Quadratic graphs
Today we will be able to construct
graphs of quadratic equations that
model real life problems
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.
Leading coefficient is positive.
Leading coefficient is negative.
The Graph of a Quadratic Function
The leading coefficient of ax2 + bx + c is a.
y
a>0
When the leading coefficient
opens f(x) = ax2 + bx + c
is positive, the parabola
upward
opens upward and the
x
vertex is a minimum.
vertex
minimum
y
x
vertex
When the leading
maximum
coefficient is negative,
f(x) = ax2 + bx + c
the parabola opens downward
a<0
opens
and the vertex is a maximum.
downward
3
Axis of symmetry
• Formula for:
b
x
2a
• We can also trace to it on the graphing
calculator!
Vertex of a Parabola
The vertex of the graph of f (x) = ax2 + bx + c (a  0)
 b
is   ,
 2a
 b 
f   
 2a  
Example: Find the vertex of the graph of f (x) = x2 – 10x + 22.
f (x) = x2 – 10x + 22 original equation
a = 1, b = –10, c = 22
 b  (10)

5
At the vertex, x 
2a
2(1)
 b
2
f
  f (5)  5  10(5)  22  3
 2a 
So, the vertex is (5, -3).
5
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:
1 2
y   x  2 x  6.
9
The path is a parabola opening downward.
The maximum height occurs at the vertex.
1 2
1
y
x  2x  6  a  , b  2
9
9
Find the maximum height of the ball.
6
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:
1 2
y   x  2 x  6.
9
The path is a parabola opening downward.
The maximum height occurs at the vertex.
1 2
1
y
x  2x  6  a  , b  2
9
9
b
At the vertex, x 
 9.
2a
 b
f
  f 9  15
 2a 
So, the vertex is (9, 15).
The maximum height of the ball is 15 feet.
7
Graph the following path of a ball
• Y=-x2+6x
• Find the
• Max height
• After how
• Seconds does the ball hit the ground.
Plot the points
Max height is 9
10
H
E
I
G
H
T
8
6
4
The ball hits the
ground
After 6 seconds
2
0
0
2
4
6
SECONDS
8
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