1. Quadratic Functions
The graph of a quadratic function is called a parabola.
The parent function is given as
This is the parent graph of all quadratic functions.

2. Characteristics of Quadratics
Line (axis) of symmetry - the line about which the parabola is symmetrical.
The graphs of all parabolas have the same general U-shape.
Vertex – the lowest (minimum) or highest (maximum) point of a parabola.

3. Quadratic Functions (-3,9) (3,9)
A table of values can be constructed to graph a quadratic function.
x y
-3 9
-2 4
-1 1
(-2,4)
(2,4)
0 0
1 1
2 4
3 9
(-1,1)
(1,1)
(0,0)

4. Quadratic Functions
All quadratic functions can be expressed either in vertex form:
(-3,9)
(3,9)
(-2,4)
or in standard form:
(2,4)
(-1,1)
(1,1)
(0,0)

5. Quadratic Functions (h,k) When quadratic equations are
written in vertex form, h k
(h,k) identifies the vertex of the parabola.
The line of symmetry is x = h

6. Quadratic Functions (1,8) In vertex form, (5,8) a
a affects the direction the parabola opens and how wide or narrow it will open.
In this problem, a=2 and it is positive, so the parabola opens up and
(2,2)
(4,2)
(3,0)
the y-values are all vertically stretched by a factor of 2. 2

7. Quadratic Functions (3,0) In vertex form, a (4,-2) (2,-2)
If a is negative, the parabola will open down.
Since a=-2 and it is negative, the parabola opens down
and
- 2
- 2
- 2
(5,-8)
(1,-8)
- 2
the y-values are all vertically stretched by a factor of 2.
- 2
- 2
- 2
- 2
- 2

8. Quadratic Functions (3,8)
The points where the parabola intersects the x-axis are called the x-intercepts, roots, zeros or solutions of the function.
(2,6)
(4,6)
These roots occur when the y-value is equal to zero.
Solving for x we get the values:
(1,0) 1
(5,0) 5 1 5 1
X= X= 1 1 5 5 1 5 1 1 5 5 5

9. The zeros of the function can be found by setting y=0.
Quadratic Functions
Now solve for x.
The roots or zeros are:
(1, 0) and (5,0)

10. Quadratic Functions Example: Graph The vertex is (5, -2) (5, -2)
The graph opens upward since 3 is positive.
(5, -2)
(5, -2)
(5, -2)
(5, -2)
(5, -2)
The y-values are multiplied by 3.
(5, -2)
(5, -2)
(5, -2)
(5, -2)
(5, -2)
(5, -2)
Pick x-values to the left and right of the vertex and plug into the equation to plot points.
(5, -2)
(5, -2)
(5, -2)
(5, -2)
(5, -2)
(5, -2)
(5, -2)

11. The zeros of the function can be found by setting y=0.
Quadratic Functions
Now solve for x.
The roots or zeros are:
(4.184, 0) and (5.816,0)