3.2 Properties of Quadratic y = ax2+bx+c
Relations
a>0
If the second difference is
positive, the graph opens up and
the graph opens down if the
second difference is negative.
The axis of symmetry is the
vertical line which passes
through the vertex.
If the coordinates of the vertex
is (h, k), the equation of the
axis of symmetry is x = h.
(h, k)
x=h
x=h
(h, k)
y = ax2+bx+c
a<0
The y-coordinate of the
vertex is called the optimum
value of the relation.
(h, k)
y=k
The optimum value is called a
minimum if the parabola
opens up and a maximum if
the parabola opens down.
(h, k)
The axis of symmetry is the
perpendicular bisector of any
horizontal line segment joining
two points on the parabola.
If the parabola crosses the x-axis,
the x-coordinates are called the
zeros or x-intercepts.
Determine the
following:
a) the coordinates of
the vertex (2, – 4)
b) the optimum value
–4
c) the equation of the
axis of symmetry.
x=2
d) the zeros of the relation.
0 and 4
(2, – 4)
Example 1: Sketch the graph of y = 3x2 + 12x
Start with a table of
values.
x
y
-5
15
-4
0
-3
-9
-2
-12
-1
-9
0
0
• The zeros are at 0 and – 4.
• Axis of symmetry (halfway between the zeros).
0   4 
x
2
x=–2
• Substitute x = – 2 into the original equation to obtain.
the vertex
Substitute x = – 2
y = 3x2 + 12x
y = 3(– 2)2 + 12(– 2)
y = – 12
The vertex is at
(–2 , -12 )
Axis of symmetry x = – 2
zeros
Ex 2: The following points lie
on a parabola. Determine the
equation of the axis of
symmetry.
a) (3, 2) and (5, 2)
The axis of symmetry lies
halfway between 3 and 5.
35
x
2
(3, 2)
(5, 2)
The following points lie on a
parabola. Determine the
equation of the axis of
symmetry.
b) (–3.25, –2) and (2.5, – 2)
The axis of symmetry
lies halfway between
–3.25 and 2.5.
3.25  2.5
= – 0.375
2
(–3.25, –2)
(2.5, –2)
The equation of the axis of
symmetry is x = – 0.375.
Properties of Quadratic Relations (2)
A golf ball is hit in the air. Its
height is given by the equation:
h = 50t – 5t2, where h is the
height in metres and t is the
time in seconds.
a) What are the zeros of the relation?
b) When does the ball hit the ground?
c) What are the coordinates of the vertex?
d) Graph the relation.
e) What is the maximum height of the golf ball?
f) After how many seconds does that occur?
a) What are the zeros of the relation?
Step 1: Set the WINDOW to the
following settings.
Reminder (–)
Press WINDOW
2nd TRACE
Use arrows to
cursor to the left
and right of the
two x-intercepts
(or zeros).

Press
Press GRAPH

Press Y= and enter the equation
C:\Documents and Settings\Cheryl Ann\My
Documents\MPM 2D1\Unit 3\Golf Example.84state
a) What are the zeros of the relation?
The zeros are 0 and 10.
b) When does the ball hit the ground?
The ball hits the ground at 10 seconds.
vertex
c) What are the coordinates
of the vertex?
V(5, 125)
d) Graph the relation
e) What is the maximum
height of the golf ball?
125 m
f) After how many seconds
does that occur?
5s
zeros
Homework: pg 145 #1 – 7, 9
– 15 (for 12 – 15, graph using
TOV)