This document provides a detailed explanation of graphing quadratic functions in the form y = ax² + bx + c. It discusses key concepts such as the opening direction of parabolas, the vertex, the line of symmetry, and steps to graph a quadratic function. The process includes finding the line of symmetry, determining the vertex coordinates, and reflecting points across the line of symmetry to complete the parabola.

1. Graphing Quadratic Functions
y = ax2 + bx + c

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3. Quadratic Functions
The graph of a quadratic function
is a parabola.
A parabola can open up or down.
If the parabola opens up, the
lowest point is called the vertex.
If the parabola opens down, the
vertex is the highest point.
NOTE: if the parabola opened
left or right it would not be a
function!
y
x
Vertex
Vertex

4. y = ax2 + bx + c
The parabola will open down
when the a value is negative.
The parabola will open up
when the a value is positive.
Standard Form
y
x
The standard form of a
quadratic function is
a > 0
a < 0

5. y
x
Line of
Symmetry
Line of Symmetry
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 line of
symmetry.
The line of symmetry ALWAYS
passes through the vertex.
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.

6. Find the line of symmetry of
y = 3x2 – 18x + 7
Finding the Line of Symmetry
When a quadratic function is in
standard form
The equation of the line of
symmetry is
y = ax2 + bx + c,
2
b
a
x 

For example…
Using the formula…
This is best read as …
the opposite of b divided by the
quantity of 2 times a.
 
18
2 3
x 18
6
 3

Thus, the line of symmetry is x = 3.

7. 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 plug the x –
value into the original equation.
STEP 1: Find the line of symmetry
STEP 2: Plug the x – value into the
original equation to find the y value.
y = –2x2 + 8x –3
8 8
2
2 2( 2) 4
b
a
x   
 
 
 
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)

8. A Quadratic Function in Standard Form
The standard form of a quadratic
function is given by
y = ax2 + bx + c
There are 3 steps to graphing a
parabola in standard form.
STEP 1: Find the line of symmetry
STEP 2: Find 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.
Plug in the line of
symmetry (x – value) to
obtain the y – value of the
vertex.
MAKE A TABLE
using x – values close to
the line of symmetry.
USE the equation
2
b
x
a
-
=

9. STEP 1: Find the line of
symmetry
Let's Graph ONE! Try …
y = 2x2 – 4x – 1
( )
4
1
2 2 2
b
x
a
-
= = =
A Quadratic Function in Standard Form
y
x
Thus the line of symmetry is x = 1

10. Let's Graph ONE! Try …
y = 2x2 – 4x – 1
STEP 2: Find the vertex
A Quadratic Function in Standard Form
y
x
( ) ( )
2
2 1 4 1 1 3
y = - - = -
Thus the vertex is (1 ,–3).
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.

11. 5
–1
Let's Graph ONE! Try …
y = 2x2 – 4x – 1
( ) ( )
2
2 3 4 3 1 5
y = - - =
STEP 3: Find two other points
and reflect them across the line
of symmetry. Then connect the
five points with a smooth curve.
A Quadratic Function in Standard Form
y
x
( ) ( )
2
2 2 4 2 1 1
y = - - = -
3
2
y
x