Name Date Class Practice A

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1 Practice A Direct Variation Tell whether each equation represents a direct variation. If so, identify the constant of variation. 1. y 4x 2. y 2 x 3. y 6.3x 3 4. y 2x 2 5. y x y 6x Tell whether each set of data represents a direct variation. If so, identify the constant of variation and write the equation. 7. x y Volume (oz) Price ($) x y Boxes of crayons Number of cartons _ x y _ Time (h) Pay ($) Tell whether each graph represents a direct variation. If so, identify the constant of variation and write the equation

2 Practice B Direct Variation Tell whether each equation represents a direct variation. If so, identify the constant of variation. 1. y 7x 2. y 0.04x 3. 3y 2x 5 4. y 13x y 2x 6. 8y 4x 12 Tell whether each set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation. 7. x y x y Times at Bat Hits School days Days Roy was out sick x y Time (Days) Distance (k) ,095 Tell whether each graph represents a direct variation. If so, identify the constant of variation and then write the direct variation equation

3 Practice C Direct Variation Tell whether each equation represents a direct variation. If so, identify the constant of variation. 1. y 1.5x 2. y 4 1 x 3. y 9x y 3x 5. 4y x y x Tell whether each set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation. 7. x y Weight (oz) Price ($) x y _ x y Tell whether each graph represents a direct variation. If so, identify the constant of variation and write the equation

4 Review for Mastery Direct Variation Direct variation is a relationship between two variables linked by a constant. Suppose you make peanut butter sandwiches. Each sandwich is made with 2 slices of bread and 1 tablespoon of peanut butter. No matter how many sandwiches you make, the ratio between peanut butter (the x variable) and the bread (the y variable) remains constant: An equation is used to express this relationship between x and y: y kx In this equation, k is called the constant of variation. It tells you how much x changes when y changes. In the example involving the sandwiches, k 2. The relationship between x and y can also be expressed in a graph: The line of a graph representing a direct variation always passes through (0, 0). 1. What is the x variable in the table above? What is the y variable? 3. What is the constant of variation in the equation y 5x? 5. What is the constant of variation in the equation 3y 6x? Explain. 2. Write an equation that expresses the direction variation in the table. _ 4. What is the constant of variation in the equation y x? _

5 Challenge Dependent and Independent Variables Reading In real-world applications involving direct variation, the x variable is known as the independent variable, or input. The y variable is known as the dependent variable, or output. Examine the graph. It shows an internet company s earnings during its first six years of operation. To find the year of operation start at year 0 and add 1 for each year. This is the independent variable. The dependent variable is the amount of the company s earnings. It can be plotted on a graph to predict, or project, future earning trends. Write the correct answer. 1. What is the constant of variation in the company s earnings from year to year? 2. Write an equation for the relationship between the dependent and independent variables. 3. Assume the company s earnings continue to grow at the rate shown. Project earnings for the ninth year of operation. Explain your answer. 4. A rival company posted these earnings for its first four years of operation: $1.5 million, $2.8 million, $4.7 million, and $8.4 million. Do these earnings represent a direct variation? Why or why not? 5. Scientists have determined that the plates supporting Earth s continents move about 8.5 m per century. What is the independent variable? The dependent variable? How much would the plates move in 500 years? In 1,300 years?

6 Problem Solving Slope-Intercept Form Write the correct answer. 1. A pipeline delivers 10,422 gallons of natural gas every other month. Tell whether this represents a direct variation. If so, identify the constant of variation and write the equation. 2. A hybrid vehicle gets 31.2 miles per gallon of gas. Complete the data table. x (gal) y (mi) 3. The temperature of a gas increases proportionally with the amount of pressure applied to it. The temperature of propane rises 30 C for every increase of 4 atm (atmospheres) of pressure. Assume that at 2 atm, the temperature of propane is -30 C. Make a graph that shows the temperature of the gas at 10 atm of pressure. Is this a direct variation? Choose the letter for the best answer. 4. The equation y 7x shows the rate at which Sara bicycled in m/hi. Which of the following is true? A When y 4, x 0. B When x 7, y 4. C At this rate, Sara would travel 28 miles in 4 hours. D At this rate, Sara would travel 4 miles in 28 hours. 6. Which equation shows the distance y Sara rode if she rode for 5 miles and then started timing her riding at 7 miles per hour, where x is the number of hours she rode? A y 5x 7 C 5y 7x B y 7x 5 D x 5y 7 5. Which of the following points must the graph of the linear equation y 7x pass through to represent a direct variation? A (0, 1) C (0, 0) B (1, 0) D (1, 1) 7. What is the y-intercept of the equation 2x 3y 6? A 2 B 2 3 C 2 3 D 2

7 Reading Strategies Understand Direct Variation When two measurements vary directly, a change in one means a corresponding change in the other. For example, consider that there are 2 socks in every pair. If you buy 2 pair, you get 4 socks. If you buy 6 pair, you get 12 socks. The list below shows this correspondence: Pairs Socks You could also state this relationship between pairs and socks as an equation: pair 2 sock A more general equation can be used: y kx In this equation, y and x are the two quantities that vary directly. k tells how the quantities vary with respect to each other. It is known as the constant of variation. In the sock example, the equation is y 1 2 x and the constant of variation is 1 2. This relationship can also be displayed in a graph. At the right is a graph of the relationship of pairs to individual socks: Identify the constant of variation. 1. y 12x 2. y 2.3x 3. y 4.5x 4. y 2 3 x 5. y 1,490x 6. y x Write an equation for each graphed line

8 Puzzles, Twisters & Teasers Code Breaker Determine whether each exercise represents a direct variation. If so, identify the constant of variation. (If not, write no direct variation.) Then use the code to convert the numerical value into a letter. Finally, unscramble the letters to solve the riddle. 1. 5y 81x 2. y 6 x 3. 2y 18x y 51x 5. 3y 37.2x 6. 1y 5x 5 7. x y x y x y _ x y Riddle: Which month has twenty-three days? Answer:

9 LESSON 5-8 Practice A 1. yes, k 4 2. yes, k yes, k no 5. no 6. yes, k 2 7. yes, k 3, y 3x 8. yes, k 0.25, y 0.25x 9. yes, k 1, y x 10. no 11. no 12. yes, k 7.5, y 7.5x 13. no 14. yes, k 6, y 6x Practice B 1. yes, k 7 2. yes, k no 4. yes, k yes, k no 7. yes, k 1 5,y 1 5 x 8. yes, k 5 2,y 5 2 x 9. yes, k 1 3, y 1 3 x 10. no 11. no 12. no 13. yes, k 4, y 4x 14. no Practice C 8. yes, k 0.8, y 0.8x 9. no 10. no 11. yes, k 5 2, y no Review for Mastery 1. The x variable is days. The y variable is hours. 2. y 24x x 5. The constant of variation is 2. You divide both sides of the equation by 3. Challenge y 0.7x 3. The earnings for the ninth year will $6.3 million. Multiply No, they do not represent a direct variation. The variation is not constant from year to year. 5. The independent variable is the time in centuries. The dependent variable is the amount of movement. The plates would move 42.5 m in 500 years. They would move m in 1,300 years. Problem Solving 1. It represents a direction variation. The constant of variation is 5,211. y 5,211x where y is the number of gallons of gas and x is the number of months yes, k yes, k no 4. yes, k no 6. yes, k no

10 This is not a direct variation. 3. y values: 187.2, 280.8, C 5. C 6. B 7. D Reading Strategies , y 3 2 x 8. y 3 4 x 9. y 1 2 x Puzzles, Twisters & Teasers 1. k k no direct variation 4. k k no direct variation 7. no direct variation 8. k 4 9. k no direct variation 11. k k 15 Answer: A-L-L O-F T-H-E-M