1. Daily Check
Factor: 3x2 + 10x + 8
Factor and Solve: 2x2 - 7x + 3 = 0

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. 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!)

5. Vertex Form (x – h)2 + k – vertex form
Each function we just looked at can be written in the form (x – h)2 + k, where (h , k) is the vertex of the parabola, and x = h is its axis of symmetry.
(x – h)2 + k – vertex form
Equation
Vertex
Axis of Symmetry
y = x2 or y = (x – 0)2 + 0
(0 , 0)
x = 0
y = x2 + 2 or y = (x – 0)2 + 2
(0 , 2)
y = (x – 3)2 or y = (x – 3)2 + 0
(3 , 0)
x = 3

6. Example 1: Graph y = (x + 2)2 + 1
Analyze y = (x + 2)2 + 1.
Step 1 Plot the vertex (-2 , 1)
Step 2 Draw the axis of symmetry, x = -2.
Step 3 Find and plot two points on one side , such as (-1, 2) and (0 , 5).
Step 4 Use symmetry to complete the graph, or find two points on the
left side of the vertex.

7. Your Turn!
Analyze and Graph:
y = (x + 4)2 - 3.
(-4,-3)

8. 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

9. Table of values with 4 points (other than the vertex?
Now you try one!
y=2(x-1)2+3
Open up or down?
Vertex?
Axis of symmetry?
Table of values with 4 points (other than the vertex?

10. (-1, 11)
(3,11)
X = 1
(0,5)
(2,5)
(1,3)

11. 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.

12. 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

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

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

15. Changing from vertex or intercepts form to standard form
The key is to FOIL! (first, outside, inside, last)
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

16. Challenge Problem
Write the equation of the graph in vertex form.