Graph of Quadratic Functions

GRAPHING
QUADRATIC
FUNCTIONS

MATHEMATICS 9

PARABOLA TRANSFORMATIONS

After studying transforming the equation of a quadratic
function to vertex form, we will now proceed to Graphing
Quadratic Functions Using Transformations.

In graphing a quadratic function using transformation, it
is important that the function is in vertex form.

𝒚 = 𝒂 𝒙 − 𝒉𝟐+ 𝒌

PARABOLA TRANSFORMATIONS

𝒚 = 𝒂 𝒙 − 𝒉𝟐+ 𝒌

The coefficient 𝒂 determines the concavity or direction of
the opening of the parabola; whether it is opening upward
or downward.

– If 𝑎 is positive, the parabola opens upward.

– If 𝑎 is negative, the parabola opens downward.

PARABOLA TRANSFORMATIONS

This is the graph of the parent function of a parabola.
When we say parent function, it means, that there is no
existing transformation yet in the graph—it is the original.

Parent Function

𝒚 = 𝒙𝟐

The coefficient

𝒂 = 𝟏

PARABOLA TRANSFORMATIONS

Notice how the graph transforms from the parent function once changes are applied in
the quadratic equation. The second graph has coefficient 𝒂 = −𝟏. Hence, the graph
reflected across the x-axis.

𝒚 = 𝒙𝟐

𝒚 = −𝒙𝟐

PARABOLA TRANSFORMATIONS

𝒚 = 𝒂 𝒙 − 𝒉𝟐+ 𝒌

Aside from the concavity, the coefficient 𝒂 determines if the
opening of the parabola will become narrower or wider compared
with the parent function. This is called horizontal stretch or
compression.
– If

𝒂 > 𝟏 (a whole number greater than 1), the parabola’s

opening becomes narrower—horizontal compression.

– If 𝟎 < 𝒂 < 𝟏 (a fraction), the parabola’s opening becomes

wider—horizontal stretch.

Technically speaking, if 𝑎 is smaller, the opening is wider; and if 𝑎 is
bigger, the opening is narrower. (An inverse relationship)
*Note: 𝑎 means “absolute value of 𝑎”.

PARABOLA TRANSFORMATIONS

Analyze the following graphs:

𝒚 = 𝒙𝟐

𝒚 = 𝟐𝒙𝟐

𝒚 =𝟏

𝟐 𝒙𝟐

Take

note

that

we

are

considering the absolute value of

𝑎.

So to compare 𝑦 = −2𝑥2and

𝑦 =

1
4𝑥2,

𝒚 =

𝟏
𝟒𝒙𝟐

will have a wider

graph than 𝒚 = −𝟐𝒙𝟐because

𝟏
𝟒< −𝟐 .

PARABOLA TRANSFORMATIONS

𝒚 = 𝒂 𝒙 − 𝒉𝟐+ 𝒌

Let us now proceed to translation.

Translation is the movement of the graph through the vertex of
the parent function (which is at the origin) to a different position
on the Cartesian plane.

Parent Function

𝒚 = 𝒙𝟐

Vertex
(𝟎, 𝟎)

PARABOLA TRANSFORMATIONS

𝒚 = 𝒂 𝒙 − 𝒉𝟐+ 𝒌

In the parent function of a parabola, the vertex is at the
origin (0,0). As soon as transformation is applied, the
vertex will translate to a certain direction depending on
the 𝒉 and 𝒌.

– 𝒉 determines the horizontal translation (to the left or to

the right)

– 𝒌 determines the vertical translation (up or down)

PARABOLA TRANSFORMATIONS

In reference with the parent function: 𝒚 = 𝒙𝟐

𝑦 = 𝑥 − 22

−ℎ means shift to the right

Remember that we need to change the sign of
h. So, ℎ = 2.
Translation: The graph shifted 2 units to the
right.

𝑦 = 𝑥 + 22

+ℎ means shift to the left

ℎ = −2

Translation: The graph shifted 2 units to the
left.

𝑦 = 𝑥2+ 1

+𝑘 means shift upward

𝑘 = 1

Translation: The graph shifted 1 unit upward.

𝑦 = 𝑥2− 1

−𝑘 means shift downward

𝑘 = −1

Translation:

The

graph

shifted

1

unit

downward.

PARABOLA TRANSFORMATIONS

Analyze the following graphs:

𝒚 = 𝒙𝟐

𝒚 = (𝒙 − 𝟐)𝟐

𝒚 = 𝒙 + 𝟐𝟐

PARABOLA TRANSFORMATIONS

Analyze the following graphs:

𝒚 = 𝒙𝟐

𝒚 = 𝒙𝟐+ 𝟏

𝒚 = 𝒙𝟐− 𝟏

PARABOLA TRANSFORMATIONS

Analyze the following graphs:

𝒚 = 𝒙𝟐

𝒚 = (𝒙 − 𝟐)𝟐+𝟏

𝒚 = (𝒙 + 𝟐)𝟐−𝟏

Observe how

the vertex
moves to a

different

position from

the origin.

PARABOLA TRANSFORMATIONS

How do you think will the graph of 𝑦 = −2 𝑥 + 32+ 4 look like? Let us start
graphing. ☺

Step 1: Identify the vertex (ℎ, 𝑘).

𝑦 = −2 𝑥 + 32+ 4

Vertex: (−𝟑, 𝟒)

The vertex gives us the translation: “The graph will shift 3 units to

the left, and 4 units upward.”

Step 2: Determine the concavity (direction of opening).

𝑎 = −2

The graph will be opening downward.

PARABOLA TRANSFORMATIONS

Step 3: Describe the stretch/compression.

𝑎 = −2 = 2

The graph has a horizontal compression of 2.

Step 4: Identify the axis of symmetry 𝑥 = ℎ.

Axis of Symmetry: 𝒙 = −𝟑

Step 5: Use table of values to find at least 2 symmetrical points of the parabola.

PARABOLA TRANSFORMATIONS

Given: 𝑦 = −2 𝑥 + 32+ 4

TRANSFORMATION:

a) Vertex: (−𝟑, 𝟒)

b) Translation: The graph shifted 3

units to the left, and 4 units
upward.

c) Concavity: Opening downward

d) Horizontally compressed by 2

e) Axis of symmetry: 𝒙 = −𝟑

PARABOLA TRANSFORMATIONS

Given: 𝑦 = −2 𝑥 + 32+ 4

TABLE OF VALUES:

In the table of values, it is efficient to
use the numbers beside the axis of
symmetry,

then

simply

reflect

the

point/s. At least 2 points will do.

x

y

(x,y)

-4

2

(-4,2)

-2

2

(-2,2)

PARABOLA TRANSFORMATIONS

Given: 𝑦 = −2 𝑥 + 32+ 4

GRAPHING:

Finally, connect the plotted points
to make a parabolic curve.

x

y

(x,y)

-5

-4

(-5,-4)

-4

2

(-4,2)

-2

2

(-2,2)

-1

-4

(-1,-4)

PARABOLA TRANSFORMATIONS

Actual picture of the graph of

𝑦 = −2 𝑥 + 32+ 4

ASSIGNMENT

Let’s apply what you learned! ☺

Graph the following quadratic functions and provide the a) vertex, b) translation of
graph, c) concavity, d) horizontal stretch/compression, and e) axis of symmetry.

1. 𝑦 =1

4𝑥 − 2 2 − 3

2. 𝑦 = 3𝑥2+ 12𝑥 + 8

3. 𝑦 = −2𝑥2+ 1

4. 𝑦 = −1

2𝑥 − 1 2

FINAL TASK: Don’t forget to smile!