Section 3.2 : Direct Variation

Learning Targets: A.CED.2

Important Terms and Definitions

Direct Variation: a relationship between variables that can be defined by the formula 𝑦 = 𝑘𝑥

Note: We say that “y varies directly with x” or “y varies directly as x”.

Constant of Variation: the number, k, that is multiplied by x in the direct variation

(ex) 𝑦 = 5𝑥 𝑦 = −3𝑥 𝑦 = 13

𝑥

so 𝑘 = 5 so 𝑘 = −3 so 𝑘 = 13

Identifying Direct Variation

If an equation is an example of direct variation, it must be able to be written in the form 𝑦 = 𝑘𝑥

(ex 1) Is the following equation an example of direct variation? If so, identify the constant of variation.

3𝑥 + 4𝑦 = 0

(ex 2) Is the following equation an example of direct variation? If so, identify the constant of variation.

−6𝑥 − 24 = 0

−3 −3

(ex 3) Is the following equation an example of direct variation? If so, identify the constant of variation.

𝑦 − 3𝑥 = 4

Writing Direct Variation Equations

Example: Suppose y varies directly with x, and 𝑦 = 12 when 𝑥 = −3. What direct variation equation relates y and x? What is the value of y when 𝑥 = 8?

Step One: Find k Step Two: Write Equation Step Three: Find y

𝑦 = 𝑘𝑥 𝑦 = 𝑘𝑥 𝑦 = −4𝑥

12 = 𝑘(−3) 𝑦 = −4𝑥 𝑦 = −4(8)

𝑦 = −32 𝑘 = −4 (ex 4) Suppose y varies directly with x, and 𝑦 = 2 when 𝑥 = −3. What direct variation equation relates y and x? What is the value of y when 𝑥 = 15?

(ex 5) Suppose y varies directly with x, and 𝑦 = 12.4 when 𝑥 = 4. What direct variation equation relates y and x? What is the value of y when 𝑥 = 3?

(ex 6) Suppose y varies directly with x, and 𝑦 = 15 13 when 𝑥 = − 1

2. What direct variation

equation relates y and x? What is the value of y when 𝑥 = 12?

Sometimes we call k the constant of proportionality.

What would happen if we solved 𝑦 = 𝑘𝑥 for k? How can we use this?

x y 4 6 8 12

10 15

x y −3 2.25 1 −0.75 4 −3

Writing Direct Variation from a Table

(ex 7) In the following table, does y vary directly with x? If it does, write an equation.

(ex 8) In the following table, does y vary directly with x? If it does, write an equation.

Graphing Direct Variation

(ex 9) The distance, d, a group of hikers travels through Mill Creek Park varies directly with amount of time they have been hiking, t. Suppose they hike 6.4 miles in 1.25 hours. What is an equation that relates d and t? Graph the equation. What is the slope of your line?

(ex 10) A recipe for 3 dozen chocolate chip cookies calls for 2 cups of flour. The number of dozen of cookies you can make varies directly with number of cups of flour you use. If you have 4.5 cups of flour, how many dozen cookies can you make?

x y 3 5.4 7 12.6

12 21.6

x y −2 1 4 8 9 15

x y −6 9 1 −1.5 8 −12

Homework – Section 3.2 : Direct Variation

Is the equation an example of direct variation? If so, identify the constant of variation.

1. 2𝑦 = 5𝑥 + 1 4. 0.7𝑥 − 1.4𝑦 = 2.8 2. −4 + 7𝑥 + 4 = 3𝑦 5. – 𝑥 = 12𝑦 3. 8𝑥 + 9𝑦 = 10 6. 1

2𝑥 + 1

3𝑦 = 0

Suppose y varies directly with x. Write the direct variation equation that relates x to y. Then find the value of y when 𝑥 = 8.

7. 𝑦 = −4 when 𝑥 = 3 10. 𝑦 = 12 when 𝑥 = 3

8. 𝑦 = −9 when 𝑥 = −5 11. 𝑦 = −1.5 when 𝑥 = 5.2 9. 𝑦 = 35 when 𝑥 = 7 12. 𝑦 = − 8

3 when 𝑥 = − 9

8

In the table, does y vary directly with x? If it does, write an equation.

13. 14. 15.

16. The amount of blood in a person’s body varies directly with body weight. A person who weighs 160 lb has about 5 qt of blood. Write an equation relating these two variables, and use that equation to estimate the amount of blood in your body.

17. The perimeter p of a regular octagon varies directly with the length l of one side of the octagon. What is an equation that relates the perimeter to the length of one side? Graph the equation.

18. The distance d you bike varies directly with the amount of time t you bike. Suppose you bike 13.2 mi in 1.25 h. What is an equation that relates the distance to the amount of time you biked? Graph the equation.