WEEK 7

Graphing Quadratic Functions

and
Analyzing the Effects on its
Graph

MRS. MARIANNE KATE C. TAJONERA

MATH Teacher
Objectives
 graphs a quadratic function and identify its
properties;
a) Vertex
b) Domain
c) Range
d) Intercepts
e) Axis of symmetry
f) Direction of the opening of the parabola
 analyzes the effects of changing the values of , and
in the equation of a quadratic function on its
graph.
The Graph of a Quadratic Function
The graph of a quadratic function is a Parabola. It
has different properties such as;
a) Vertex
b) Domain
c) Range
d) Intercepts
e) Axis of symmetry
f) Opening of the parabola
The Graph of a Quadratic Function
The Parabola and Its Properties
a) Vertex
 The vertex of the parabola is the point . It is the
minimum point of the parabola if (opens upward) and
maximum point of the parabola if (opens downward).

b) Domain
 It is the set of all possible values of . The domain of a
quadratic function is the set of real numbers and can be
written as .
c) Range
 It is the set of all possible values of . The range of a
quadratic function is;
(if , the parabola opens upward)
(if , the parabola opens downward)
The Graph of a Quadratic Function
The Parabola and Its Properties
d) Intercepts
• x-intercept/s
 It is the value of coordinate when the graph touches
the axis. The x-intercepts is determined by setting ,
and then solve for .
Note: if , there are 2 x-intercepts;
if , there is only 1 x-intercept; and
if , there is no x-intercept.
• y-intercept
 It is the value of coordinate when the graph touches
the axis. The y-intercept is determined by setting ,
and then solve for .
The Graph of a Quadratic Function
The Parabola and Its Properties
e) Axis of Symmetry (AOS)
 The axis of symmetry of the parabola is the vertical line
that divides the parabola into two equal parts. It is
defined as .
f) Opening of the Parabola
 The value of indicates the opening of the parabola. If
the value of is;
• (positive), then the parabola opens upward;
• (negative), the parabola opens downward.
 Examples
1) Given the graph, identify the following properties of the parabola.

Vertex ( 𝟏 ,− 𝟒 )
Domain 𝒙∈𝑹
Range 𝒚 ≥− 𝟒
x-intercepts and x-intercepts
y-intercept −𝟑
Axis of
Symmetry 𝒙 =𝟏
Opening of
Parabola Upward

Vertex ¿ ( 𝟏 ,− 𝟒 ) y-intercept
Vertex

Since the graph opens upward, then the range is . If , therefore .

The equation for the AOS is . Since , therefore the AOS is .

The Graph of a Quadratic Function
How to Sketch the Graph of a Quadratic Function
STEPS:
1. Find the coordinate of the vertex by transforming the
equation into vertex form (), or you may use the formula;
and
2. Make a table of values for the quadratic function with 5
ordered pairs including the vertex at the center. Choose 4
more x-values (2 integers larger than , and 2 integers
smaller than ).
3. Complete the table of values by substituting your chosen
values of to the equation and solve for .
4. Plot the ordered pairs in a cartesian plane then connect all
points.
Example:
1) Sketch the graph of the equation and determine the properties of
parabola.
Solution:
(Step 1. Find the coordinate of the vertex.)
 and

 , ,


 Vertex
(Step 2. Make a table of values for the quadratic function with 5
ordered pairs including the vertex at the center.)
Examples:
1) Sketch the graph of the equation and determine the properties of
parabola.
Solution:
(Step 3. Complete the table of values by substituting the values of to
the equation, then solve for .)

−𝟏 𝟐 𝟐 −𝟏
 or
 or
¿ −1+ 4 −1 ¿ 𝟐
 or
¿ −9+12 −1 ¿ 𝟐
 or
¿ −16+16 − 1 ¿ −𝟏
Examples:
1) Sketch the graph of the equation and determine the properties of
parabola.
Solution:
(Step 4. Plot the ordered
pairs in a cartesian plane
then connect all points.)
Examples:
1) Sketch the graph of the equation and determine the properties of
parabola.
Properties of the Parabola:
a) Vertex: ( 𝟐 ,𝟑 ) (See solution on Step 1.)

b) Domain: 𝒙∈𝑹 (The domain of all QF are set of real nos.)

(Since (negative) and the parabola opens
c) Range: 𝒚 ≤𝟑 downward, the range is . If , then .)

d) x-intercept/s:
(It is the value of , when is 0. From the given function, change to , then
solve for using any of the four methods.)
𝑦 =− 𝑥 2+ 4 𝑥 − 1 → or
𝑥2 − 4 𝑥+1=0 (Solve using Quadratic Formula)
 , ,



𝟐 ± √𝟓
Examples:
1) Sketch the graph of the equation and determine the properties of
parabola.
Properties of the Parabola:
e) y-intercept: (It is the value of , when is . From the
given function, change to , then solve
𝑦 =− 𝑥 2+ 4 𝑥 − 1 for .)
2
𝑦 =− ( 0 ) +4 (0) −1
𝑦 =0+ 0 − 1

e) Axis of Symmetry:
 , since (From the value of the vertex)


f) Opening of Parabola:
Since (negative), the parabola opens Downward.
The Graph of a Quadratic Function
How to Sketch the Graph of a Quadratic Function
The table below indicates the different forms of quadratic
functions and its properties.
The Graph of a Quadratic Function
Effects of Changing the Value of .
 The value of affects the graph of a quadratic function.
 It tells the opening of the graph;
• If is positive, the parabola opens upward.
• If is negative, the parabola opens downward.
 It tells how narrow or wide is the parabola;
• The larger the value of , the narrower the graph.
• The smaller the value of , the wider the graph
The Graph of a Quadratic Function
Effects of Changing the Value of .
Observe the graph of the following functions.
(Use desmos.com)
1)
2)

3)

4)
5)
6)
The Graph of a Quadratic Function
Effects of Changing the Value of .
 The value of affects the graph of a quadratic function.
 It translate the graph horizontally;
• If is positive, the parabola moves to the right.
• If is negative, the parabola moves to the left.

Observe the graph of the following functions.

(Use desmos.com)
1) V: ;
2) V: ;

3) V: ;
4) V: ;
5) V: ;
The Graph of a Quadratic Function
Effects of Changing the Value of .
 The value of affects the graph of a quadratic function.
 It translate the graph vertically;
• If is positive, the parabola moves upward.
• If is negative, the parabola moves downward.

Observe the graph of the following functions.

(Use desmos.com)
1) V: ;
2) V: ;

3) V: ;
4) V: ;
5) V: ;
 WEEK 8
Determining the Equation
of a
Quadratic Function
MRS. MARIANNE KATE C. TAJONERA
MATH Teacher
Objectives
 determines the equation of a quadratic function
given;
a) a table of values;
b) graph;
c) Zeros
 solves problems involving quadratic functions.
Determining the Equation of a
Quadratic Function
a) Given a Table of Values
STEPS
:
1. Choose any 3 ordered pairs from the table of values.
2. Substitute the values of and in the quadratic function of
the form to obtain 3 systems of linear equations.
3. Solve for the values of , , and using Substitution
method and/or Elimination method by Addition or
Subtraction.
4. Substitute the obtained values of , , and in the form of
the quadratic function.
Determining the Equation of a
Quadratic Function
a) Given a Table of Values
Example:
1) Given: Solution:
(Step 1. Choose any 3 ordered pairs.)
 , , and
(Step 2. Substitute the values of and in the form to obtain 3 systems of linear
equations.)
 Equation 1:

 Equation 2:

 Equation 3:

(Step 3. Solve for the values of , , and )

 (From Equation 1, we get .
 Substitute the value of to Equation 2
 Equation 4 & 3 to obtain Equation 4 & 5.)
Determining the Equation of a
Quadratic Function
a) Given a Table of Values
Example:
1) Given: Solution:
(Step 3. Solve for the values of , ,
and .)

Equation 5
 Eq. 4: (Solve for the value of by using
Eq. 5: Elimination method in Equation 4 & 5.)
 (To eliminate the variable , make sure they
have the same numerical coefficient.)
 (Solve using Elimination by Subtraction.
− Subtract the terms in Equation 4 & 5.)
−𝟐=−𝟐 𝒂+𝟎 (Simplify to get the value of .)
 or
Determining the Equation of a
Quadratic Function
a) Given a Table of Values
Example:
1) Given: Solution:
(Step 3. Solve for the values of , ,
and .)

 ; (Substitute the obtain value of in either of

 Eq.4: the Equation 4 and 5.)

or
(Step 4. Substitute the obtained values of , , and in the standard form of the
quadratic equation.)
 , , and

Answer
Determining the Equation of a
Quadratic Function
a) Given a Graph
STEPS
:
1. Determine the coordinates of the vertex .
2. Identify any point on the graph and determine its
coordinates .
3. Substitute the coordinates of the vertex and the point
to the vertex form and solve for the value of .
4. Write the equation of the quadratic function in
general/standard form .
Determining the Equation of a
Quadratic Function
a) Given a Graph
Example:
1) Given: Solution:
(Step 1. Determine the coordinates
of the vertex .)


(Step 2. Identify any point on the

𝒉=𝟏 graph and determine its
P:
coordinates.)


𝒌=−𝟒

Vertex:
Determining the Equation of a
Quadratic Function
a) Given a Graph
Example:
1) Given: Solution:
(Step 3. Substitute the coordinates
of the vertex and the point to the
vertex form and solve for the value
of .)

or
Determining the Equation of a
Quadratic Function
a) Given a Graph
Example:
1) Given: Solution:
(Step 4. Write the equation of the
quadratic function in general
(standard) form .)

 and



Answer
Determining the Equation of a
Quadratic Function
a) Given Its Zeros
 Zeros of a quadratic function are also called as roots, solutions,
and -intercepts.
STEPS
:
1. Substitute the zeros in the formula;
or ,
where and are the roots/zeros of the quadratic
function.
2. Simplify by using the FOIL method.
3. Combine similar terms. Express your answer in general
(standard) form of the quadratic function.
Determining the Equation of a
Quadratic Function
a) Given Its Zeros
Example:
1) Find the equation of the quadratic function whose zeros are and .
Solution:
(Step 1. Substitute the zeros in the formula .)
 and
 → 𝑦 =(𝑥 +3)(𝑥 −2)
 (Step 2. Simplify by using the FOIL method.)
(Step 3. Combine similar terms. Express your
answer in general form)
Answer
Solving Problems Involving
Quadratic Function
STEPS
:
1. Read, understand, and analyze the problem.
2. Identify the given and represent them using variables and
algebraic expressions.
3. Formulate an equation.
4. Solve for the unknown and what is asked in the problem.
5. Check your answer by substituting in the equation.
Solving Problems Involving
Quadratic Function
Example:
1) The sum of two numbers is . What is the minimum sum of their
squares.
Solution:
 Let, be the first number, and
be the second number
 Their squares are and
 Then, the sum of their squares can be express into equation;
 or
Solving Problems Involving
Quadratic Function
Example:
1) The sum of two numbers is . What is the minimum sum of their
squares.
Solution:

 , , and
 Solve for ; Remember:
 In a graph of a quadratic function, if
(positive), the vertex is the minimum
point and the y-coordinate of the
vertex or is the minimum value.

Answer: 288 is the minimum sum of the squares of

the two numbers whose sum is 24.
Solving Problems Involving
Quadratic Function
Example:
2) A rectangular field has a perimeter of 80 m. Find the largest possible
area for the field.
Solution:
(The formula for the perimeter of a rectangle)

(Substitute the given value of the perimeter,
then express in terms of .)

or

 Then, let be the width and be the length.



 or 𝑨=− 𝒘 𝟐+𝟒𝟎 𝒘
Solving Problems Involving
Quadratic Function
Example:
2) A rectangular field has a perimeter of 80 m. Find the largest possible
area for the field.
Solution:

 , , and
 Solve for ; Remember:
 In a graph of a quadratic function, if
(negative), the vertex is the maximum
point and the y-coordinate of the vertex
or is the maximum value.

𝒌=𝟒𝟎𝟎 Answer: 400 is the largest possible area of the

rectangular field.