Notes 9.1
Graphing Quadratic Functions
y = ax2 + bx + c
Standard form
of a
Quadratic
Function
I. Quadratic Functions
A. The basics
The graph of a quadratic function is a parabola. A parabola for a quadratic
function can open up or down, but not left or right.
The vertex is either the highest or lowest
point on the graph depending on whether it
opens up or down. If the parabola opens
down, the vertex is the highest point.
Parabolas have a symmetric property to
them. If we drew a line down the middle of
the parabola, we could fold the parabola in
half. We call this line the axis of
symmetry.
Or, if we graphed one side of the
parabola, we could “fold” (or
REFLECT) it over, the line of
symmetry to graph the other side.
The axis of symmetry ALWAYS
passes through the vertex.
Vertex/
Maximum
Axis of
Symmetry
Vertex/Minimum
y
x
We can graph a quadratic equation if we know the
following:
-The location of the vertex
-The location of the axis of symmetry (a.o.s.)
-Whether it opens up or down
-A few points (including y-intercept)
In the following slides, we will discuss strategies for
finding each of these and we will try graphing one
function.
B. Up or Down?
y
The standard form of a
quadratic function is
a>0
y = ax2 + bx + c
The parabola will open up
when the a value is positive.
Ex: y = 3x2+2x+1
x
The parabola will open down
when the a value is negative.
a<0
Ex y = -3x2+2x+1
C. Finding the y-intercept
Finding the y-intercept is perhaps the easiest one of allit is the value of ‘c’.
For example, the function y = 6x2+3x+5 has a yintercept of 5, or at (0, 5).
D. Finding the Axis of Symmetry
When a quadratic function is in
standard form
y=
ax2
+ bx + c,
The equation of the line of
symmetry is
x  b
2a
For example…
Find the line of symmetry of
y = 3x2 – 18x + 7
Using the formula…
x  18  18  3
2 3 6
This is best read as …
the opposite of b divided by the
quantity of 2 times a.
Thus, the line of symmetry is x = 3.
E. Finding the Vertex
Remember that the vertex is a point on the graphthe maximum or minimum point depending on
whether the function opens up or down. Also recall
that the axis of symmetry always goes through the
vertex, the a.o.s. gives us the x-value of the vertex.
Once you find the a.o.s., substitute the value in for
in the function to get the y-value of the vertex.
Example of finding the vertex.
We know the line of symmetry
always goes through the vertex.
Thus, the line of symmetry
gives us the x – coordinate of
the vertex.
To find the y – coordinate of the
vertex, we need to substitute the x –
value into the original equation.
y = –2x2 + 8x –3
STEP 1: Find the line of symmetry
x  b  8  8  2
2a
2(2)
4
STEP 2: Plug the x – value into the
original equation to find the y value.
y = –2(2)2 + 8(2) –3
y = –2(4)+ 8(2) –3
y = –8+ 16 –3
y=5
Therefore, the vertex is (2 , 5)
F. Graphing A Quadratic Function
in Standard Form
The standard form of a quadratic
function is given by
There are 3 main steps to graphing a
parabola in standard form.
y = ax2 + bx + c
x
STEP 1: Find the axis of symmetry
STEP 2: Find the vertex
b
2a
Substitute into the line of symmetry (x –
value) to obtain the y – value of the
vertex.
STEP 3: Find two other points and reflect
them across the line of symmetry. Then
connect the five points with a smooth
curve.
MAKE A TABLE
using x – values close to the line
of symmetry.
Example 1:
Let's Graph ONE! Try …
y
y = 2x2 – 4x – 1
STEP 1: Find the line of
symmetry
x
b
2a
4
22
1
x
Thus the line of symmetry is x = 1
y
y = 2x2 – 4x – 1
STEP 2: Find the vertex
Since the x – value of the
vertex is given by the line of
symmetry, we need to plug
in x = 1 to find the y – value
of the vertex.
y
212
41
1
x
3
Thus the vertex is (1 ,–3).
y
y = 2x2 – 4x – 1
STEP 3: Find two other points
and reflect them across the line
of symmetry. Then connect the
five points with a smooth curve.
x
y
x
2 –1
3
5
y
2 2
2
4 2
1
y
23
2
43
1
1
5
5. Graphing A Quadratic Function
in Standard Form
Let's Graph ONE! Try …
y
y = 2x2 – 4x – 1
y-intercept: The y-intercept
will always be the value of ‘c’.
c = -1
Y-intercept is -1
x