1. Parabolas
Objective:
Understand Properties of Parabolas, Translate Parabolas, and Quadratic Functions

2. What is a “quadratic” function?

3. The standard form for a quadratic function:
y = ax2 + bx + c, where “a” is not = 0.
Why can’t “a” be zero??

4. The graph of any quadratic function is called a parabola.
What are some real life examples of a parabola?

5. Parabolas in Real Life

6. Vertex
Axis of Symmetry
Minimum Value
Maximum Value

7. Identify the vertex, axis of symmetry, and the maximum or minimum value:

8. The graph of y= ax2 + bx + c has the following properties:
a > 0 means it opens up
a < it opens down
x = is the axis of symmetry
(0, c) is the y-intercept

9. Examples: Graph in Calculator – Do not copy in notes!
y = 5x2
y = ½x2 3.

10. Remember……….. If “a” is negative, the parabola opens down.
If “a” is positive, the parabola opens up.
If “a” is negative, the parabola opens down.

11. As the value of |a| decreases, the parabola gets wider.
As the value of |a| increases, the parabola gets narrower.
As the value of |a| decreases, the parabola gets wider.

12. Graphing Quadratic Functions
For standard form:
Find the x coordinate of vertex:
Make a table of values
Graph

13. Vertex form of a parabola:
y = a (x – h)2 + k
where the vertex is the point (h, k) and the axis of symmetry is the vertical line x = h.

14. Examples: Find the vertex & the axis of symmetry: 1
Examples: Find the vertex & the axis of symmetry: 1. y = 3 (x – 2) y = (x – 2)2 3. y = -0.3 (x +1) y = (x +1)2 + 9

15. y = x2 vertex: axis of symmetry: direction of opening:
Problems for you to do! Find the vertex, axis of symmetry and direction of opening.
y = x2
vertex:
axis of symmetry:
direction of opening:

16. y = ( x – 2 )2
vertex:
axis of symmetry:
direction of opening:

17. 3. y = -2( x – 2 )2 + 3
vertex:
axis of symmetry:
direction of opening:
Graph:

18. Math 3 in Italian!

19. More complicated stuff……. The “focus” and the “directrix”

20. Focus and Directrix Facts:
The focus lies ON the axis of symmetry at the point
The vertex, of course, is the point
The directrix is a horizontal line on the side of the vertex opposite the focus; its equation is given by
So, we need to figure out the value of “p”!
“p” is given by the formula

21. Let’s Recap