Transcript Course 3
12-5 Direct Variation Warm Up Problem of the Day Lesson Presentation Course 3 12-5 Direct Variation Warm Up Use the point-slope form of each equation to identify a point the line passes through and the slope of the line. 1 1. y – 3 = – 1 (x – 9) (9, 3), – 7 7 2. y + 2 = 2(x – 5) 3 2 (5, –2), 3 3. y – 9 = –2(x + 4) (–4, 9), –2 4. y – 5 = – 1 (x + 7) (–7, 5), – 1 4 4 Course 3 12-5 Direct Variation Problem of the Day Where do the lines defined by the equations y = –5x + 20 and y = 5x – 20 intersect? (4, 0) Course 3 12-5 Direct Variation Learn to recognize direct variation by graphing tables of data and checking for constant ratios. Course 3 12-5 Direct Insert Variation Lesson Title Here Vocabulary direct variation constant of proportionality Course 3 12-5 Direct Variation Course 3 12-5 Direct Variation Helpful Hint 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. Course 3 12-5 Direct Variation Additional Example 1A: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation. Course 3 12-5 Direct Variation Additional Example 1A Continued Make a graph that shows the relationship between Adam’s age and his length. The graph is not linear. Course 3 12-5 Direct Variation Additional Example 1A Continued You can also compare ratios to see if a direct variation occurs. 22 ? 27 3 = 12 81 81 ≠ 264 264 The ratios are not proportional. The relationship of the data is not a direct variation. Course 3 12-5 Direct Variation Additional Example 1B: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation. Course 3 12-5 Direct Variation Additional Example 1B 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. Course 3 12-5 Direct Variation Additional Example 1B Continued You can also compare ratios to see if a direct variation occurs. 25 = 50= 75 = 100 10 20 30 40 Compare ratios. The ratios are proportional. The relationship is a direct variation. Course 3 12-5 Direct Variation Check It Out: Example 1A Determine whether the data set shows direct variation. Kyle's Basketball Shots Distance (ft) Number of Baskets Course 3 20 30 40 5 3 0 12-5 Direct Variation Check It Out: Example 1A Continued Make a graph that shows the relationship between number of baskets and distance. The graph is not linear. Number of Baskets 5 4 3 2 1 20 30 40 Distance (ft) Course 3 12-5 Direct Variation Check It Out: Example 1A Continued You can also compare ratios to see if a direct variation occurs. 5 ? 3 20= 30 60 150 150 60. The ratios are not proportional. The relationship of the data is not a direct variation. Course 3 12-5 Direct Variation Check It Out: Example 1B Determine whether the data set shows direct variation. Ounces in a Cup Ounces (oz) 8 16 24 32 Cup (c) 1 2 3 4 Course 3 12-5 Direct Variation Check It Out: Example 1B Continued Number of Cups Make a graph that shows the relationship between ounces and cups. Plot the points. 4 3 The points lie in a straight line. 2 (0, 0) is included. 1 8 16 24 32 Number of Ounces Course 3 12-5 Direct Variation Check It Out: Example 1B Continued You can also compare ratios to see if a direct variation occurs. 1= 2 = 3 = 4 8 16 24 32 Compare ratios. The ratios are proportional. The relationship is a direct variation. Course 3 12-5 Direct Variation Additional Example 2A: Finding Equations of Direct Variation Find each equation of direct variation, given that y varies directly with x. y is 54 when x is 6 y = kx 54 = k 6 Course 3 y varies directly with x. Substitute for x and y. 9=k Solve for k. y = 9x Substitute 9 for k in the original equation. 12-5 Direct Variation Additional Example 2B: Finding Equations of Direct Variation x is 12 when y is 15 y = kx 15 = k 5=k 4 y =5 4x Course 3 y varies directly with x. 12 Substitute for x and y. Solve for k. 5 Substitute 4 for k in the original equation. 12-5 Direct Variation Check It Out: Example 2A Find each equation of direct variation, given that y varies directly with x. y is 24 when x is 4 y = kx 24 = k Course 3 y varies directly with x. 4 Substitute for x and y. 6=k Solve for k. y = 6x Substitute 6 for k in the original equation. 12-5 Direct Variation Check It Out: Example 2B x is 28 when y is 14 y = kx 14 = k 1=k 2 y =1 2x Course 3 y varies directly with x. 28 Substitute for x and y. Solve for k. 1 Substitute 2 for k in the original equation. 12-5 Direct Variation 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. Course 3 12-5 Direct Variation Additional Example 3 Continued interest from CD and time interest from CD 17 interest from CD 34 = 1 = 2 = 17 time time 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. interest from CD = 17 = 34 = 51 = 68 = 17 time 1 2 3 4 The variables are related by a constant ratio of 17 to 1, and (0, 0) is included. The equation of direct variation is y = 17x, where x is the time, y is the interest from the CD, and 17 is the constant of proportionality. Course 3 12-5 Direct Variation Additional Example 3 Continued interest from money market and time interest from money market = 19 = 19 time 1 interest from money market = 37 =18.5 time 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. Course 3 12-5 Direct Variation Check It Out: Example 3 Mr. Ortega has $2000 in a CD and $2000 in a money market account. The amount of interest he has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation. Course 3 Interest Interest from Time (mo) from CD ($) Money Market ($) 0 0 0 1 12 15 2 30 40 3 40 45 4 50 50 12-5 Direct Variation Check It Out: Example 3 Continued A. interest from CD and time interest from CD = 12 time 1 interest from CD 30 = 2 = 15 time 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. Course 3 12-5 Direct Variation Check It Out: Example 3 Continued B. interest from money market and time interest from money market = 15 = 15 time 1 interest from money market = 40 =20 time 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. Course 3 12-5 Direct Insert Variation Lesson Title Here Lesson Quiz: Part I 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. Course 3 y = 26x y =1 x 9 y =6 x 5 12-5 Direct Insert Variation Lesson Title Here Lesson Quiz: Part II 4. The table shows the amount of money Bob makes for different amounts of time he works. Determine whether there is a direct variation between the two sets of data. If so, find the equation of direct variation. direct variation; y = 12x Course 3