Transcript 12-5 Direct Variation

12-5 Direct Variation
Warm Up
Problem of the Day
Lesson Presentation
Course 3
12-5 Direct Variation
Warm Up
Use the point-slope form of each
equation to identify a point the line
passes through and the slope of the
line.
1
1. y – 3 = – 1
(x
–
9)
(9, 3), – 7
7
2. y + 2 = 2(x – 5)
3
2
(5, –2), 3
3. y – 9 = –2(x + 4)
(–4, 9), –2
4. y – 5 = – 1 (x + 7)
(–7, 5), – 1
4
4
Course 3
12-5 Direct Variation
Problem of the Day
Where do the lines defined by the
equations y = –5x + 20 and y = 5x –
20 intersect?
(4, 0)
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12-5 Direct Variation
Learn to recognize direct variation by
graphing tables of data and checking for
constant ratios.
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12-5 Direct
Insert Variation
Lesson Title Here
Vocabulary
direct variation
constant of proportionality
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12-5 Direct Variation
Course 3
12-5 Direct Variation
Helpful Hint
The graph of a direct-variation equation is always
linear and always contains the point (0, 0). The
variables x and y either increase together or
decrease together.
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12-5 Direct Variation
Additional Example 1A: Determining Whether a Data
Set Varies Directly
Determine whether the data set shows direct
variation.
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12-5 Direct Variation
Additional Example 1A Continued
Make a graph that shows the relationship between
Adam’s age and his length. The graph is not linear.
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12-5 Direct Variation
Additional Example 1A Continued
You can also compare ratios to see if a direct
variation occurs.
22 ? 27
3 = 12
81 81 ≠ 264
264
The ratios are not proportional.
The relationship of the data is not a direct
variation.
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12-5 Direct Variation
Additional Example 1B: Determining Whether a Data
Set Varies Directly
Determine whether the data set shows direct
variation.
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12-5 Direct Variation
Additional Example 1B Continued
Make a graph that shows the relationship between
the number of minutes and the distance the train
travels.
Plot the points.
The points lie in
a straight line.
(0, 0) is included.
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12-5 Direct Variation
Additional Example 1B Continued
You can also compare ratios to see if a direct
variation occurs.
25 = 50= 75 = 100
10 20 30 40
Compare ratios.
The ratios are proportional. The relationship is
a direct variation.
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12-5 Direct Variation
Check It Out: Example 1A
Determine whether the data set shows direct
variation.
Kyle's Basketball Shots
Distance (ft)
Number of Baskets
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20
30
40
5
3
0
12-5 Direct Variation
Check It Out: Example 1A Continued
Make a graph that shows the relationship between
number of baskets and distance. The graph is not
linear.
Number of Baskets
5
4
3
2
1
20 30 40
Distance (ft)
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12-5 Direct Variation
Check It Out: Example 1A Continued
You can also compare ratios to see if a direct
variation occurs.
5 ? 3
20= 30
60
150
150  60.
The ratios are not proportional.
The relationship of the data is not a direct
variation.
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12-5 Direct Variation
Check It Out: Example 1B
Determine whether the data set shows direct
variation.
Ounces in a Cup
Ounces (oz)
8
16
24
32
Cup (c)
1
2
3
4
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12-5 Direct Variation
Check It Out: Example 1B Continued
Number of Cups
Make a graph that shows the relationship between
ounces and cups.
Plot the points.
4
3
The points lie in
a straight line.
2
(0, 0) is included.
1
8
16 24 32
Number of Ounces
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12-5 Direct Variation
Check It Out: Example 1B Continued
You can also compare ratios to see if a direct
variation occurs.
1= 2 = 3 = 4
8 16 24 32
Compare ratios.
The ratios are proportional. The relationship is
a direct variation.
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12-5 Direct Variation
Additional Example 2A: Finding Equations of Direct
Variation
Find each equation of direct variation, given
that y varies directly with x.
y is 54 when x is 6
y = kx
54 = k  6
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y varies directly with x.
Substitute for x and y.
9=k
Solve for k.
y = 9x
Substitute 9 for k in the original
equation.
12-5 Direct Variation
Additional Example 2B: Finding Equations of Direct
Variation
x is 12 when y is 15
y = kx
15 = k

5=k
4
y =5
4x
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y varies directly with x.
12
Substitute for x and y.
Solve for k.
5
Substitute 4 for k in the original
equation.
12-5 Direct Variation
Check It Out: Example 2A
Find each equation of direct variation, given
that y varies directly with x.
y is 24 when x is 4
y = kx
24 = k
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
y varies directly with x.
4
Substitute for x and y.
6=k
Solve for k.
y = 6x
Substitute 6 for k in the original
equation.
12-5 Direct Variation
Check It Out: Example 2B
x is 28 when y is 14
y = kx
14 = k

1=k
2
y =1
2x
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y varies directly with x.
28
Substitute for x and y.
Solve for k.
1
Substitute 2 for k in the original
equation.
12-5 Direct Variation
Additional Example 3: Money Application
Mrs. Perez has $4000 in a CD and $4000 in a
money market account. The amount of
interest she has earned since the beginning of
the year is organized in the following table.
Determine whether there is a direct variation
between either of the data sets and time. If
so, find the equation of direct variation.
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12-5 Direct Variation
Additional Example 3 Continued
interest from CD and time
interest from CD 17 interest from CD 34
= 1
= 2 = 17
time
time
The second and third pairs of data result in a common
ratio. In fact, all of the nonzero interest from CD to
time ratios are equivalent to 17.
interest from CD = 17 = 34 = 51 = 68 = 17
time
1
2
3
4
The variables are related by a constant ratio of 17 to
1, and (0, 0) is included. The equation of direct
variation is y = 17x, where x is the time, y is the
interest from the CD, and 17 is the constant of
proportionality.
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12-5 Direct Variation
Additional Example 3 Continued
interest from money market and time
interest from money market = 19 = 19
time
1
interest from money market = 37 =18.5
time
2
19 ≠ 18.5
If any of the ratios are not equal, then there
is no direct variation. It is not necessary to
compute additional ratios or to determine
whether (0, 0) is included.
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12-5 Direct Variation
Check It Out: Example 3
Mr. Ortega has $2000 in a CD and $2000 in a
money market account. The amount of interest he
has earned since the beginning of the year is
organized in the following table. Determine
whether there is a direct variation between either
of the data sets and time. If so, find the equation
of direct variation.
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Interest
Interest from
Time (mo)
from CD ($)
Money Market ($)
0
0
0
1
12
15
2
30
40
3
40
45
4
50
50
12-5 Direct Variation
Check It Out: Example 3 Continued
A. interest from CD and time
interest from CD = 12
time
1
interest from CD 30
= 2 = 15
time
The second and third pairs of data do not result in a
common ratio.
If any of the ratios are not equal, then there
is no direct variation. It is not necessary to
compute additional ratios or to determine
whether (0, 0) is included.
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12-5 Direct Variation
Check It Out: Example 3 Continued
B. interest from money market and time
interest from money market = 15 = 15
time
1
interest from money market = 40 =20
time
2
15 ≠ 20
If any of the ratios are not equal, then there
is no direct variation. It is not necessary to
compute additional ratios or to determine
whether (0, 0) is included.
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12-5 Direct
Insert Variation
Lesson Title Here
Lesson Quiz: Part I
Find each equation of direct variation, given
that y varies directly with x.
1. y is 78 when x is 3.
2. x is 45 when y is 5.
3. y is 6 when x is 5.
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y = 26x
y =1 x
9
y =6 x
5
12-5 Direct
Insert Variation
Lesson Title Here
Lesson Quiz: Part II
4. The table shows the amount of money Bob
makes for different amounts of time he works.
Determine whether there is a direct variation
between the two sets of data. If so, find the
equation of direct variation.
direct variation; y = 12x
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