Teacher Most Extraordinaire Direct Variation Taught very directly and directed to you by: Mr. Richard Teacher Most Extraordinaire

Tell whether the ratios are proportional. Warm Up Tell whether the ratios are proportional. 1. = 2. = 3. = 4. = 6 9 ? 24 36 yes 56 68 ? 14 17 yes 12 13 ? 60 78 no 45 6 ? 30 4 yes

Vocabulary Direct Variation Constant of Proportionality Varies Directly

A direct variation is a linear function that can be written as y = kx, where k is a nonzero constant called the constant of variation.

Additional Example 1: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation. A.

Additional Example 1A Continued Method 1: Make a graph. The graph is not linear.

Additional Example 1A Continued Method 2: Compare ratios. 81 81 ≠ 264 The ratios are not equivalent. 22 3 27 12 = ? 264 Both methods show the relationship is not a direct variation.

Additional Example 1: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation. Are ratios equivalent?

Additional Example 1B Continued Method 1: Make a graph. Plot the points. The points lie in a straight line. (0, 0) is included.

Additional Example 1B Continued Method 2: Compare ratios. 25 10 50 20 75 30 100 40 The ratio is constant. = = = Both methods show the relationship is a direct variation.

Kyle's Basketball Shots Check It Out! Example 1 Determine whether the data set shows direct variation. A. Kyle's Basketball Shots  Distance (ft) 20 30 40 Number of Baskets 5 3

Check It Out! Example 1A Continued Method 1: Make a graph. 5 4 3 Number of Baskets 2 1 20 30 40 Distance (ft)

Check It Out! Example 1A Continued Method 2: Compare ratios. 60 150  60. The ratios are not equivalent. 5 20 3 30 = ? 150 Both methods show the relationship is not a direct variation.

Check It Out! Example 1 Determine whether the data set shows direct variation. B. Ounces in a Cup Ounces (oz) 8 16 24 32 Cup (c) 1 2 3 4

Check It Out! Example 1B Continued Method 1: Make a graph. Number of Cups Number of Ounces 2 3 4 8 16 24 1 32 Plot the points. The points lie in a straight line. (0, 0) is included.

Check It Out! Example 1B Continued Method 2: compare ratios. = 1 8 2 16 3 24 4 32 The ratio is constant. Both methods show the relationship is a direct variation.

Additional Example 2: Finding Equations of Direct Variation Rachel rents space in a salon to cut and style hair. She paid the salon owner $24 for 3 cut and styles. Write a direct variation function for this situation. If Rachel does 7 cut and styles, how much will she pay the salon owner? Step 1 Write the direct variation function. Think: The amount owed varies directly with the amount of cuts given. x = 3 and y = 24 y = kx 24 = k  3 Substitute 24 for y and 3 for x. 8 = k Solve for k. y = 8x Substitute 8 for k in the original equation.

Additional Example 2 Continued Step 2 Find how much Rachel will pay the salon owner for 7 cut and styles. Substitute 7 for x in the direct variation function. y = 8(7) y = 56 Multiply. Rachel will pay the salon owner $56 for 7 cut and styles.

Check It Out! Example 2 Rinny cuts and styles hair in a salon. She earns $120 for 4 cut and styles. Write a direct variation function for this situation. If Rinny does 9 cut and styles, how much will she earn? Step 1 Write the direct variation function. Think: The amount owed varies directly with the amount of cuts given. x = 4 and y = 120 y = kx 120 = k  4 Substitute 120 for y and 4 for x. 30 = k Solve for k. y = 30x Substitute 30 for k in the original equation.

Check It Out! Example 2 Continued Step 2 Find how much Rinny will earn for 9 cut and styles. Substitute 9 for x in the direct variation function. y = 30(9) y = 270 Multiply. Rinny will earn $270 for 9 cut and styles.

Additional Example 3: Money Application Mrs. Perez has $4000 in a CD and $4000 in a money market account. The amount of interest she has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.

Additional Example 3 Continued A. interest from CD and time interest from CD time = 17 1 = = 17 interest from CD time 34 2 The second and third pairs of data result in a common ratio. In fact, all of the nonzero interest from CD to time ratios are equivalent to 17. = = = 17 interest from CD time = = 17 1 34 2 51 3 68 4 The variables are related by a constant ratio of 17 to 1.

Additional Example 3 Continued B. interest from money market and time interest from money market time = = 19 19 1 interest from money market time = =18.5 37 2 19 ≠ 18.5 If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

Check It Out! Example 3 Mr. Ortega has $2000 in a CD and $2000 in a money market account. The amount of interest he has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.   Interest Interest from Time (mo) from CD ($) Money Market ($) 1 12 15 2 30 40 3 45 4 50

Check It Out! Example 3 Continued A. interest from CD and time interest from CD time = 12 1 interest from CD time = = 15 30 2 The second and third pairs of data do not result in a common ratio. If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

Check It Out! Example 3 Continued B. interest from money market and time interest from money market time = = 15 15 1 interest from money market time = =20 40 2 15 ≠ 20 If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

Find the Missing coordinate if the following points are in direct variation: (2, 7) , (6, y). (Use Proportions) 2 = 6 7 y 2y = 42 Y = 21

Direct Variation Quiz Quiz: Page 281-283 # 24, 26, 28, 48, 50 Homework # 25, 49, 51, 53, 61