1. Graphing Quadratic Functions
Definitions
3 forms for a quad. function
Steps for graphing each form
Examples
Changing between eqn. forms

2. Quadratic Function
A function of the form y=ax2+bx+c where a≠0 making a u-shaped graph called a parabola.
Example quadratic equation:

3. Vertex- Axis of symmetry- The lowest or highest point of a parabola.
The vertical line through the vertex of the parabola.
Axis of
Symmetry

4. Standard Form Equation
y=ax2 + bx + c
If a is positive, u opens up
If a is negative, u opens down
The x-coordinate of the vertex is at
To find the y-coordinate of the vertex, plug the x-coordinate into the given eqn.
The axis of symmetry is the vertical line x=
Choose 2 x-values on either side of the vertex x-coordinate. Use the eqn to find the corresponding y-values.
Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.

5. Vertex Form Equation y=a(x-h)2+k If a is positive, parabola opens up
If a is negative, parabola opens down.
The vertex is the point (h,k).
The axis of symmetry is the vertical line x=h.
Don’t forget about 2 points on either side of the vertex! (5 points total!)

6. Intercept Form Equation
y=a(x-p)(x-q)
The x-intercepts are the points (p,0) and (q,0).
The axis of symmetry is the vertical line x=
The x-coordinate of the vertex is
To find the y-coordinate of the vertex, plug the x-coord. into the equation and solve for y.
If a is positive, parabola opens up
If a is negative, parabola opens down.

7. Example 1: Graph y=2x2-8x+6 Axis of symmetry is the vertical line x=2
Table of values for other points: x y
0 6
1 0
2 -2
3 0
4 6
* Graph!
a=2 Since a is positive the parabola will open up.
Vertex: use
b=-8 and a=2
Vertex is: (2,-2)
x=2

8. Now you try one. y=-x2+x+12. Open up or down. Vertex. Axis of symmetry
Now you try one! y=-x2+x+12 * Open up or down? * Vertex? * Axis of symmetry? * Table of values with 5 points?

9. (.5,12)
(-1,10)
(2,10)
(-2,6)
(3,6)
X = .5

10. Example 2: Graph y=-.5(x+3)2+4
a is negative (a = -.5), so parabola opens down.
Vertex is (h,k) or (-3,4)
Axis of symmetry is the vertical line x = -3
Table of values x y
-1 2
-3 4
-5 2
Vertex (-3,4)
(-4,3.5)
(-2,3.5)
(-5,2)
(-1,2)
x=-3

11. Table of values with 5 points?
Now you try one!
y=2(x-1)2+3
Open up or down?
Vertex?
Axis of symmetry?
Table of values with 5 points?

12. (-1, 11)
(3,11)
X = 1
(0,5)
(2,5)
(1,3)

13. Example 3: Graph y=-(x+2)(x-4)
Since a is negative, parabola opens down.
The x-intercepts are (-2,0) and (4,0)
To find the x-coord. of the vertex, use
To find the y-coord., plug 1 in for x.
Vertex (1,9)
The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex)
(1,9)
(-2,0)
(4,0)
x=1

14. Now you try one! y=2(x-3)(x+1) Open up or down? X-intercepts? Vertex?
Axis of symmetry?

15. x=1
(-1,0)
(3,0)
(1,-8)

16. Changing from vertex or intercepts form to standard form
The key is to Distribute!
Ex: y=-(x+4)(x-9) Ex: y=3(x-1)2+8
=-(x2-9x+4x-36) =3(x-1)(x-1)+8
=-(x2-5x-36) =3(x2-x-x+1)+8
y=-x2+5x =3(x2-2x+1)+8
=3x2-6x+3+8
y=3x2-6x+11