Quadratic models Warm Up Solve each system of equations. 1. 3a + b = 5 2a 6b = 30 2. 9a + 3b = 24 a+b=6 3. 4a 2b = 8 2a 5b = 16 a = 0, b = 5 a = 1, b = 5 a = , b = 3

Objectives Use quadratic functions to model data. Use quadratic models to analyze and predict. Vocabulary quadratic model quadratic regression Recall that you can use differences to analyze patterns in data. For a set of ordered parts with equally spaced x-values, a quadratic function has constant nonzero second differences, as shown below. Example 1A: Identifying Quadratic Data Determine whether the data set could represent a quadratic function. Explain. x y 1

3 5 7 9 1 1 7 17 31 Find the first and second differences. Equally spaced x-values x

y 1st 2nd 1 3 5 7 9 1 1 7 17 31

2 6 4 10 4 14 4 Quadratic function: second differences are constant for equally spaced xvalues Example 1B: Identifying Quadratic Data Determine whether the data set could represent a quadratic function. Explain. x y 3

4 5 6 7 1 3 9 27 81 Equally spaced x-values x y 1st 2nd

3 4 5 6 7 1 3 9 27 81 2 6

4 18 12 54 36 Find the first and second differences. Not a Quadratic function: second differences are not constant for equally spaced x-values Check It Out! Example 1a Determine whether the data set could represent a quadratic function. Explain. x y 3

4 5 6 7 11 21 35 53 75 Find the first and second differences. Equally spaced x-values

x y 1st 2nd 3 4 5 6 7 11 21 35 53 75

10 14 4 18 4 22 4 Quadratic function: second differences are constant for equally spaced xvalues Check It Out! Example 1b Determine whether the data set could represent a quadratic function. Explain. x y 10

9 8 7 6 6 8 10 12 14 Find the first and second differences. Equally spaced x-values

x y 1st 2nd 10 9 8 7 6 6 8 10 12 14

2 2 0 2 0 2 0 Not a quadratic function: first differences are constant so the function is linear. Just as two points define a linear function, three noncollinear points define a quadratic function. You can find three coefficients a, b, and c, of f(x) = ax2

+ bx + c by using a system of three equations, one for each point. The points do not need to have equally spaced x-values. Reading Math Collinear points lie on the same line. Noncollinear points do not all lie on the same line. Example 2: Writing a Quadratic Function from Data Write a quadratic function that fits the points (1, 5), (3, 5) and (4, 16). Use each point to write a system of equations to find a, b, and c in f(x) = ax2 + bx + c. (x, y) (1, 5) f(x) = ax2 + bx + c 5 = a(1)2 + b(1) + c System in a, b, c 1 a + b + c = 5 (3, 5)

5 = a(3)2 + b(3) + c 9a + 3b + c = 5 2 (4, 16) 16 = a(4)2 + b(4) + c 16a + 4b + c = 16 3 1 Example 2 Continued Subtract equation 2 by equation 1 to get 4 . 2 1

4 9a + 3b + c = 5 a + b + c = 5 8a + 2b + 0c = 10 Subtract equation 3 by equation 1 to get 5 . 16a + 4b + c = 16 a + b + c = 5 3 1 5 15a + 3b + 0c = 21 Example 2 Continued Solve equation elimination. 5 4

4 and equation 5 for a and b using 2(15a + 3b = 21) 3(8a + 2b = 10) 30a + 6b = 42 24a 6b = 30 6a + 0b = 12 a =2 Multiply by 2. Multiply by 3. Subtract. Solve for a. Example 2 Continued Substitute 2 for a into equation 4 or equation 5 to get b. 8(2) +2b = 10

15(2) +3b = 21 2b = 6 b = 3 3b = 9 b = 3 Example 2 Continued Substitute a = 2 and b = 3 into equation solve for c. 1 to (2) +(3) + c = 5 1 + c = 5 c = 4 Write the function using a = 2, b = 3 and c = 4. f(x) = ax2 + bx + c f(x)= 2x2 3x 4

Example 2 Continued Check Substitute or create a table to verify that (1, 5), (3, 5), and (4, 16) satisfy the function rule. Check It Out! Example 2 Write a quadratic function that fits the points (0, 3), (1, 0) and (2, 1). Use each point to write a system of equations to find a, b, and c in f(x) = ax2 + bx + c. (x,y) (0, 3) f(x) = ax2 + bx + c 3 = a(0)2 + b(0) + c System in a, b, c 1 c = 3 (1, 0) 0 = a(1)2 + b(1) + c

a+b+c=0 2 (2, 1) 1 = a(2)2 + b(2) + c 4a + 2b + c = 1 3 1 Check It Out! Example 2 Continued Substitute c = 3 from equation equation 2 and equation 3 . 2 a+b+c=0 a+b3=0 a+b=3

3 4 1 into both 4a + 2b + c = 1 4a + 2b 3 = 1 4a + 2b = 4 5 Check It Out! Example 2 Continued Solve equation elimination. 4 5 4

and equation 4(a + b) = 4(3) 4a + 2b = 4 5 for b using 4a + 4b = 12 (4a + 2b = 4) 0a + 2b = 8 b=4 Multiply by 4. Subtract. Solve for b. Check It Out! Example 2 Continued Substitute 4 for b into equation to find a. 4

a+b=3 a+4=3 a = 1 4 or equation 5 5 4a + 2b = 4 4a + 2(4) = 4 4a = 4 a = 1 Write the function using a = 1, b = 4, and c = 3. f(x) = ax2 + bx + c f(x)= x2 + 4x 3 Check It Out! Example 2 Continued

Check Substitute or create a table to verify that (0, 3), (1, 0), and (2, 1) satisfy the function rule. You may use any method that you studied in Chapters 3 or 4 to solve the system of three equations in three variables. For example, you can use a matrix equation as shown. A quadratic model is a quadratic function that represents a real data set. Models are useful for making estimates. In Chapter 2, you used a graphing calculator to perform a linear regression and make predictions. You can apply a similar statistical method to make a quadratic model for a given data set using quadratic regression. Helpful Hint The coefficient of determination R2 shows how well a quadratic function model fits the data. The closer R2 is to 1, the better the fit. In a model with R2 0.996, which is very close to 1, the quadratic model is a good fit.

Example 3: Consumer Application The table shows the cost of circular plastic wading pools based on the pools diameter. Find a quadratic model for the cost of the pool, given its diameter. Use the model to estimate the cost of the pool with a diameter of 8 ft. Diameter (ft) Cost 4 \$19.95 5 6 7 \$20.25 \$25.00 \$34.95 Example 3 Continued Step 1 Enter the data into two lists in a

graphing calculator. Step 2 Use the quadratic regression feature. Example 3 Continued Step 3 Graph the data and function model to verify that the model fits the data. Step 4 Use the table feature to find the function value x = 8. Example 3 Continued A quadratic model is f(x) 2.4x2 21.6x + 67.6, where x is the diameter in feet and f(x) is the cost in dollars. For a diameter of 8 ft, the model estimates a cost of about \$49.54. Check It Out! Example 3

The tables shows Film Run Times (16 mm) approximate run Diameter Reel Length Run Time times for 16 mm (in) (ft) (min) films, given the diameter of the film 5 200 5.55 on the reel. Find a 7 400 11.12 quadratic model for the reel length given 9.25 600 16.67 the diameter of the 10.5

800 22.22 film. Use the model to estimate the reel 12.25 1200 33.33 length for an 8-inch13.75 1600 44.25 diameter film. Check It Out! Example 4 Continued Step 1 Enter the data into two lists in a graphing calculator. Step 2 Use the quadratic regression feature. Check It Out! Example 4 Continued Step 3 Graph the data

and function model to verify that the model fits the data. Step 4 Use the table feature to find the function value x = 8. Check It Out! Example 4 Continued A quadratic model is L(d) 14.3d2 112.4d + 430.1, where d is the diameter in inches and L(d) is the reel length. For a diameter of 8 in., the model estimates the reel length to be about 446 ft. Lesson Quiz: Part I Determine whether each data set could represent a quadratic function. 1. 2. x

y 5 6 7 8 9 5 8 13 21 34 x

2 3 4 5 6 y 1 11 25 43 65 not quadratic

quadratic 3. Write a quadratic function that fits the points (2, 0), (3, 2), and (5, 12). f(x) = x2 + 3x 2 Lesson Quiz: Part II 4. The table shows the prices of an ice cream cake, depending on its side. Find a quadratic model for the cost of an ice cream cake, given the diameter. Then use the model to predict the cost of an ice cream cake with a diameter of 18 in. Diameter (in.) Cost 6 \$7.50 10

\$12.50 15 \$18.50 f(x) 0.011x2 + 1.43x 0.67; \$21.51

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