Learn to recognize direct variation by graphing tables of data and checking for constant ratios.

The graph of a direct-variation equation is always linear and always contains the point (0, 0). The variables x and y either increase together or decrease together. Helpful Hint

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

Example 1 Continued Make a graph that shows the relationship between Adam’s age and his length.

Example 1 Continued You can also compare ratios to see if a direct variation occurs. 81 81 ≠ 264 The ratios are not proportional. 22 3 27 12 = ? 264 The relationship of the data is not a direct variation.

Example 2: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation. B.

Example 2 Continued Make a graph that shows the relationship between the number of minutes and the distance the train travels. Plot the points. The points lie in a straight line. (0, 0) is included.

Example 2 Continued You can also compare ratios to see if a direct variation occurs. 25 10 50 20 75 30 100 40 Compare ratios. = = = The ratios are proportional. The relationship is a direct variation.

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

Example 3 Continued Make a graph that shows the relationship between number of baskets and distance. 5 4 3 Number of Baskets 2 1 20 30 40 Distance (ft)

Example 3 Continued You can also compare ratios to see if a direct variation occurs. 60 150  60. The ratios are not proportional. 5 20 3 30 = ? 150 The relationship of the data is not a direct variation.

Determine whether the data set shows direct variation. B. Example 4 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

Make a graph that shows the relationship between ounces and cups. Example 4 Continued Make a graph that shows the relationship between ounces and cups. 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.

Example 4 Continued You can also compare ratios to see if a direct variation occurs. = 1 8 2 16 3 24 4 32 Compare ratios. The ratios are proportional. The relationship is a direct variation.

Example 3: Finding Equations of Direct Variation Find each equation of direct variation, given that y varies directly with x. 3. y is 54 when x is 6

Example 4: Finding Equations of Direct Variation 4. x is 12 when y is 15 y = kx .

Example 5: Finding Equations of Direct Variation 5. y is 8 when x is 5

Example 6 Find each equation of direct variation, given that y varies directly with x. 6. y is 24 when x is 4

Example 9 B. x is 28 when y is 14

Example 10 C. y is 7 when x is 3

Lesson Review: Part 1 Find each equation of direct variation, given that y varies directly with x. 1. y is 78 when x is 3. 2. x is 45 when y is 5. 3. y is 6 when x is 5. y = 26x y = x 1 9 y = x 6 5