5 5 Direct Variation Vocabulary direct variation constant

5 -5 Direct Variation Vocabulary direct variation constant of variation

5 -5 Direct Variation Direct variation is a linear relationship between two variable that can be written in the form y = kx or k = y , where k 0. The fixed number k in a direct x variation equation is the constant of variation.

5 -5 Direct Variation Reading Math You can read direct variation as “y varies directly as x” or “y is directly proportional to x” or “y varies with x. ”

5 -5 Direct Variation Additional Example 1 A: Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. y+8=x – 8= – 8 Solve the equation for y. Subtract 8 from both sides. y=x– 8 The equation is not in the form y = kx, so y + 8 = x is not a direct variation.

5 -5 Direct Variation Additional Example 1 B: Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. 3 y = 2 x 3 3 Solve the equation for y. Divide both sides by 3. 2 x 2 2 Write as x. y= x 3 3 3 The equation is in the form y = kx, so the original equation 3 y = 2 x is a direct variation. The constant of variation is 2. 3

5 -5 Direct Variation Check It Out: Example 1 Tell whether each equation represents a direct variation. If so, identify the constant of variation. A. 3 y + 4 x = 0 yes; k = – 4 3 B. y – 12 x = 2 x no

5 -5 Direct Variation Additional Example 2 A: Identifying a Direct Variation from a Table Tell whether each set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation. Price (¢) Weight (oz) 69 2 99 3 129 4 yx Find for each ordered pair. y = 2 x 69 y = 3 = 1 x 33 99 y = 4 x 129 k is not the same for each ordered pair. The data do not represent a direct variation.

5 -5 Direct Variation Helpful Hint In a direct variation where k is positive, when x increases, y also increases; when x decreases, y also decreases.

5 -5 Direct Variation Additional Example 2 B: Identifying a Direct Variation from a Table Tell whether each set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation. Inches 1 2 Centimeters 2. 54 5. 08 yx Find for each ordered pair. 5 12. 70 y = 2. 54 y = 5. 08 = 2. 54 y = 12. 7 = 2. 54 x 1 x 2 x 5 k = 2. 54 for each ordered pair. The data represent a direct variation where k = 2. 54. The equation is y = 2. 54 x

5 -5 Direct Variation Check It Out: Example 2 Tell whether the set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation. Tickets Prices ($) 5 37. 50 Yes; k = 7. 5; y = 7. 5 x 9 67. 5 14 105

5 -5 Direct Variation Additional Example 3: Identifying a Direct Variation from a Graph Tell whether each graph represents a direct variation. If so, identify the constant of variation and then write the direct variation equation. y The graph is a line through (0, 0). This is a direct variation. The Slope of the line is – 1 , 1 2 so k = –. The equation 2 1 is y = – x. 2 4 2 x – 4 – 2 – 4 0 2 4

5 -5 Direct Variation Helpful Hint In a direct variation, the slope, k, represents a constant rate of change.

5 -5 Direct Variation Check It Out: Example 3 A Tell whether each graph represents a direct variation. If so, identify the constant of variation and then write the direct variation y equation. 4 2 x – 4 – 2 0 2 4 – 4 Yes; k = 1 ; y = 1 x 4 4

5 -5 Direct Variation Check It Out: Example 3 B Tell whether each graph represents a direct variation. If so, identify the constant of variation and then write the direct variation y equation. 4 2 x – 4 – 2 0 2 4 – 4 no, not a direct variation

5 -5 Direct Variation Additional Example 4 A: Application A truck travels at a speed of 55 miles per hour. a. Write a direct variation equation for the distance y the truck travels in x hours. distance = 55 miles per hour times number of hours Use the formula y = kx. k = 55 y = 55 x 55 x

5 -5 Direct Variation Additional Example 4 B: Application A truck travels at a speed of 55 miles per hour. b. Graph the data. Make a table. Since time cannot be negative, use nonnegative number for I.

5 -5 Direct Variation Additional Example 4 Continued Use the ordered pairs top plot the points on a coordinate plane. Connect the points in a straight line. Label the axes. Check 75 Distance (mi) y = 55 x is in slope-intercept form with m = 55 and b = 0. The graph shows a slope of 55 and a y-intercept of 0. 100 50 25 2 4 6 Time (h) 8

5 -5 Direct Variation Additional Example 4 Continued c. How long does it take the truck to travel 660 miles? Find the value of x when y = 660 y = 55 x 660 = 55 x 55 55 Write the equation for the direct variation. Substitute 660 for y. Divide both sides by 660. 12 = x It will take the truck 12 hours to travel 660 miles.

5 -5 Direct Variation Check It Out: Example 4 A Peter has decided to save $30 each week to buy a new stereo system. Write a direct variation equation for the amount of money d that Peter has saved in w weeks. d = 30 w

5 -5 Direct Variation Check It Out: Example 4 B Graph the data.

5 -5 Direct Variation Check It Out: Example 4 C How many weeks will it take Peter to save $270? d = 30 w 270 = 30 w 30 30 9=w 9 weeks

5 -5 Direct Variation Lesson Quiz: Part I Tell whether each of the following represents a direct variation. If so, identify the constant of variation. 1. 12 y = 6 x 2. no yes; k = 1 2

5 -5 Direct Variation Lesson Quiz: Part II 3. A cheetah runs at a speed of 0. 75 mile per minute. a. Write a direct variation equation for the distance y the cheetah runs in x minutes. y = 0. 75 x b. Graph the data. cheetah run in 5 minutes? 3. 75 miles Distance (mi) c. How far does the 8 6 4 2 2 4 6 8 Time (min)