GRAPH OF
QUADRATIC
FUNCTIONS
PRAYER
ATTENDANCE
CLASSROOM
MANAGEMENT
REVIEW
Complete the table of values for x and y.
2
y  2x
x -3 -2 -1 0 1 2 3 4 5
18 8
2
8
2
0
18 32 50
y
PRE-TEST:
Lance, a Grade 9 student is a Sepak Takraw player in the
City Meet. In one of his games he kicked the ball forming a
2
f
(
x
)


x
 5.
parabolic path with an equation of
Draw the graph and determine the following:
1. equation of the axis of symmetry
2. vertex
3. highest / lowest point obtained by the graph
4. direction of the opening of the parabola
5. Opening of the graph
MOTIVATION
INTRODUCTION TO PARABOLAS
MOTIVATIONAL ACTIVITY
Given the graph of a
quadratic function , can
you determine the
intercepts, axis of
symmetry, vertex, and
direction of the opening
of the parabola?
LESSON
PROPER
Quadratic Functions
The graph of a quadratic function is a: parabola
y
A parabola can open up
or down.
Vertex
If the parabola opens up,
the lowest point is called
the vertex (minimum).
If the parabola opens down,
the vertex is the highest point
(maximum).
Vertex
NOTE: if the parabola opens left or right it is not a function!
x
Standard Form
The standard form of a quadratic function is:
y = ax2 + bx + c
y
The parabola will
open up when the a
value is positive.
a>0
The parabola will
open down when the
a value is negative.
a¹ 0
x
a<0
Axis of Symmetry
Parabolas are symmetric.
If we drew a line down
the middle of the
parabola, we could fold
the parabola in half.
Axisy of
Symmetry
We call this line the
Axis of symmetry.
x
If we graph one side of
the parabola, we could
REFLECT it over the
Axis of symmetry to
graph the other side.
The Axis of symmetry ALWAYS
passes through the vertex.
Finding the Axis of Symmetry
When a quadratic function is in standard form
y = ax2 + bx + c,

b
the equation of the Axis of symmetry is x 
2a
This is best read as …
‘the opposite of b divided by the quantity of 2 times
a.’
2
Find the Axis of symmetry for y = 3x – 18x + 7
a = 3 b = -18
The Axis of
symmetry is
x=
Finding the Vertex
The Axis of symmetry always goes through the
Vertex Thus, the Axis of symmetry gives
_______.
X-coordinate of the vertex.
us the ____________
Find the vertex of
y = -2x2 + 8x - 3
STEP 1: Find the Axis of symmetry
b
x
2a
a = -2
b=8
The xcoordinate
of the vertex
is 2
Finding the Vertex
Find the vertex of
y = -2x2 + 8x - 3
STEP 1: Find the Axis of symmetry
STEP 2: Substitute the x – value into the original equation
to find the y –coordinate of the vertex.
y = -2 ( 2 ) +8 ( 2 ) - 3
2
= - 2 ( 4 ) + 16 - 3
= - 8 + 16 - 3
= 5
The
vertex is
(2 , 5)
Graphing a Quadratic Function
There are 3 steps to graphing a parabola in
standard form.
b
STEP 1: Find the Axis of symmetry using: x 
2a
STEP 2: Find the vertex
STEP 3: Find two other points and reflect them across
the Axis of symmetry. Then connect the five points
with a smooth curve.
MAKE A TABLE
using x – values close to
the Axis of symmetry.
Graphing a Quadratic Function
Graph : y = 2x - 4x -1
2
y
x =1
STEP 1: Find the Axis of
symmetry
- b
4
x=
=
=1
2 a 2(2)
STEP 2: Find the vertex
Substitute in x = 1 to
find the y – value of the
vertex.
2
y = 2 (1) - 4 (1)- 1 = - 3
x
Vertex : (1, - 3)
Graphing a Quadratic Function
Graph : y = 2x - 4x -1
2
STEP 3: Find two other
points and reflect them
across the Axis of
symmetry. Then connect
the five points with a
smooth curve.
x
y
2 –1
3
5
2
y = 2 (2) - 4 (2)- 1 = - 1
2
y = 2 (3) - 4 (3)- 1 = 5
y
x
Y-intercept of a Quadratic Function
y = 2x - 4x -1
2
Y-axis
y
The y-intercept of a
Quadratic function can
Be found when x = 0.
y = 2x 2 - 4 x - 1
= 2(0) - 4(0) - 1
2
= 0 - 0 -1
= -1
The constant term is always the y- intercept
x
Solving a Quadratic
The x-intercepts (when y = 0) of a quadratic function
are the solutions to the related quadratic equation.
The number of real solutions is at
most two.
No solutions
One solution
Two solutions
X=3
X= -2 or X = 2
Identifying Solutions
Find the solutions of 2x - x2 = 0
The solutions of this
quadratic equation can be
found by looking at the
graph of f(x) = 2x – x2
The xintercepts(or
Zero’s) of
f(x)= 2x – x2
are the solutions
to 2x - x2 = 0
X = 0 or X = 2
Analyze the graph of a quadratic
function , and determine the following:
1. Intercepts (points where the graph
passes the x and y axes) X=0, Y=0
2. equation of the axis of symmetry X=0
3. vertex (turning point of the graph) (0,0)
4. highest / lowest point obtained by the
graph (0,0)
5. direction of the opening of the
parabola UPWARD
GUIDED PRACTICE
Draw the graph of a quadratic function f ( x)  x  2 x  1
and identify the vertex, and opening of the graph. State
whether the vertex is a minimum or a maximum point, and
write the equation of its axis of symmetry
2
(-1,0)
Vertex _______
UPWARD
Opening of the graph __
MINIMUM point
Vertex is a ___
X=-1
Equation of the axis of symmetry _____
The graph of a quadratic function is called parabola.
The parabola opens upward or downward. It has a turning
point called vertex which is either the lowest point or the
highest point of the graph.
If the value of a>0, it opens upward and has a minimum
point but if a<0, the parabola opens downward and has a
maximum point.
There is a line called the axis of symmetry which divides
the graph into two parts such that one-half of the graph is a
reflection of the other half.
If the quadratic function is expressed in the form , the
vertex is the point (h,k). The line x = h is the axis of symmetry
and k is the minimum or maximum value of the function.
Graphing a Quadratic Function
There are 3 steps to graphing a parabola in
standard form.
b
STEP 1: Find the Axis of symmetry using: x 
2a
STEP 2: Find the vertex
STEP 3: Find two other points and reflect them across
the Axis of symmetry. Then connect the five points
with a smooth curve.
MAKE A TABLE
using x – values close to
the Axis of symmetry.
APPLICATION
Draw the graph of a quadratic function and determine
the vertex, domain, range, and opening of the graph.
State whether the vertex is a minimum or a maximum
point, and write the equation of its axis of symmetry.
2
GroupsGroup
1 and 12:: f ( x)  x  5 x  6
2
GroupsGroup
3 and 24:: f ( x)   x  4 x  4
2
Group
3
:
Groups 5 and 6: f ( x)  x  2
Group 4 : 𝑓 𝑥 = 𝑥 2 − 4𝑥 − 6
EVALUATION:
Jaime, a Grade 10 student is a Sepak Takraw player in the
City Meet. In one of his games he kicked the ball forming a
2
parabolic path with an equation of f ( x)   x  5.
Draw the graph and determine the following:
1. equation of the axis of symmetry
2. vertex
3. highest / lowest point obtained by the graph
4. direction of the opening of the parabola
5. Opening of the graph
ASSIGNMENT
1. Take a picture on your
neighborhood that shows a
parabola.
2.Find the vertex, axis of
symmetry.
THANK
YOU