1. Quadratic graphs
Today we will be able to construct graphs of quadratic equations that model real life problems

2. The Graph of a Quadratic Function
All parabolas are symmetric with respect to a line called the axis of symmetry.
The point where the axis intersects the parabola is the vertex of the parabola.
The Graph of a Quadratic Function
Leading coefficient is positive.
Leading coefficient is negative.

3. The leading coefficient of ax2 + bx + c is a.y
a > 0 opens upward
When the leading coefficient is positive, the parabola opens upward and the vertex is a minimum.
f(x) = ax2 + bx + c
vertex minimum x y
vertex maximum
When the leading
coefficient is negative, the parabola opens downward and the vertex is a maximum.
f(x) = ax2 + bx + c
a < 0 opens downward
Leading Coefficient

4. Axis of symmetry Formula for:
We can also trace to it on the graphing calculator!

5. The vertex of the graph of f (x) = ax2 + bx + c (a  0)
Vertex of a Parabola
The vertex of the graph of f (x) = ax2 + bx + c (a  0)
Example: Find the vertex of the graph of f (x) = x2 – 10x + 22.
f (x) = x2 – 10x original equation
a = 1, b = –10, c = 22
At the vertex,
So, the vertex is (5, -3).
Vertex of a Parabola

6. Find the maximum height of the ball.
Example: A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is:
The path is a parabola opening downward. The maximum height occurs at the vertex.
Find the maximum height of the ball.
Example: Basketball

7. The maximum height of the ball is 15 feet.
Example: A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is:
The path is a parabola opening downward. The maximum height occurs at the vertex.
At the vertex,
So, the vertex is (9, 15).
The maximum height of the ball is 15 feet.
Example: Basketball

8. Graph the following path of a ball
Y=-x2+6x
Find the
Max height
After how
Seconds does the ball hit the ground.

9. Plot the points 10 8 6 4 2 0 2 4 6 8 10 Max height is 9H E I G T
The ball hits the ground
After 6 seconds
SECONDS

10. THINGS TO KEEP IN MIND WHEN GRAPHING
CHANGE THE VIEWING WINDOW
BY CHANGING THE Y MIN AND Y MAX.
REMEMBER THAT HEIGHT IS THE VERTICAL
AXIS AND TIME IS THE HORIZONTAL AXIS