10 8 Compare Linear Exponential and Quadratic Models

10. 8 Compare Linear, Exponential, and Quadratic Models Warm Up Lesson Presentation

10. 8 Warm-Up 1. Graph y = 3 x. ANSWER 2. Tell whether the ordered pairs (0, 0), (1, 2), (2, 4), and (3, 6) represent a linear function. ANSWER yes 3. For y = x 2 – 3 x – 5, find corresponding y-values for the x-values – 2, 1, and 3. ANSWER 5, – 7, – 5

10. 8 Example 1 Use a graph to tell whether the ordered pairs represent a linear function, an exponential function, or a quadratic function. 1 a. – 4, 32 , 1 – 2, 8 SOLUTION a. Exponential function , 0, 1 2 , 2, 2 , 4, 8

10. 8 Example 1 b. – 4, 1 , – 2, 2 b. Linear function , 0, 3 , 2, 4 , 4, 5

10. 8 Example 1 c. – 4, 5 , – 2, 2 c. Quadratic function , 0, 1 , 2, 2 , 4, 5

10. 8 Guided Practice 1. Tell whether the ordered pairs represent a linear function, an exponential function, or a quadratic function: (0, – 1. 5), (1, – 0. 5), (2, 2. 5), (3, 7. 5). ANSWER quadratic function

10. 8 Example 2 Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function. a. x y – 2 – 6 – 1 – 6 First differences: 0 Second differences: 2 0 – 4 2 1 0 4 2 2 6 6 2 ANSWER The table of values represents a quadratic function .

10. 8 Example 2 b. x – 2 – 1 0 1 2 y – 2 1 4 7 10 Differences: 3 3 ANSWER The table of values represents a linear function.

10. 8 Guided Practice 2. Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. x – 2 – 1 0 1 y 0. 08 0. 4 2 10 ANSWER exponential function

10. 8 Example 3 Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Then write an equation for the function. SOLUTION Determine which type of function the table of values represents.

10. 8 Example 3 The table of values represents a quadratic function because the second differences are equal.

10. 8 Example 3 STEP 2 Write an equation for the quadratic function. The equation has the form y = ax 2. Find the value of a by using the coordinates of a point that lies on the graph, such as (1, 0. 5). y = ax 2 Write equation for quadratic function. 0. 5 = a(1)2 Substitute 1 for x and 0. 5 for y. 0. 5 = a Solve for a. ANSWER The equation is y = 0. 5 x 2.

10. 8 Example 3 CHECK Plot the ordered pairs from the table. Then graph y = 0. 5 x 2 to see that the graph passes through the plotted points.

10. 8 Guided Practice Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Then write an equation for the function. 3. ANSWER linear function, y = 2 x 1 4. ANSWER quadratic function, y = 2 x 2

10. 8 Example 5 CYCLING The table shows the breathing rates y (in liters of air per minute) of a cyclist traveling at different speeds x (in miles per hour). Tell whether the data can be modeled by a linear function, an exponential function, or a quadratic function. Then write an equation for the function.

10. 8 Example 5 SOLUTION STEP 1 Graph the data. The graph has a slight curve. So, a linear function does not appear to model the data.

10. 8 Example 5 STEP 2 Decide which function models the data. In the 63. 3 57. 1 table below, notice that 1. 11, 57. 1 51. 4 70. 3 1. 11, 78. 0 1. 11, and 86. 6 1. 11. 63. 3 70. 3 78. 0 So, the ratios are all approximately equal. An exponential function models the data.

10. 8 Example 5 STEP 3 Write an equation for the exponential function. The breathing rate increases by a factor of 1. 11 liters per minute, so b = 1. 11. Find the value of a by using one of the data pairs, such as (20, 51. 4). y = abx 51. 4 = a(1. 11)20 51. 4 =a (1. 11)20 6. 38 ANSWER a Write equation for exponential function. Substitute 1. 11 for b, 20 for x, and 51. 4 for y. Solve for a. Use a calculator. The equation is y = 6. 38(1. 11)x.

10. 8 Guided Practice 5. In Example 4, suppose the cyclist is traveling at 15 miles per hour. Find the breathing rate of the cyclist at this speed. ANSWER about 30. 5 liters of air per minute