1. Quadratic Functions
Lesson 3.1

2. Applications of Parabolas
Today we look at functions which describe parabolas.
Solar rays reflect off a parabolic mirror and focus at a point
This could make a good solar powered cooker (also wireless antenna reflector) YouTube view

3. Finding Zeros
Often with quadratic functions     f(x) = a*x2 + bx + c   we speak of “finding the zeros”
This means we wish to find all possible values of x for which    a*x2 + bx + c = 0

4. Finding Zeros
Another way to say this is that we are seeking the x-axis intercepts
This is shown on the graph below
Here we see two zeros – what other possibilities exist?

5. Factoring Given the function x2 - 2x - 8 = 0
 Factor the left side of the equation    (x - 4)(x + 2) = 0
We know that if the product of two numbers   a * b = 0     then either ...
a = 0     or
b = 0
Thus either
x - 4 = 0    ==> x = 4     or
x + 2 = 0    ==> x = -2

6. Warning!!
Problem ... many (most) quadratic functions are NOT easily factored!! 
 Example:

7. The Quadratic Formula
 It is possible to create two functions on your calculator to use the quadratic formula.
quad1 (a,b,c)           which uses the    -b + ...
quad2 (a,b,c)           which uses the    -b -

8. Click to view Spreadsheet Solution
The Quadratic Formula
Try it for the quadratic functions
4x2 - 7x + 3 = 0                          
6x2 - 2x + 5 = 0
Click to view Spreadsheet Solution

9. The Quadratic Formula
4x2 - 7x + 3 = 0  

10. The Quadratic Formula
Why does the second function give "non-real result?“
6x2 - 2x + 5 = 0

11. Concavity and Quadratic Functions
Quadratic function graphs as a parabola
Will be either concave up
Or Concave Down

12. Applications Consider a ball thrown into the air
It's height (in feet) given by h(t) = 80t – 16t 2
Evaluate and interpret h(2)
Solve the equation h(t) = 80
Interpret the solution
Illustrate solution on a graph of h(t)

13. Quadratic Regression
Our calculators will determine quadratic modeling functions
Enter numbers into Data Matrix
Choose F5, Calc
Then choose Quadratic Regression
Set x, y as before
Send results to Y= screen

14. WindSurfer Curve

15. Assignment
Lesson 3.1
Page 108
Exercises 1 – 35 Odd