Download Describing Data - Harrison High School

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”.
QUADRATIC REGRESSION EXAMPLE
The table shows the monthly sales (thousands) for a new hair
salon since its grand opening in March.
Month, 0
t
1
2
3
4
5
Sales.
S
5.8
6.2
6.9
7.9
9.0
5.6
1) Find the best fitting quadratic model. y=0.12x2+0.07x+5.6
2) What will be the total sales in September?
$10,450
QUADRATIC REGRESSION BY HAND
Case 1: Given the vertex and a point.
1) Substitute given info into: y  a( x  h)  k
2) Solve for a.
3) Rewrite the equation using the a that you
find and the vertex point.
2
QUADRATIC REGRESSION BY HAND
Case 1: Given the vertex and a point.
Ex 1: Vertex (-2,3) and Point (-1,1)
y  a ( x  h) 2  k
y  a( x  (2))2  3
y  a( x  2)2  3
1  a(1  2)2  3
1 a 3
a  2
y  2( x  2)2  3
QUADRATIC REGRESSION BY HAND
Case 1: Given the vertex and a point.
Ex 2: Vertex (1,-4) and Point (0,-3)
y  a ( x  h) 2  k
y  a( x  1)2  4
3  a(0  1)2  4
3  a  4
a 1
y  ( x  1)2  4
QUADRATIC REGRESSION BY HAND
Case 2: Given 2 zeroes and a point
1) Substitute given info into: y  a( x  p)( x  q)
2) Solve for a.
3) Rewrite the equation using the a that you
find and the vertex point.
QUADRATIC REGRESSION BY HAND
Case 1: Given 2 zeroes and a point.
Ex 1: Zeroes at -2 and 1, and Point (-1,-4)
y  a( x  p)( x  q )
y  a ( x  2)( x  1)
4  a(1  2)(1  1)
4  a (1)(2)
4  2a
a2
y  2( x  2)( x  1)
Class work
Workbook Page 267 #10-11
Homework
Page 257 #17-20