Name: __________________________Date: __________

Grade: _____________________ Section: ___________

Activity 13
The Graph of a Quadratic Function
Learning Competency: Graphs a quadratic function: (a) domain; (b) range; (c)
intercepts; (d) axis of symmetry; (e) vertex; (f) direction of the opening of the parabola.
(M9AL-Ii-2)

What I Need to Do
In this lesson, you will learn how to identify the different characteristics of a
quadratic function and how to graph it. The following exercises will help you to
understand the lesson really well.

ACTIVITY 1: PART OF ME!

Identify the different parts of the graph or parabola by writing the name inside the
box.

1.

4.
2.

3.
5.

ACTIVITY 2: POINT OF VIEW!

From the given graph, determine the highest or the lowest point which is pointed by
the arrow. Write the coordinates inside the parenthesis.

1. (__, __) 3. (__, __)

4. (__, __)
2. (__, __)
 Gearing Up

In order to sketch the graph of a quadratic function, the following should be determined:
a) vertex
b) axis of symmetry
c) orientation (upward or downward)
d) x – intercept/s
e) y – intercept

From the graph of the quadratic function, the following can also be determined:
f) maximum / minimum point of the graph
g) domain and range of the function

Parabola - graph of a quadratic

function.

a. Vertex – the maximum(highest)

or the minimum(lowest) point of a
parabola. Vertex = (h, k)

b. Axis of symmetry – the line that

intersects with the vertex and for which
all points on opposite sides of the line are
mirror images of each other.

c. Orientation - The graph of a quadratic

function, which is a parabola, opens
upward or downward, depending on the
value of a in the function. If the value of a
is positive, the parabola opens upward. If
the value of a is negative, the parabola
opens downward.

d. x-intercepts – the points at which the

parabola intersects or touches the x-axis

e. y-intercepts – the points at which the

parabola passes through the y-axis

g. Range – depends upon the orientation of the f. The minimum/maximum point -

parabola when the parabola opens up, the
➢ If the parabola opens upward, the range vertex is the lowest point on the graph
is {𝑦|𝑦 ≥ 𝑘} — called the minimum, or min. When
➢ If the parabola opens downward, the the parabola opens down, the vertex is
range is {𝑦|𝑦 ≤ 𝑘} the highest point on the graph —
called the maximum, or max.
h. Domain – the set of all x-coordinates. In
symbol, {𝑥│𝑥 ∈ 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 }

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I. IDENTIFYING THE PROPERTIES OF THE QUADRATIC FUNCTION.
A. VERTEX
−𝑏 4𝑎𝑐 − 𝑏2
The vertex (h,k) can be found by (1) using the formula ℎ= and 𝑘= or by
2𝑎 4𝑎
(2) transforming the function to its vertex form. You can just choose either of the two,
whichever is more convenient to you
Example No. 1: Find the values of h and k in f(x) = x2 – 1 using the formula
−𝑏 4𝑎𝑐 − 𝑏2
ℎ= and 𝑘= to identify the vertex of its graph.
2𝑎 4𝑎

In f(x) = x2 – 1; a = 1, b = 0, c = -1
−𝑏 4𝑎𝑐 − 𝑏2
ℎ= 𝑘=
2𝑎 4𝑎
−(0) 4(1)(−1) − (0)2
h= 𝑘=
2(1) 4(1)
0 −4 − 0
h= 𝑘=
2 4(1)
−4
h=0 𝑘= 4

𝒌 = −𝟏
The vertex of the graph of f(x) = x2 - 1 is (0, -1).

Example No. 2: Find the values of h and k in f(x) = x2 – 2x – 3 by completing the square.
f(x) = x2 – 2x – 3
f(x) = (x2 – 2x) – 3 -isolate terms with x
f(x) = (x – 2x + 1) – 3 – 1 -complete the square
f(x) = (x – 1)2 – 4 -simplify
h = 1, k = -4
The vertex of the graph of f(x) = x2 – 2x – 1 is (1, -4).

B. AXIS OF SYMMETRY
Every parabola has an axis of symmetry which is the vertical line that divides the
graph into two perfect halves. It can be expressed as an equation in the form x = h,
where h can be found in the vertex (h, k)
Example No. 3: Find the axis of symmetry of the graph of f(x) = x2 – 1 using the
−𝑏
formula ℎ= . 5
2𝑎
4
f(x) = x2 – 1, a = 1, b = 0, c = -1 3
−𝑏 2
ℎ= 2𝑎
1
−(0)
h= 2(1)
-5 -4 -3 -2 -1 0 1 2 3 4 5

-1
0
h= -2
2
Axis of
h=0 -3
symmetry
The axis of symmetry is x = 0. -4

-5

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C. ORIENTATION
The graph of a quadratic function, which is a parabola, opens upward or downward,
depending on the value of a in the function. If the value of a is positive, the parabola
opens upward. If the value of a is negative, the parabola opens downward.
Example No. 4: Determine whether the graph of f(x) = x2 – 1 opens upward or downward.

f(x) = x2 – 1, a = 1, b = 0, c = -1
Since the value of a is positive then the parabola opens upward.

Example No. 4: Determine whether the graph of 𝑓(𝑥) = −𝑥 2 − 4𝑥 + 1 opens upward or

downward.

f(x) = −𝒙𝟐 − 𝟒𝒙 + 𝟏, a = -1, b = -4, c = 1

Since the value of a is negative then the parabola opens downward.

D. X – INTERCEPT/S
The x – intercept/s are the points at which the parabola intersects or touches the x-
axis. The value of the x-intercept is actually the roots/zeros of the quadratic function.
Thus, in order for you to determine the x-intercepts of the graph, you need to solve for
the roots of the function. A parabola may have no, one or two x-intercepts.

Example No. 5: Determine the x-intercept/s of the graph of 𝑓(𝑥) = 𝑥 2 − 1 by solving for
the roots.

𝒂 = 𝟏, 𝒃 = 𝟎, 𝒄 = −𝟏
−𝑏±√𝑏2 −4𝑎𝑐
𝑥=
2𝑎

−(0)±√(−0)2 −4(1)(−1)
𝑥= 2(1)

0±√0+4
𝑥= 2

0±√4
𝑥= 2
0±2
𝑥= 2
0+2 2 0−2 −2
𝑥= = =1 𝑥= = = -1
2 2 2 2
Since the roots are 1 and -1, then the x-intercepts are (1, 0) and (-1, 0).

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E. Y – INTERCEPT
The y – intercept is the point at which the parabola passes through the y-axis. To get
the y-intercept, set the value of x into 0.

Example No. 6: Determine the y-intercept of the graph of f(x) = 𝒙𝟐 − 𝟏.

Let x = 0.

𝑓(𝑥) = 𝑥 2 − 1
𝑓(0) = (0)2 − 1
𝒇(𝟎) = −𝟏
The y – intercept is (0, -1)

F. MAXIMUM / MINIMUM POINT OF THE GRAPH

When the parabola opens up, the vertex is the lowest point on the graph — called
the minimum, or min. When the parabola opens down, the vertex is the highest point on
the graph — called the maximum, or max. The maximum or the minimum value of the
4𝑎𝑐 − 𝑏2
graph is the k value of the vertex. Use the formula 𝑘 = to get the value of k.
4𝑎

Example No. 7: Find the minimum/maximum point of the graph of 𝒇(𝒙) = 𝒙𝟐 − 𝟏

In f(x) = x2 – 1, a = 1, b = 0, c = -1. Since a >1, the graph opens upward. Thus, it attains
a minimum value.
4𝑎𝑐 − 𝑏2
𝑘= 4𝑎

4(1)(−1) − (0)2
𝑘=
4(1)
−4 − 0
𝑘=
4
−4
𝑘=
4
𝒌 = −𝟏
The k value is -1, thus the minimum point is (0, -1)

G. DOMAIN OF THE FUNCTION

The domain of the function is all x - values. there are no restrictions on the domain of
this function. The domain is the set of real numbers.

Example No. 8: Find the domain of the function f(x) = x2 – 1

Solution: Using the table of values

x -2 -1 0 1 2
f(x) 3 0 -1 0 3

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If x = -2 If x = -1 If x = 0 If x = 1 If x = 2
f(x) = (-2)2 – 1 f(x) = (-1)2 – 1 f(x) = (0)2 – 1 f(x) = (1)2 – 1 f(x) = (2)2 – 1
f(x) = 3 f(x) = 0 f(x) = -1 f(x) = 0 f(x) = 3

Any real number may be squared and then lowered by one, so there are no restrictions
on the domain of this function. The domain is the set of real numbers.

The domain of the function is the set of real numbers or can be written as
{𝒙 │𝒙 ∈ 𝒂𝒍𝒍 𝒓𝒆𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓𝒔}

H. RANGE OF THE FUNCTION

The range of the function are all y- values of the function. Take note that:
The range is {𝒚|𝒚 ≥ 𝒌}, if the graph opens upward.
The range is {𝒚|𝒚 ≤ 𝒌}, if the graph opens downward.

Example No. 9: Find the range of the function f(x) = x2 – 1

In f(x) = x2 – 1, a = 1, b = 0, c = -1.
Since a >1, the graph opens upward. Also, the k value is -1. Thus, the range is
{𝒚|𝒚 ≥ −𝟏}.

II. SKETCHING THE GRAPH OF A QUADRATIC FUNCTION

To sketch the graph of f(x) = x2 – 1, you need to summarize the characteristics of the
graph as shown on the table below.

A. Vertex V(1, -4)

B. Axis of symmetry 1
C. Orientation (upward or
upward
downward)
D. x - intercept (-1, 0); (3, 0)

E. y-intercept (0, -3)

F. Maximum/minimum point Minimum point: (1, -4)

G. Domain of the functions {𝑥 │𝑥 ∈ 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠}

H. Range of the function {𝑦 │𝑦 ≥ −4}

To sketch the graph, plot the intercepts and the vertex. Also take note the opening of
the graph, whether upward or downward. Then, make a smooth curve. There is no need
to find other points since you are only sketching the graph.

in this example, the y-axis

is the axis of symmetry

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 Getting Better

ACTIVITY 3: YOU KNOW IT!

Describe the properties of the graph of the quadratic function. Know each property by
writing the value or the point in the space given.

Quadratic function f(x) = -x2 -4x + 12

A. Vertex
B. Axis of symmetry
C. Orientation
D. x - intercept
E. y-intercept
F. Maximum/minimum point
G. Domain
H. Range

ACTIVITY 4: SKETCH IT
Complete the table below and sketch the graph of f(x)=x2 + 4x + 3.

1. Vertex
2. Axis of
symmetry
3. Orientation
4. x - int
5. y-int

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 Gaining Mastery
Choose the BEST answer from the choices.

1. Which of the following is the vertex form of y = x2 – 2x – 8?

a. y = (x – 2)2 – 8 c. y = -(x – 2)2 – 8
b. y = (x – 1) – 9
2 d. y = -(x – 1)2 – 9

2. Which of the following is the axis of symmetry of the graph of f(x) = x 2 + 6x – 8?

a. x = 6 b. x = -6 c. y = 6 d. y = -6

3. Which of the following is the graph of f(x) = (x – 2)2 + 4?

4. Identify the vertex and the y-intercept of the graph of the function f(x) = 2(x + 2)2 – 2
a. vertex: (2, 2) / y-intercept: (0, 8) c. vertex: (2, -2) / y-intercept: (0, 6)
b. vertex: (-2. -2) / y-intercept: (0, 6) d. vertex: (-2, 2) / y-intercept: (0, 2)

5. Which of the following functions has a maximum value?

a. f(x)= x2 – 3x + 4 c. f(x) = 3 – 5x – x2
b. f(x) = 2x2 + 5x + 3 d. f(x) = 4x2

6. Where is the vertex located in the graph of f(x) = x 2 + 4x – 8?

a. (2, -12) b. (2, 4) c. (2, -4) d. (-2, -12)

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7. Which of the following statements is/are TRUE?
I. The graph of a quadratic function is a parabola.
II. A parabola may not intersect the x-axis.
III. When the parabola opens downward, it attains a minimum value.
IV. When the parabola opens upward, it attains a minimum value.

a. I, II, III b. I, II, IV c. I, III, IV d. II, III, IV

8. Which of the following statements is/are FALSE?

I. The axis of symmetry is a vertical line.
II. A parabola may intersect the y-axis twice.
III. The range for any quadratic function is the set of all real numbers.
IV. The parabola opens upward if the value of a is greater than 0.

a. II, b. III c, II, III d. I, IV

9. Which function has a graph that opens upward?

a. f (x) = 4x – x2 – 7 c. f (x) = – 5x2 + 8x + 3
b. f (x) = – x + 3x – 12
2 d. f (x) = 5x2 – 4x + 6

10. Which of the following is the graph of f(x) = 2x2 – 1?

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 What I Need to Remember

The properties of a quadratic function:

1. Parabola – graph of a quadratic function
2. Vertex – the point on a parabola where a minimum or maximum y-value occurs
3. Axis of symmetry – a line which a parabola is reflected onto itself.
4. The value of a in y = ax2 + bx + c, has something to do with the opening of the
parabola.
5. If a > 0, the parabola opens upward but if a < 0, the parabola opens downward.
6. The domain of a function is the set of all x-coordinates. In symbol, {𝑥Ι𝑥 ∈ ℝ}
7. The range of a function depends upon the orientation of the parabola
• If the parabola opens upward, the range is {𝑦Ι𝑦 ≥ 𝑘} or [𝑘, ∞)
• If the parabola opens downward, the range is {𝑦Ι𝑦 ≤ 𝑘} or (−∞, 𝑘]

WRITER: JENNILYN A. EDILO

SCHOOL: F. BUSTAMANTE NATIONAL HIGH SCHOOL
DIVISION: DAVAO CITY
EVALUATOR: ROMAN JOHN C. LARA
SCHOOL: DAVAO CITY NATIONAL HIGH SCHOOL
DIVISION: DAVAO CITY

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 Answer Key
ACTIVITY 1: PART OF ME!

1. axis of symmetry
2. x-intercept
3. x-intercept
4. y-intercept
5. vertex

ACTIVITY 2: POINT OF VIEW!

1. (2, -4)
2. (-4, 2)
3. (6, 4)
4. (-2, 0)
ACTIVITY 3: YOU KNOW IT!

Quadratic function f(x) = -x2 -4x + 12

I. Vertex v(-2, 16)
J. Axis of symmetry x = -2
K. Orientation (upward or
the graph opens downward
downward)
L. x - intercept (2, 0) and (-6, 0)
M. y-intercept (0, 12)
N. Maximum/minimum point (2, 16)
O. Domain set of all real numbers
P. Range {𝒚|𝒚 ≤ 𝟏𝟔},

ACTIVITY 4: SKETCH IT!

1. Vertex v(-2, -1)

2. Axis of
x = -2
symmetry
3. Orientation upward
(-3, 0) and
4. x - int
(-1, 0)
5. y-int (0, 3)

GAINING MASTERY

1. B 3. A 5. C 7. B 9. A
2. B 4. B 6. D 8. C 10. C

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