1. The Graph of a Quadratic Function
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

2. 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.
The function f (x) = ax +bx + c can be written in the form
f (x) = a (x + h) + k
by completing the square

3. Graphing the Quadratic Function Using Transformations
Given the Quadratic function
We complete the square and get
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

4. Graph of a Quadratic Function
f (x)=(x -1) - 4
(1,-4)
vertex

5. Graphing the Quadratic Function Using Transformations
To find the vertex of the given function
f (x) = 3x + 6x +2
We write f (x) as
3(x + 2x ) then
3(x + 2x +1 ) (1)
3(x + 1)

6. Graph of a Quadratic Function
y = 3x + 6x +2
Vertex(-1, -1)
y = (x + 1) -1

7. 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
The y coordinate of the vertex is
The Axis of Symmetry is the x =

8. The Quadratic function
Opens up when a>o
opens down a < 0
Vertex
axis of symmetry

9. Identify the Vertex and Axis of Symmetry of a Quadratic Function
Vertex =(x, y). thus
Vertex =
Axis of Symmetry: the line x =
Vertex is minimum point if parabola opens up
Vertex is maximum point if parabola opens down

10. Identify the Vertex and Axis of Symmetry
Vertex x =
y =
Vertex = (-1, -3)
Axis of Symmetry is x =
= -1 = -3
= -1

11. Steps for Graphing Quadratic Function
Method I
Given the Quadratic
f (x) = ax + bx + c = a 0
1. Complete the square to write the function as
f (x) = a (x – h) + k
2. Graph function in stages using transformation

12. Steps for Graphing Quadratic Function
Method 2
Determine
1. the vertex
2. the Axis of Symmetry: the line x =
3. the y intercept. That is f(0)

13. Steps for Graphing Quadratic Function(cont.)
4. Determine the x–intercept, that is, f (x) = 0
a. If there are 2 x-intercepts
and the graph crosses the x axis at 2 points
b. If there is 1 x-intercept
and the graph crosses the x axis at 1 point
c. If there are no x-intercepts
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

14. 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 point of the quadratic function
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 point of the quadratic function

15. Finding the Maximum and Minimum Points
Page 153 # 60
Find the maximum or minimum
f (x) = -2x + 12x a = -2 < 0.
Thus the parabola opens down and has a maximum
Vertex x = = = 3
y = f(3) = 18
Vertex = (3, 18)
Maximum is y = 18

16. Finding the Maximum and Minimum Points
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