Math II

UNIT QUESTION: Can real world data be modeled by algebraic functions?Standard: MM2D1, D2

Today’s Question:What methods can be used to find the equation of a quadratic?Standard: MM2D2c

QUADRATIC REGRESSION

Given a set of discrete data we want to determine a quadratic equation model that approximates the data as “close” as possible. The phrase quadratic of best fit or quadratic regression is often used.

Since we want the quadratic of best fit we want the equation of the quadratic that minimizes the sum of the squares of the distances from the data points to the parabola defined by the quadratic equation.

How is the computation performed?For our course there are formulas for the coefficients a, b, and c of the quadratic of best fit given by

f(x) = ax2 + bx + c.

We will use calculators to get the numeric values.

The data set appears to have a parabolic form. A “trial” parabola is shown.

To get a “better fit” we try to adjust the coefficients of f(x) = ax2 + bx + c

The objective is to minimize the sum of the squares of the vertical line segments shown.

Example: Approximate the path of the projectile the “Human Cannon Ball”.

The table shows the monthly sales (thousands) for a new hair salon since its grand opening in March.

1) Find the best fitting quadratic model.2) What will be the total sales in September?

QUADRATIC REGRESSION EXAMPLE

Month, t

0 1 2 3 4 5

Sales. S

5.6 5.8 6.2 6.9 7.9 9.0

y=0.12x2+0.07x+5.6

$10,450

Case 1: Given the vertex and a point.

1) Substitute given info into:

2) Solve for a.

3) Rewrite the equation using the a that you find and the vertex point.

QUADRATIC REGRESSION BY HAND

2( )y a x h k

Case 1: Given the vertex and a point.

Ex 1: Vertex (-2,3) and Point (-1,1)

QUADRATIC REGRESSION BY HAND

2( )y a x h k 2( ( 2)) 3y a x

2( 2) 3y a x 21 ( 1 2) 3a

1 3a 2a

22( 2) 3y x

Case 1: Given the vertex and a point.

Ex 2: Vertex (1,-4) and Point (0,-3)

QUADRATIC REGRESSION BY HAND

2( )y a x h k 2( 1) 4y a x 23 (0 1) 4a

3 4a

1a 2( 1) 4y x

Case 2: Given 2 zeroes and a point

1) Substitute given info into:

2) Solve for a.

3) Rewrite the equation using the a that you find and the vertex point.

QUADRATIC REGRESSION BY HAND

( )( )y a x p x q

Case 1: Given 2 zeroes and a point.

Ex 1: Zeroes at -2 and 1, and Point (-1,-4)

QUADRATIC REGRESSION BY HAND

( )( )y a x p x q

( 2)( 1)y a x x

4 ( 1 2)( 1 1)a

4 (1)( 2)a

4 2a 2a 2( 2)( 1)y x x

Workbook Page 267 #10-11

Class work

Page 257 #17-20

Homework