5.7 Curve Fitting with Quadratic Models Learning Objective: To find a quadratic function that exactly fits three data

points and to find a quadratic model to represent a data set.

Warm-up (IN)

Notes

Learning Objective: To find a quadratic function that exactly fits three data points and to find a quadratic model to represent a data set.

There are 2 ways to fit a curve to a set of data points - 1 – enter data into lists on calc, the find quadreg

2 – use a system of equations

Ex 1 – Find a quadratic function whose graph contains the points (-3,16), (2,6) and (1,-4)

Step 1 Write a system of 3 equations in 3 variables using 2y ax bx c

(-3,16) 16 23a 3b c 9a 3b c 16

(2,6) 6 22a 2b c 4a 2b c 6

(1,-4) 4 21a 1b c a b c 4

9a 3b c 16

4a 2b c 6a b c 4

Learning Objective: To find a quadratic function that exactly fits three data points and to find a quadratic model to represent a data set.

Step 2

Use matrices to solve the system for a, b and c

9 3 14 2 11 1 1

a

b

c

166

4

1A B

(3,1,-8)

y 23x 1x 8

Learning Objective: To find a quadratic function that exactly fits three data points and to find a quadratic model to represent a data set.

Ex 2 – On a trip to St. Louis you visit the Gateway Arch. Since you have plenty of time on your hands, you decide to estimate its height. You walk the distance across the base of the arch and find that it is 162 meters. You assume the arch is in the shape of a parabola and you set up a coordinate system with one end of the arch at the origin. To find a third point, you measure the vertical distance to the arch is 4.55 meters when you are one meter from the base.

162

(0,0) (162,0) (1,4.55)

(0,0) (162,0) (1,4.55)

Learning Objective: To find a quadratic function that exactly fits three data points and to find a quadratic model to represent a data set.

2y ax bx c 0 2

0a 0b c c 0

0 2162a 162b c 26244a 162b c 0

4.55 21a 1b c a b c 4.55

0 0 126244 162 1

1 1 1

a

b

c

00

4.55

1A B

(-0.03,4.58,0)

y 20.03x 4.58xUse calc to find max! 174.8 meters

HW –

Out – Explain why you can always find a quadratic functi0n to fit any three noncollinear points in the coordinate plane.

Summary – I can use this when I…

POW!!