```The Graph of a Quadratic Function
The graph of a quadratic function
2
•
f ( x)  ax  bx  c
• is called a Parabola
• The parabola opens up when a > 0
• The parabola opens down when a < 0
• When the parabola opens up the lowest
point on the graph is called the Vertex
• When the parabola opens down the highest
point on the graph is called the Vertex
The Graph of a Quadratic Function (cont)
• The Vertex of a quadratic function is a pair
(x, y) where the x coordinate is sometimes
denoted by h and the y by k.
2
The function f (x) = ax +bx + c can be written
in the form
2
f (x) = a (x + h) + k
by completing the square
Using Transformations
x 2  2x  3
We complete the square and get
2
f (x)=(x -1) – 4. Here h = 1 and k = -4
The graph is y = x shifted horizontal right 1
unit and vertically down 4 units. The
result is given below
2
2
f (x)=(x -1) - 4
(1,-4)
vertex
Using Transformations
To find the vertex of the given function
2
f (x) = 3x + 6x +2
We write f (x) as
2
3(x + 2x
) + 2 then
2
3(x + 2x +1 ) + 2 -3(1)
3(x + 1)
2
-1
2
2
y = 3x + 6x +2
2
Vertex(-1, -1)
2
y = (x + 1) -1
The Graph of a Quadratic Function
The vertical line which passes through the vertex
is called the Axis of Symmetry or the Axis
Recall that the equation of a vertical line is x =c
For some constant c
x coordinate of the vertex  b
2a
4ac  b 2
The y coordinate of the vertex is
2a
The Axis of Symmetry is the x =
b
2a
•
Opens up when a>o
opens down a < 0
Vertex
axis of symmetry
Identify the Vertex and Axis of
• Vertex =(x, y). thus
• Vertex =   b , f   b

2a


 
2a  
• Axis of Symmetry: the line x =
b
2a
• Vertex is minimum point if parabola opens up
• Vertex is maximum point if parabola opens
down
Identify the Vertex and Axis of
Symmetry
f ( x)  3x  6x
2
• Vertex x =
y=
b
2a
=

6
2(3)
= -1
 b 
f     f  1  -3
 2a 
• Vertex = (-1, -3)
b
• Axis of Symmetry is x = 2a
= -1
Function
Method I
2
f (x) = ax + bx + c = 0
a 0
1. Complete the square to write the function as
2
f (x) = a (x – h) + k
2. Graph function in stages using transformation
Function
Method 2
Determine
1. the vertex
 b  b 
  , f    
 2a  2a  
2. the Axis of Symmetry: the line x =
3. the y intercept. That is f(0)
b
2a
Function(cont.)
4. Determine the x–intercept, that is, f (x) = 0
a. If b 2  4ac  0
there are 2 x-intercepts
and the graph crosses the x axis at 2 points
b. If
there is 1 x-intercept
b 2  4ac  0
and the graph crosses the x axis at 1 point
c. If
there are no x-intercepts
b 2  4ac  0
and the graph does not cross the x axis.
5. Use the Axis of Symmetry and y –intercept to get
an additional point and plot the points
Finding the Maximum and
Minimum Points
• If the parabola opens up, that is, if a > 0
• the vertex is the lowest point on the graph and
the y coordinate of the vertex is the minimum
If the parabola opens down, that is, if a < 0
the vertex is the highest point on the graph and
the y coordinate of the vertex is the maximum
Finding the Maximum and
Minimum Points
Page 153 # 60
Find the maximum or minimum
2
f (x) = -2x + 12x
a = -2 < 0.
Thus the parabola opens down and has a maximum
b
12
Vertex x = 2 a = 
= 3
2(2)
y = f(3) = 18
Vertex = (3, 18)
Maximum is y = 18
Finding the Maximum and
Minimum Points
•
•
•
•
•
2
f(x) = 4x – 8x + 3 a = 4 >0.
Thus the parabola opens up and has a minimum
Vertex (x = -(-8)/2(4) = 1, y=f(1)= -1)
Vertex(1, -1)
Minimum = y=f(1)= -1
• Note the range is y  -1
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