1. Direct Variation

2. Vocabulary Direct Variation
Constant of variation for a direct variation

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. Definition
Direct Variation – a special type of linear relationship that can represented by a function in the form y = kx, where k ≠ 0.
Constant of Variation – is k, the coefficient of x, in the function y = kx.
By dividing each side of y = kx by x, you can see that the ratio of the variables is constant: y/x = k.
Direct Variation Equations ALWAYS go through the origin!!!
Constant of Variation = Slope, so k = m!

5. Direct Variation

6. Direct Variation

7. Determining Direct Variation
To determine whether an equation represents a direct variation;
Given an equation.
Solve for y.
If you can write the equation in the form y = kx, where k ≠ 0, then it is a direct variation.
Given a table of values.
Find y/x for each ordered pair.
If the ratio y/x is constant for all ordered pairs, then it is a direct variation.
Given a graph.
The graph is a line and passes through the origin, then it is a direct variation.
The slope of the line is k.

8. 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.

9. Ex. 1 Graph y = 4x Write the slope as a ratio. 4 = 4 1 Graph (0,0)
Use slope to find next
point.
Draw line through points 1 4

10. Example: Determine Direct Variation from a Table
Tell whether the relationship is a direct variation. Explain.
Find for each ordered pair. …
This is not direct variation because is the not the same for all ordered pairs.

11. Example: Writing a Direct Variation Equation
The value of y varies directly with x, and y = 3, when x = 9. Find y when x = 21.
Find the value of k and then write the equation.
y = kx
Write the equation for a direct variation.
3 = k(9)
Substitute 3 for y and 9 for x. Solve for k.
Since k is multiplied by 9, divide both sides by 9.
The equation is y = x. When x = 21, y = (21) = 7.

12. Example: Graphing Direct Variation
A group of people are tubing down a river at an average speed of 2 mi/h. Write a direct variation equation that gives the number of miles y that the people will float in x hours. Then graph.
Step 1 Write a direct variation equation.
distance =
2 mi/h
times
hours y 2  x

13. 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