Quadratic Functions

1
Quadratic Functions

• Lesson 2.6

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

3
Finding Zeros

• Often with quadratic functions    
  f(x) ax2 bx c   we speak of finding the
  zeros
• This means we wish to find all possible values of
  x for which    ax2 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    gt x 4     or
• x 2 0    gt 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
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
Assignment

• Lesson 2.6
• Page 92
• Exercises 1 31 Odd