5 8 Curve Fitting with Quadratic Models Warm

5 -8 Curve Fitting with Quadratic Models Warm. Up: Get your HW ? s ready Answers to Pg 370 evens: 48) 50) 66) B A Yes; examples will vary [ 0. 5 , 3 ] Holt Algebra 2 You’ll need a calculator today!

5 -8 Curve Fitting with Quadratic Models Objectives Use quadratic functions to model data. Use quadratic models to analyze and predict. Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Vocabulary quadratic model quadratic regression Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models For a set of ordered parts with equally spaced x-values, a quadratic function has constant nonzero second differences, as shown below. Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Example 1: Identifying Quadratic Data Determine whether the data set could represent a quadratic function. Explain. x y Holt Algebra 2 1 3 5 7 9 – 1 1 7 17 31 Find the first and second differences.

5 -8 Curve Fitting with Quadratic Models Example 1: Identifying Quadratic Data Determine whether the data set could represent a quadratic function. Explain. Equally spaced x-values x y 1 st 2 nd Holt Algebra 2 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

5 -8 Curve Fitting with Quadratic Models On Your Own Determine whether the data set could represent a quadratic function. Explain. x y Holt Algebra 2 3 4 5 6 7 11 21 35 53 75 Find the first and second differences.

5 -8 Curve Fitting with Quadratic Models On Your Own Determine whether the data set could represent a quadratic function. Explain. Equally spaced x-values x y 1 st 2 nd Holt Algebra 2 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

5 -8 Curve Fitting with Quadratic Models On Your Own Determine whether the data set could represent a quadratic function. Explain. x y Holt Algebra 2 10 9 8 7 6 6 8 10 12 14 Find the first and second differences.

5 -8 Curve Fitting with Quadratic Models On Your Own Determine whether the data set could represent a quadratic function. Explain. Equally spaced x-values x y 1 st 2 nd Holt Algebra 2 10 9 8 7 6 6 8 10 12 14 2 2 0 2 0 Not a quadratic function: first differences are constant so the function is linear.

5 -8 Curve Fitting with Quadratic Models What if the data came from the real world? Would it be okay if the 2 nd differences weren’t exactly equal? Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models 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. Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Example 3: Consumer Application The table shows the cost of circular plastic wading pools based on the pool’s 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 Holt Algebra 2 4 $19. 95 5 6 7 $20. 25 $25. 00 $34. 95

5 -8 Curve Fitting with Quadratic Models Example 3 Continued Step 1 Enter the data into two lists in a graphing calculator. Press: S e to Edit… Diameter (ft) Cost Holt Algebra 2 4 $19. 95 5 6 7 $20. 25 $25. 00 $34. 95

5 -8 Curve Fitting with Quadratic Models Example 3 Continued Step 2 Use the quadratic regression feature. Press: î to get back to the main screen S> CALC 5: Quad. Reg e You may not see R 2, depending on your calculator’s settings. It tells you how good the model is for the data. R 2 = 1 is perfect. Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Example 3 Continued A quadratic model is f(x) ≈ 2. 4 x 2 – 21. 6 x + 67. 6, where x is the diameter in feet and f(x) is the cost in dollars. Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Example 3 Continued Step 3 Type the model into !. DO NOT ROUND! You may have to adjust the @ settings. Use $ to verify that the model fits the data. Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Example 3: Consumer Application The table shows the cost of circular plastic wading pools based on the pool’s 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 Holt Algebra 2 4 $19. 95 5 6 7 $20. 25 $25. 00 $34. 95

5 -8 Curve Fitting with Quadratic Models Example 3 Continued Step 4 Use the table feature to find the function value for x = 8. Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Example 3 Continued A quadratic model is f(x) ≈ 2. 4 x 2 – 21. 6 x + 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. Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Example 3 Continued What if the largest pool for sale cost $195. 00? What would be its diameter, according to the model? Use the equation (rounding to the tenth is okay): f(x) ≈ 2. 4 x 2 – 21. 6 x + 67. 6, where x is the diameter in feet and f(x) is the cost in dollars. Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Example 3 Continued 195 = 2. 4 x 2 – 21. 6 x + 67. 6 0 = 2. 4 x 2 – 21. 6 x – 127. 4 Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Example 3 Continued What if the largest pool for sale cost $195. 00? What would be its diameter, according to the model? The diameter of a pool that costs $195 would be about 13. 1 feet. Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models On Your Own The table shows the prices of an ice cream cake, depending on its size. 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. Holt Algebra 2 Diameter (in. ) Cost 6 $7. 50 10 $12. 50 15 $18. 50

5 -8 Curve Fitting with Quadratic Models On Your Own f(x) – 0. 011 x 2 + 1. 43 x – 0. 67; f(18) $21. 51 Holt Algebra 2 Diameter (in. ) Cost 6 $7. 50 10 $12. 50 15 $18. 50

5 -8 Curve Fitting with Quadratic Models there’s one more example if there’s time… we can also start homework in class Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Assignment: Pg 377 #13, 29 -35 odd, 43 Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Assignment: Pg 377 #13, 29 -35 odd, 43 Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Assignment: Pg 377 #13, 29 -35 odd, 43 Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Assignment: Pg 377 #13, 29 -35 odd, 43 Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Check It Out! Example 3 The tables shows approximate run times for 16 mm films, given the diameter of the film on the reel. Find a quadratic model for the reel length given the diameter of the film. Use the model to estimate the reel length for an 8 -inchdiameter film. Holt Algebra 2 Film Run Times (16 mm) Diameter (in) 5 Reel Length Run Time (ft) (min) 200 5. 55 7 400 11. 12 9. 25 600 16. 67 10. 5 800 22. 22 12. 25 1200 33. 33 13. 75 1600 44. 25

5 -8 Curve Fitting with Quadratic Models Check It Out! Example 4 Continued Step 1 Enter the data into two lists in a graphing calculator. Holt Algebra 2 Step 2 Use the quadratic regression feature.

5 -8 Curve Fitting with Quadratic Models Check It Out! Example 4 Continued Step 3 Graph the data and function model to verify that the model fits the data. Holt Algebra 2 Step 4 Use the table feature to find the function value x = 8.

5 -8 Curve Fitting with Quadratic Models Check It Out! Example 4 Continued A quadratic model is L(d) 14. 3 d 2 – 112. 4 d + 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. Holt Algebra 2

5 -8 Curve Fitting with Quadratic Models Lesson Quiz: Part I Determine whether each data set could represent a quadratic function. 1. x y 5 6 7 8 9 5 8 13 21 34 2. x y 2 3 4 5 6 1 11 25 43 65 Holt Algebra 2 not quadratic