Direct Variation

1. Direct Variation Learn to recognize direct variation and identify the constant of proportionality.

2. Spiders • How many legs does a spider have? • 8 legs • Therefore • 2 spiders have a total of 16 legs • 3 spiders have a total of 24 legs • 4 spiders have a total of 32 legs • And so on. . . • This type of relationship is called . . .

3. Direct Variation • The relationship between the amount of spiders and how many legs they have can be said to vary directly! • We will be learning about equations, tables, and graphs of direct variations. • Sometimes this is called direct proportion rather than direct variation but it is the same thing.

4. Direct Variation • A direct variationrelationship can be represented by a linear equation in the form y = kx, where k is a positive number called the constant of proportionality. • The constant of proportionality can sometimes be referred to as the constant of variation.

5. y = kx • When two variable quantities have a constant (unchanged) ratio, their relationship is called a direct proportion. • We say, “y varies directly as x.” • k is the constant of proportionality which means it never changes within a problem.

6. Finding Values Using a Direct Variation Equation • At a frog jumping contest, Edward’s frog jumped 60 inches. Bella’s frog jumped 72 inches. Jacob’s frog jumped 6.5 feet. Use the equation y = 12x, where y is the number of inches and x is the number of feet to find the missing values of the table.

7. Using Equations and Tables 78 5 Substitute Solve Simplify

8. Now You Try • Identify the constant of proportionality: 1. y = 15x 15 2. y = .72x .72 3. y = ¼x ¼

9. How about solving for k? • When we are given the x and y values, we can solve for the constant of proportionality. • Example • If y varies directly with x, and y = 8 when x = 12, find k and write an equation that expresses this variation. • Steps: • 8 = k× 12 Substitute numbers into y = kx • 8/12 = (k× 12)/12 Divide both sides by 12 • 2/3 = k Simplify • y = 2/3x Plug k back into the equation

10. Now You Try • If y varies directly as x, and x = 12 when y = 9, what is the equation that describes this direct variation? • y = kx • 9 = k× 12 • 9/12 = k • ¾ = k • y = ¾x

11. Now You Try • If y varies directly as x, and x = 5 when y = 10, what is the equation that describes this direct variation? • y = kx • 10 = k× 5 • 10/5 = k • 2 = k • y = 2x

12. Copy and fill out tables: y = x y = 4x 1 4 2 8 3 12 4 16 5 20

13. Copy and fill out tables: y = 10x y = ½x 10 ½ 20 1 30 3/2 40 2 50 5/2

14. Sometimes we will have to put an equation into y = kx form and solve for y. Then we will be asked to identify k, the constant of proportionality. Examples: Tell whether each equation or relationship is a direct variation. If so identify the constant of proportionality. 4y = 2x ½y – ¾x = 0 7y – 5 = 3x Direct Variation Equation

15. More Examples If y varies directly as x and y = 24 when x = 16, find y when x = 12. Solution: Set up a proportion since the ratios of corresponding values of x to y are always the same.