10.8 Warm Up Warm Up Lesson Presentation Lesson Presentation Compare Linear, Exponential, and Quadratic Models

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

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

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

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

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. ANSWER The table of values represents a quadratic function. x–2–1012 y–6 –406 First differences: Second differences: a

10.8 Example 2 ANSWER The table of values represents a linear function. x – 2– y – Differences: b. 3333

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

10.8 Bell Ringer 4/21/15 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 = ax2y = ax2 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 = 0.5x 2.

10.8 Example 3 CHECK Plot the ordered pairs from the table. Then graph y = 0.5x 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. ANSWER quadratic function, y = 2x ANSWER linear function, y = 2x  1

10.8 Bell Ringer 4/22/15 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. Calculate the difference or ratio. CYCLING

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 table below, notice that 1.11, 1.11, 1.11, 1.11, and 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 = Find the value of a by using one of the data pairs, such as (20, 51.4). Write equation for exponential function. Substitute 1.11 for b, 20 for x, and 51.4 for y. y = ab x 51.4 = a(1.11) 20 Solve for a (1.11) 20 = a Use a calculator a ANSWERThe 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. about 30.5 liters of air per minute ANSWER

10.8 Quiz Write the equation for the data shown.

10.8 Lesson Quiz 1. Tell whether the data can be modeled by a linear function, an exponential function, or a quadratic function. ANSWER quadratic

10.8 Lesson Quiz ANSWER exponential; y = 0.025(2) x 2. 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.