1. What is it and how do I know when I see it?
Direct Variation
What is it and how do I know when I see it?

2. Definition
Direct Variation – a special type of linear relationship that can represented by a function in the form y = kx,
Constant of Variation – is k, the coefficient of x, in the function y = kx.

3. Direct Variation
A recipe for paella calls for 1 cup of rice to make 5 servings. In other words, a chef needs 1 cup of rice for every 5 servings.
The equation y = 5x describes this relationship. In this relationship, the number of servings varies directly with the number of cups of rice.

4. Determine Direct Variation from a Table
What happens if you solve y = kx for k?
y = kx
Divide both sides by x (x ≠ 0).
So, in a direct variation, the ratio is equal to the constant of variation.

5. Example: Determine Direct Variation from a Table
Tell whether the relationship is a direct variation. Explain.
Find for each ordered pair.
This is a direct variation because is the same for each ordered pair.

6. Examples of Direct Variation:
What is the constant of variation of the table above?
Since y = kx we can say Therefore:
12/6=k or k = 2 14/7=k or k = 2
16/8=k or k =2 Note k stays constant.
y = 2x is the equation!

7. Examples of Direct Variation:
What is the constant of variation of the table above?
Since y = kx we can say Therefore:
30/10=k or k = 3 15/5=k or k = 3
9/3=k or k =3 Note k stays constant.
y = 3x is the equation!

8. What is the constant of variation for the following direct variation?2 -2 -½
Answer Now

9. The k values are different!
Is this a direct variation? If yes, give the constant of variation (k) and the equation.
No!
The k values are different!

10. Which is the equation that describes the following table of values?
y = -2x
y = 2x
y = ½ x
xy = 200
Answer Now

11. In a table, simplify any one of the ratios.
We can find the constant of proportionality from a table of values, equation and a graph.
In a table, simplify any one of the ratios.
Chaperones 1 2 3 4 5
Students 12 24 36 48 60

12. Find the constant of proportionality:
Example:
Find the constant of proportionality:
Apples (lbs) 2
2.5 3
3.5 4
Cost ($)
3.96
4.95
5.94
6.93
7.92

13. Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = 28 when x=7, Find x when y = HOW???
2 step process
1. Find the constant variation
k = y/x or k = 28/7 = 4
k=4
2. Use y = kx. Find the unknown (x).
52= 4x or 52/4 = x
x= 13
Therefore:
X =13 when Y=52

14. In a table, simplify any one of the ratios.
We can find the constant of proportionality from a table of values, equation and a graph.
In a table, simplify any one of the ratios.
Chaperones 1 2 3 4 5
Students 12 24 36 48 60

15. Find the constant of proportionality:
Apples (lbs) 2
2.5 3
3.5 4
Cost ($)
3.96
4.95
5.94
6.93
7.92
Click

16. Find the constant of proportionality:X Y 3
4.5 4 6 5
7.5 8 12 9
13.5
Click

17. Joke Time How would you describe a frog with a broken leg? Unhoppy
What did the horse say when he got to the bottom of his feed bag?
That’s the last straw!
What kind of music do chiropractors listen to?
Hip - Pop

18. the ORIGIN!!!!!

19. A Graph is a direct variation if:
It is a straight line that passes through the origin (0,0)

20. Tell if the following graph is a Direct Variation or not.
Yes No No No

21. Tell if the following graph is a Direct Variation or not.
Yes No No
Yes

22. Is the relationship shown in the graph proportional?41
Is the relationship shown in the graph proportional?
Yes No
Hours
Salary ($) 5 10 15 20 25 30 35 40 45 50
Answer: Yes

23. Is the relationship shown in the graph proportional?42
Is the relationship shown in the graph proportional?
Yes No 50 45
Cost ($) 40 35 30 25 20 15
Answer: No 10 5
Toppings

24. Is the relationship shown in the graph proportional?43
Is the relationship shown in the graph proportional?
Yes No
Feet
Seconds
0.5 1
1.5 2
2.5 3
3.5 4
4.5 5
Answer: Yes

25. Is the relationship shown in the graph proportional?44
Is the relationship shown in the graph proportional?
Yes No
Text Messages
Cost ($) 5 10 15 20 25 30 35 40 45 50
Answer: No

26. Constant of Proportionality
The graph of a proportional relationship is a straight line that passes through the origin.
Proportional quantities can be described by the equation y= kx, where k is a constant ratio.

27. Constant of Proportionality
Equation for constant of proportionality is
y = kx
We can tell that the relationship is directly proportional by looking at the graph. The graph is a straight line that passes through the origin.

28. Constant of Proportionality
Equation for constant of proportionality is
y = kx
Create a table using the points from the graph:
Total price (y) 20 40 60 80
100
Total pounds (x) 2 4 6 8 10
Divide total price by total pounds

29. Constant of Proportionality
Equation for constant of proportionality is
y = kx
Create a table using the points from the graph:
Total price (y) 20 40 60 80
100
Total pounds (x) 2 4 6 8 10
Divide total price by total pounds

30. In a graph, choose a point (x, y) to find and simplify the ratio.
(2,24)
Chaperones
Students 6 12 18 24 30 36 42 48 54 60