1. Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes

2. 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. y – 3 = – (x – 9)
2. y + 2 = (x – 5)
3. y – 9 = –2(x + 4)
4. y – 5 = – (x + 7) 1 7
(9, 3), – 1 7 2 3
(5, –2), 2 3
(–4, 9), –2 1 4
(–7, 5), – 1 4

3. Problem of the Day
Where do the lines defined by the equations y = –5x + 20 and y = 5x – 20 intersect?
(4, 0)

4. Learn to recognize direct variation by graphing tables of data and checking for constant ratios.

5. Vocabulary
direct variation
constant of proportionality

6. A direct variation is a linear function that can be written as y = kx, where k is a nonzero constant called the constant of variation.

7. The constant of variation is also called the constant of proportionality.
Reading Math

8. Additional Example 1A: Determining Whether a Data Set Varies Directly
Determine whether the data set shows direct variation.

9. Additional Example 1A Continued
Make a graph that shows the relationship between Adam’s age and his length. The graph is not linear.

10. Additional Example 1A Continued
You can also compare ratios to see if a direct variation occurs. 81
81 ≠ 264
The ratios are not proportional. 22 3 27 12 = ?
264
The relationship of the data is not a direct variation.

11. Additional Example 1B: Determining Whether a Data Set Varies Directly
Determine whether the data set shows direct variation.

12. 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.

13. Additional Example 1B Continued
You can also compare ratios to see if a direct variation occurs. 25 10 50 20 75 30
100 40
Compare ratios. = = =
The ratios are proportional. The relationship is a direct variation.

14. Kyle's Basketball Shots
Check It Out: Example 1A
Determine whether the data set shows direct variation.
Kyle's Basketball Shots 
Distance (ft) 20 30 40
Number of Baskets 5 3

15. Check It Out: Example 1A Continued
Make a graph that shows the relationship between number of baskets and distance. The graph is not linear. 5 4 3
Number of Baskets 2 1 20 30 40
Distance (ft)

16. Check It Out: Example 1A Continued
You can also compare ratios to see if a direct variation occurs. 60
150  60.
The ratios are not proportional. 5 20 3 30 = ?
150
The relationship of the data is not a direct variation.

17. 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

18. Check It Out: Example 1B Continued
Make a graph that shows the relationship between ounces and cups.
Number of Cups
Number of Ounces 2 3 4 8 16 24 1 32
Plot the points.
The points lie in a straight line.
(0, 0) is included.

19. Check It Out: Example 1B Continued
You can also compare ratios to see if a direct variation occurs. = 1 8 2 16 3 24 4 32
Compare ratios.
The ratios are proportional. The relationship is a direct variation.

20. 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
y varies directly with x.
54 = k  6
Substitute for x and y.
9 = k
Solve for k.
Substitute 9 for k in the original equation.
y = 9x

21. Additional Example 2B: Finding Equations of Direct Variation
x is 12 when y is 15
y = kx
y varies directly with x.
15 = k  12
Substitute for x and y.
= k 5 4
Solve for k.
Substitute for k in the original equation. 5 4
y = x 5 4

22. 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
y varies directly with x.
24 = k  4
Substitute for x and y.
6 = k
Solve for k.
Substitute 6 for k in the original equation.
y = 6x

23. Check It Out: Example 2B
x is 28 when y is 14
y = kx
y varies directly with x.
14 = k  28
Substitute for x and y.
= k 1 2
Solve for k.
Substitute for k in the original equation. 1 2
y = x 1 2

24. 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.

25. Additional Example 3A Continued
interest from CD and time
interest from CD
time = 17 1
= = 17
interest from CD
time 34 2
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.
= = = 17
interest from CD
time
= = 17 1 34 2 51 3 68 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.

26. Additional Example 3B Continued
interest from money market and time
interest from money market
time
= = 19 19 1
interest from money market
time
= =18.5 37 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.

27. Check It Out: Example 3
Mr. Ortega has $2000 in a CD and $2000 in a money market account. The amount of interest he has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.
Interest
Interest from
Time (mo)
from CD ($)
Money Market ($) 1 12 15 2 30 40 3 45 4 50

28. Check It Out: Example 3A Continued
interest from CD and time
interest from CD
time = 12 1
interest from CD
time
= = 15 30 2
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.

29. Check It Out: Example 3B Continued
interest from money market and time
interest from money market
time
= = 15 15 1
interest from money market
time
= =20 40 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.

30. Lesson Quiz for Student Response Systems
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems 30 30

31. 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.
y = 26x
y = x 1 9
y = x 6 5

32. 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

33. Lesson Quiz for Student Response Systems
1. Identify the equation of direct variation given that y varies directly with x.
y is 75 when x is 5. A.
B. y = 15x
C. y = 15x D. 33 33

34. Lesson Quiz for Student Response Systems
2. Identify the equation of direct variation given that y varies directly with x.
x is 66 when y is 11.
A. y = 66x
B. y = 6x C. D. 34 34

35. Lesson Quiz for Student Response Systems
3. The table shows the approximate value of a fruit depending on the weight. Determine whether there is a direct variation between the two sets of data. If so, identify the equation of direct variation.
A. direct variation; y = 1.5x
B. direct variation; y = x
C. direct variation; y = 0.75x
D. no direct variation 35 35