Section 2.2
Quadratic Functions
Graphs of Quadratic Functions
Graphs of Quadratic Functions
2
Parabolas f ( x)  ax  bx  c
Vertex
Minimum
Maximum
Axis of symmetry
y

y

















x
x















































Quadratic functions are any function of the form
f(x)=ax +bx+c where a  0, and a,b and c are
real numbers. The graph of any quadratic
function is called a parabola. Parabolas are
shaped like cups. Parabolas are symmetic with
respect to a line called the axis of symmetry.
If a parabola is folded along its axis of symmetry,
the two halves match exactly.
2
Graphing Quadratic Functions
in Standard Form
Seeing the Transformations
Using Standard Form
f ( x)  2( x  3)2  8
y


1.
vertex(h, k ) V(3,8)


2. axis of symmetry x=3
3.

finding the x intercept, let y=0

0=-2(x-3)2  8

8 2( x  3)

2
2
4  ( x  3)


2
x


















2


 4  ( x  3) 2

2  x  3
3  2  x, (5,0) (1,0)
4. finding the y intercept let x=0
y=-2(0-3)  8
2
y=-10 (0,-10)




a<0 so parabola has
a minimum, opens
down




Example
Graph the quadratic function f(x) = - (x+2)2 + 4.
y






x


























Example
Graph the quadratic function f(x) = (x-3)2 - 4
y






x


























Graphing Quadratic Functions
in the Form f(x)=ax2+bx+c
Using the form f(x)=ax2+bx+c
f ( x)  x2  2 x  1 a=1, b=2, c=1
 -b
,f
 2a
1. Vertex 
y
-2
 b  
 2a   x= 2 1  1
 
f (1)  (1)2  2(1)  1  0





V(-1,0)


2. Axis of symmetry x=-1
3. Finding x intercept
0=x 2  2 x  1
0  ( x  1)( x  1)
x 1  0
x  1
(-1,0) x intercept
4. Finding y intercept
y=02  2  0  1
y 1
(0,1) y intercept












x













a>0 so parabola has a
minimum, opens up






Example
Find the vertex of the function f(x)=-x2-3x+7
Example
Graph the function f(x)= - x2 - 3x + 7. Use the
graph to identify the domain and range.
y






x


























Minimum and Maximum Values
of Quadratic Functions
Example
For the function f(x)= - 3x2 + 2x - 5
Without graphing determine whether it has a
minimum or maximum and find it.
Identify the function’s domain and range.
Applications of Quadratic
Functions
Example
You have 64 yards of fencing to enclose a
rectangular region. Find the dimensions of the
rectangle that maximize the enclosed area. What is
the maximum area?
Find the coordinates of the vertex for
the parabola defined by the given equation.
f(x)=2(x-4) 2  5
(2,5)
(b) ( 2, 5)
(c) ( 4,5)
(d) (4,5)
(a)
Use the graph of the parabola to
determine the domain and range of
the function. f(x)=x 2  6 x  4
(a) D: 3,   R:  ,  
(b) D:  ,   R: 13,  
(c) D:  ,3 R: 13,  
(d) D:  ,-3 R:  -13, 