the length of
1
the rod increases
mm. Each time the temperature
increases 1 8C, the length of the rod increases
The slope is
1
50
1
50
mm.
Hint
The slope is the
amount that y
changes each time
that x increases
by 1.
Reflecting
.
c) What equation describes this situation?
1
1
Change in rod length 5
(temperature increase), or y 5
x
50
50
Could a non-linear
relation be a direct
variation or a
partial variation?
Explain.
d) Suppose the temperature increases by 120 8C. How much
will the length of the rod increase?
1
Change in rod length 5
(120), or 2.4 The length of the rod will increase 2.4 mm.
50
6. Ben is a sales representative. This equation describes his
long-distance phone plan:
y 5 0.05x 1 35, where y is his total monthly bill and x is the
number of long-distance minutes he uses
a) Is this a direct variation or a partial variation? Explain.
It is a partial variation. e.g., x is multiplied by a number,
and then a constant is added to get y.
b) What is the rate of change per month? $0.05/min
c) What does the rate of change mean? e.g., long-distance charge per minute
d) What is the monthly fee for the phone plan?
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