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Preview Warm Up California Standards Lesson Presentation

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**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

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California Standards AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation. Also covered: AF3.3, AF3.4

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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.

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**Additional Example 1: Determining Whether a Data Set Varies Directly**

Determine whether the data set shows direct variation. A.

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**Additional Example 1A Continued**

Method 1: Make a graph. The graph is not linear.

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**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.

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**Additional Example 1: Determining Whether a Data Set Varies Directly**

Determine whether the data set shows direct variation. B.

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**Additional Example 1B Continued**

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

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**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.

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**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.

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**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.

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**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.

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**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.

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**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.

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**Amount of Water in a Rain Gauge**

Lesson Quiz: Part I Determine whether the data sets show direct variation. 1. 2. Amount of Water in a Rain Gauge Time (h) 1 2 3 4 5 Rain (in) 6 8 10 direct variation Driving Time Speed (mi/h) 30 40 50 60 80 Time (h) 10 7.5 6 5 3.75 no direct variation

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Lesson Quiz: Part II 3. Roy’s income varies directly with the number of dogs that he walks. He earned $8.50 for walking 2 dogs. Write a direct variation function for this situation. If Roy walks 5 dogs, how much will he earn? y = 4.25x; $21.25

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