1. Essential Question: What does the k represent in a direct variation?

2. 5-5: Direct Variation
Direct Variation: A function in the form y = kx, where k ≠ 0
The k is called the constant of variation
For y = kx, y is a function of x
If x = 0, then y = 0, which means the graph of every direct variation will go through the point (0, 0)
To tell whether an equation represents a direct variation, solve for y. If the equation can be written in the form y = kx, it is a direct variation.

3. 5-5: Direct Variation Example 1: Is an equation a direct variation?
5x + 2y = x - 5x
2y = -5x ÷2 ÷2
y = -5/2 x
Direct variation, -5/2 is the constant of variation
5x + 2y = x - 5x
2y = -5x + 9 ÷2 ÷2 ÷2
y = -5/2x + 4.5
Not a direct variation

4. 5-5: Direct Variation Your Turn
Is each equation a direct variation? If so, find the constant of variation.
7y = 2x
Yes, k = 2/7
3x + 4y = 8 No
y – 7.5x = 0
Yes, k = 7.5

5. 5-5: Direct Variation
To write an equation for a direct variation, find k first.
If you are given any other point on the line:
Substitute in for x and y
Solve for k
Use the value of k to make an equation
Example 2: Writing an Equation Given a Point
Write an equation of the direct variation that includes the point (4, -3)
y = kx
-3 = k(4) Substitute ÷4 ÷4 Divide each side by 4
-3/4 = k
y = -3/4 x Substitute k in the direct variation

6. 5-5: Direct Variation Your Turn
Write an equation of the direct variation that includes the point (-3, -6)
y = 2x

7. 5-5: Direct Variation
Example 3: Identifying direct variation from a table.
For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation.
Because every y/x gives the Because the ratios are different same answer, this is a direct y/x is not constant, and this is NOT variation. a direct variation. The constant of variation is The equation is y = 4x. x y 2 8 3 12 5 20 x y 1 4 2 7 5 16
8/2 = 4
12/3 = 4
20/5 = 4
4/1 = 4.0
7/2 = 3.5
16/5 = 3.2

8. 5-5: Direct Variation Your Turn
Which of the following functions are direct variations? For the ones that are, find the constant of variation and write the equation x y -9 5 3
1 2/3 6
3 5/8 x y -1 -2 3 4 6 7 x y -6 -2 3 1 12 4
5/-9 = -5/9
1 2/3  3 = 5/9
3 5/8  6 = 29/48
Not a DV
-2/-1 = 2
4/3 = 1 1`/3
7/6 = 1 1/6
Not a DV
-2/-6 = 1/3
1/3 = 1/3
4/12 = 1/3
k = 1/3
y = 1/3 x

9. 5-5: Direct Variation
Your distance from lightning varies directly with the time it takes you to hear thunder. If you hear thunder 10 seconds after you see lightning, you are about 2 miles from the lightning. Write an equation for the relationship between time and distance.
Distance varies directly with time
When x = 10, y = 2
The formula for a direct variation is: y = kx
2 = k(10) Divide both sides by 10 ÷10 ÷10
1/5 = k
y = 1/5 x

10. 5-5: Direct Variation Your Turn
A recipe for a dozen corn muffins calls for 1 cup of flour. The number of muffins varies directly with the amount of flour you use. Write a direct variation for the relationship between the number of cups of flour and the number of muffins.
y = 12x

11. 5-5: Direct Variation
Assignment
Worksheet #5-5
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