2Tell whether the ratios are proportional. Warm UpTell whether the ratios are proportional.1. =2. =3. =4. =69?2436yes5668?1417yes1213?6078no456?304yes
3CaliforniaStandardsAF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation.Also covered: AF3.3, AF3.4
4A direct variation is a linear function that can be written as y = kx, where k is a nonzero constant called the constant of variation.
5Additional Example 1: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation.A.
6Additional Example 1A Continued Method 1: Make a graph.The graph is not linear.
7Additional Example 1A Continued Method 2: Compare ratios.8181 ≠ 264The ratios are not equivalent.2232712=?264Both methods show the relationship is not a direct variation.
8Additional Example 1: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation.B.
9Additional Example 1B Continued Method 1: Make a graph.Plot the points.The points lie in a straight line.(0, 0) is included.
10Additional Example 1B Continued Method 2: Compare ratios.25105020753010040The ratio is constant.===Both methods show the relationship is a direct variation.
11Additional 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 = 24y = kx24 = k 3Substitute 24 for y and 3 for x.8 = kSolve for k.y = 8xSubstitute 8 for k in the original equation.
12Additional 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 = 56Multiply.Rachel will pay the salon owner $56 for 7 cut and styles.
13Additional 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.
14Additional Example 3 Continued A. interest from CD and timeinterest from CDtime=171= = 17interest from CDtime342The 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.= = = 17interest from CDtime= =171342513684The variables are related by a constant ratio of 17 to 1.
15Additional Example 3 Continued B. interest from money market and timeinterest from money markettime= = 19191interest from money markettime= =18.537219 ≠ 18.5If 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.
16Amount of Water in a Rain Gauge Lesson Quiz: Part IDetermine whether the data sets show direct variation.1.2.Amount of Water in a Rain GaugeTime (h)12345Rain (in)6810direct variationDriving TimeSpeed (mi/h)3040506080Time (h)107.5653.75no direct variation
17Lesson Quiz: Part II3. 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