E. Comparing Linear, Exponential, and Quadratic Models (pp. 96 99) The linear, exponential, and quadratic functions you have studied can be used to model data. The following examples describe methods for determining which type of function best models a set of ordered pairs. 1. Choose Functions Using Sets of Ordered Pairs Use a graph to tell whether the ordered pairs represent a linear function, an exponential function, or a quadratic function. a. (4, 10), (, 4), (0, ), (, 4), (4, 10) b. (4, 6), (, 4), (0, ), (, 0), (4, ) Linear function: y 5 mx 1 b Exponential function: y 5 ab x Quadratic function: y 5 ax 1 bx 1 c Draw a smooth curve through the points you plot. c. 1, 1 } 7, 11, 1 } 9, 10, 1 } 3, (1, 1), (, 3) Solution: a. c. 1 y 10 8 6 4 864 4 6 8 x 4 Quadratic function 3 1 43 1 1 1 Exponential function y x b. 6 5 4 3 1 43 1 1 3 4 x 1 y Linear function Use a graph to tell whether the ordered pairs represent a linear function, an exponential function, or a quadratic function. 1. (, 1), (1, 0), (0, 1), (1, ), (, 3). 1, 3 } 8, 11, 3 } 4, 10, 3 }, (1, 3), (, 6) 3. (, 3), 11, 3 }, (0, 1), 11, 3 }, (, 3) 96 Benchmark 5 Chapters 9 and 10
The table of values represents: an exponential function if the ratios of successive y-values are all equal a linear function if the differences of successive y-values are all equal a quadratic function if the second differences are all equal. Identify Functions Using Differences or Ratios a. Ratios: y 0.065 0.5 1 4 16 b. y 1 3 5 7 9 0.5 } 0.065 5 4 4 4 4 Differences: c. Exponential function Linear function y 4 4 10 0 First differences: 6 10 Second differences: 4 4 4 The table of values represents a quadratic function. Check your function by plotting the ordered pairs from the table on the same grid as a graph of the function. The graph should pass through the plotted points. 4. 6. y 3 0 3 6 9 5. y 0.5 8 3 18 3. Write an Equation for the Function y 7 4 3 4 7 Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Then write an equation for the function. Step 1: Determine which type of function the table of values represents. First differences: 1 4 4 1 Second differences: 8 8 8 The table of values represents a quadratic function because the second differences are equal. Benchmark 5 Chapters 9 and 10 97
Step : Write an equation for the quadratic function. The equation has the form y 5 ax. Find the value of a by using the coordinates of a point that lies on the graph, such as (1, 4). y5 ax Write equation for quadratic function. 4 5 a(1) Substitute 1 for x and 4 for y. 4 5 a Solve for a. The equation is y 5 4x. Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Then write an equation for the function. 7. 8. 9. y 0. 0 0. 0.8 1.8 y 8 5 1 4 y 1 3 5 7 9 10. Quiz y 1 3 0 3 1 Use a graph to tell whether the ordered pairs represent a linear function, an exponential function, or a quadratic function. 1. (4, 9), (, 3), (0, 1), (, 3), (4, 9). (4, 10), (, 4), (0, ), (, 8), (4, 14) 3. (4, 0.065), (, 0.5), (0, 1), (, 4), (4, 16) 98 Benchmark 5 Chapters 9 and 10
4. 5. y 3 7 1 17 y 1 } 9 1 }3 1 3 9 6. y 6 3 4 9 18 Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Then write an equation for the function. 7. 8. y 3 1 1 3 5 9. 10. y 0.1 0 0.1 0.4 0.9 y 6 6 10 Benchmark 5 Chapters 9 and 10 99