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3. Quadratic Functions
y
The graph of a quadratic function
is a parabola.
Vertex
A parabola can open up or down.
If the parabola opens up, the
lowest point is called the vertex.
x
If the parabola opens down, the
vertex is the highest point.
NOTE: if the parabola opened Vertex
left or right it would not be a
function!
4. Standard Form
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.
x
The parabola will open down
when the a value is negative.
a<0
5. Line of Symmetry
Lineyof
Parabolas have a symmetric
Symmetry
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 x
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 The line of symmetry ALWAYS
side. passes through the vertex.
6. Finding the Line of Symmetry
When a quadratic function is in For example…
standard form
Find the line of symmetry of
y = ax + bx + c,
2
y = 3x2 – 18x + 7
The equation of the line of
symmetry is Using the formula…
x = −b x = 18 = 18 = 3
2a 2( 3) 6
This is best read as …
the opposite of b divided by the Thus, the line of symmetry is x = 3.
quantity of 2 times a.
7. Finding the Vertex
We know the line of symmetry y = –2x2 + 8x –3
always goes through the vertex.
STEP 1: Find the line of symmetry
Thus, the line of symmetry
gives us the x – coordinate of x = −b = −8 = −8 = 2
2a 2(−2) −4
the vertex. STEP 2: Plug the x – value into the
original equation to find the y value.
To find the y – coordinate of the
y = –2(2)2 + 8(2) –3
vertex, we need to plug the x –
value into the original equation. 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 There are 3 steps to graphing a
function is given by parabola in standard form.
y = ax2 + bx + c
USE the A TABLE
MAKE equationof
Plug in the line
STEP 1: Find the line of symmetry
symmetry (x – value) to
b
- value of the
using x – values close to
obtain x = –
the y
STEP 2: Find the vertex the line of symmetry.
2a
vertex.
STEP 3: Find two other points and reflect
them across the line of symmetry. Then
connect the five points with a smooth
curve.
9. A Quadratic Function in Standard Form
Let's Graph ONE! Try … y
y = 2x2 – 4x – 1
STEP 1: Find the line of
symmetry
-b 4
x= = =1 x
2a 2( 2)
Thus the line of symmetry is x = 1
10. A Quadratic Function in Standard Form
Let's Graph ONE! Try … 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
x
of the vertex.
y = 2( 1) 2 - 4( 1) - 1 = - 3
Thus the vertex is (1 ,–3).
11. A Quadratic Function in Standard Form
Let's Graph ONE! Try … 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 = - 1
y = 2( 3) 2 - 4( 3) - 1 = 5
