3.1 Quadratic Functions and Models
A quadratic function is any function of the form
f(x)=
The graph is a ____________ .
It either has a minimum or a maximum value (vertically).
Vertex form of a quadratic function:
f(x)=a(x­h)2+k
(h,k) is the ___________ .
Graph:
Vertex is a minimum.
If a>0, the parabola opens ____ .
Vertex is a maximum.
If a<0, the parabola opens ____ .
f(x)=(x-3)2+1
x=h is the axis of ___________ .
1
To convert a quadratic function into vertex form, complete the square.
Suppose f(x)=ax2+bx+c.
1) Divide both sides by "a."
2) Complete the square: Take half of the "b," square, then add to both sides.
3) Solve for f(x).
Let f(x)=2x2+8x+7
Graph:
f(x)=3x2-18x+22
First find the vertex form:
2
Vertex form: f(x)=a(x­h)2+k
Suppose f(x) is a quadratic function with a vertex at (1,­2) and passes through (3,­6). Find an equation for f(x). Let f(x)=x2­6x­7.
Find the vertex form:
Where is the vertex? Does it correspond to a minimum or a maximum?
Does it open up or down?
Find x­intercepts, if any (set f(x)=0):
Find the y­intercept (set x=0):
What is the domain? What is the range?
3
Formula for the vertex:
For f(x)=ax2+b+c the vertex occurs at:
For example, the vertex of f(x)=2x2­4x+3 is:
A farmer wants to fence in a rectangular garden against a side of a long barn using 200 feet of fencing. What are the dimensions that give the farmer maximum area for his garden? What is that area?
Area: A(x)=
x
200­2x
x
BARN
The graph of this function is a
parabola that opens _____.
So the vertex corresponds to a _______ .
4
You try:
1) Find the vertex form of f(x)=x2+10x­8.
2) Graph g(x)= ­(x­1)2+3.
3) Find the maximum value for h(x)=­x2+12x­13.
5