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This lesson will discuss how to apply different transformations to quadratic functions.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Transforming the Graph of a Function

The graph of the quadratic function is drawn on the coordinate plane. By applying transformations to the parabola that corresponds to draw the graph of

Parabola
Discussion

Reflection of Quadratic Functions

A reflection of a function is a transformation that flips a graph over a line called the line of reflection. A reflection in the axis is achieved by changing the sign of every output value of the function rule. In other words, the sign of the coordinate of every point on the graph of a function should be changed. Consider the quadratic parent function
The reflection of the corresponding parabola can be shown on a coordinate plane.
Reflection of the parabola in x-axis
A reflection in the axis is instead achieved by changing the sign of every input value. However, since is equivalent to reflecting in the axis does not change the graph. For this reason, the reflection of the graph of another quadratic function will be shown.
This transformation can also be shown on a coordinate plane.
Reflection of y=(x-1)^2 in y-axis
Example

Reflecting a Quadratic Function

Kriz is given an extra credit math assignment about quadratic functions and parabolas. The assignment consists of four tasks. For the first task, Kriz is given the graph of the function

Parabola of the given function

Help Kriz with their extra credit assignment.

a Reflect the given parabola in the axis and write its corresponding equation.
b Reflect the given parabola in the axis and write its corresponding equation.

Answer
a Graph:
reflection in the x-axis

Equation:

b Graph:
reflection in the y-axis

Equation:

Hint
a Recall that the graph of is a reflection of the graph of in the axis.
b The graph of is a reflection of the graph of in the axis.

Solution
a The reflection of the graph of in the axis is given by the equation of
Therefore, the graph of is a reflection of the graph of in the axis.
The given function is being reflected over the x-axis
b The reflection of the graph of in the axis is given by the equation of
To obtained equation can be simplified.
NegBaseToPosPow

Distr

Distribute

Therefore, the graph of is a reflection of the graph of in the axis.
The given function is being reflected over the y-axis
Explore

Stretching and Shrinking a Parabola

In the coordinate plane, the parabola that corresponds to the quadratic function can be seen. Observe how the graph is vertically and horizontally stretched and shrunk by changing the values of and

Illustration of how changes of the coefficients of a and b affect the graph of the parabola
Discussion

Stretch and Shrink of Quadratic Functions

A function graph is vertically stretched or shrunk by multiplying the output of a function rule by some constant where This constant must be positive, otherwise a reflection is involved. Consider the quadratic function
If is greater than the parabola is vertically stretched by a factor of Conversely, if is less than the graph is vertically shrunk by a factor of If then there is no stretch nor shrink. Here, the coordinates of all points on the graph are multiplied by the factor
Different values of the coefficient a lead to the graph stretching or shrinking vertically
Similarly, a function graph is horizontally stretched or shrunk by multiplying the input of a function rule by some constant where Again, the constant must be positive because if it was negative, a reflection would be required.
In this case, if is greater than the graph is horizontally shrunk by a factor of Conversely, if is less than the graph is horizontally stretched by a factor of If then there is neither a stretch nor shrink of the graph. Here, the coordinates of all points on the graph are multiplied by the factor
Different values of the coefficient b lead to the graph stretching or shrinking horizontally
The information about vertical and horizontal stretches and shrinks of the graph of a function can be summed up in a table.
Vertical Horizontal
Stretch with with
Shrink with with
Example

Finding the Equation of a Stretch and a Shrink

Kriz's second task is about horizontal and vertical stretches and shrinks of parabolas.

Kriz studying

They are given the quadratic function and want to write the function rules of two related functions.

a A quadratic function whose graph is a vertical stretch of the graph of by a factor of
b A quadratic function whose graph is a horizontal shrink of the graph of by a factor of

Hint
a A function graph is vertically stretched or shrunk by multiplying the function rule by a positive constant.
b A function graph is horizontally stretched or shrunk by multiplying the input of a function rule by a positive constant.

Solution
a A function graph is vertically stretched or shrunk by multiplying the function rule by a positive constant. If the constant is greater than the graph is vertically stretched. Therefore, if the graph of the given function is to be vertically stretched by a factor of the function rule must be multiplied by
Recall that stretching a graph by a factor of means multiplying the coordinates of all the points on the curve by a factor of This can be seen in a coordinate plane.
vertical stretch
It is worth noting that the resulting function can be simplified by distributing
b A function graph is horizontally stretched or shrunk by multiplying the input of the function rule by a positive constant. If the constant is greater than the graph is horizontally shrunk. Therefore, to horizontally shrink the graph by a factor of the input of the function rule must be multiplied by
This can be seen in a coordinate plane.
horizontal shrink
It is worth noting that the resulting function can be simplified by using the Power of a Product Property.
Pop Quiz

Stating the Factor of a Stretch or a Shrink

The graph of the quadratic function is shown in the coordinate plane. The graph of a horizontal or vertical stretch or shrink is also shown.

Find the value of the constant
Example

Combining Transformations of Quadratic Functions

Kriz's assignment is getting more interesting, as the third task is about combining reflections with vertical and horizontal stretches and shrinks.

Kriz studying

This time, they are given the quadratic function and want to write the function rules of two other functions.

a A quadratic function whose graph is a vertical stretch by a factor of followed by a reflection in the axis of the graph of
b A quadratic function whose graph is a horizontal shrink by a factor of followed by a reflection in the axis of the graph of

Hint
a A function graph is vertically stretched or shrunk by multiplying the function rule by a positive constant. Furthermore, a function graph is reflected in the axis by changing the sign of the input.
b A function graph is horizontally stretched or shrunk by multiplying the input of the function rule by a positive constant. Furthermore, a function graph is reflected in the axis by changing the sign of the output.

Solution
a A function graph is vertically stretched or shrunk by multiplying the function rule by a positive constant. If the constant is greater than the graph is vertically stretched. Therefore, if the graph of the given function is to be vertically stretched by a factor of the function rule must be multiplied by
Furthermore, a function graph is reflected in the axis by changing the of the input.
These transformations are illustrated by the following diagram.
Combined transformations of y=(x-1)^2
The resulting function can be simplified by squaring the binomial and then distributing the
▼
Simplify right-hand side
NegBaseToPosPow

Distr

Distribute

ExpandPosPerfectSquare

BaseOne

IdPropMult

Identity Property of Multiplication

Distr

Distribute

b A function graph is horizontally stretched or shrunk by multiplying the input of the function rule by a positive constant. If the constant is greater than the graph is horizontally shrunk. Therefore, to horizontally shrink the graph by a factor of the input of the function rule must be multiplied by
Furthermore, a function graph is reflected in the axis by changing the of the output.
The described transformations are demonstrated by the following diagram.
Combined transformations
The resulting function can be simplified by squaring the binomial and then distributing the
▼
Simplify right-hand side
Explore

Translating a Graph

In the coordinate plane, the graph of the quadratic function can be seen. Observe how the graph is horizontally and vertically translated by changing the values of and

Changing values of h and k affects the graph of the function
Discussion

Translation of Quadratic Functions

A translation of a function is a transformation that shifts a graph vertically or horizontally. A vertical translation is achieved by adding some number to every output value of a function rule. Consider the quadratic function
If is a positive number, the translation is performed upwards. Conversely, if is negative, the translation is performed downwards. If then there is no translation. This transformation can be shown on a coordinate plane.
Translating a parabola vertically
A horizontal translation is instead achieved by subtracting a number from every input value.
In this case, if is a positive number, the translation is performed to the right. Conversely, if is negative, the translation is performed to the left. If then there is no translation. It is worth noting that since is subtracted from if is positive, then a number is subtracted from On the other hand, if is negative, a number is added to the variable
Translating a parabola horizontally
The vertical and horizontal translations of the graph of a function can be summarized in a table.
Translation
Vertical Horizontal
Upwards
with 		To the Right
with
Downwards
with 		To the Left
with
Example

Translating a Quadratic Function

To finally finish the assignment and get the extra credit they need, Kriz has to finish the fourth task of the math assignment. This time, the graph of the quadratic parent function is given.

quadratic function

By translating this quadratic function, Kriz wants to draw the graphs and write the equations of the following functions.

a A translation of the graph of three units up.
b A translation of the graph of two units to the right.
c A translation of the graph of one unit down and three units to the left.

Answer
a Equation:

Graph:

translation 3 units up
b Equation:

Graph:

translation 2 units to the right
c Equation:

Graph:

translation 1 unit down and 3 units to the left

Hint
a The graph of is a vertical translation of the graph of by units. If is positive, the translation is upwards. If is negative, the translation is downwards.
b The graph of is a horizontal translation of the graph of by units. If is positive, the translation is to the right. If is negative, the translation is to the left.
c The graph of is a horizontal translation followed by a vertical translation by and units, respectively.

Solution
a The graph of is a vertical translation of the graph of by units. If is positive, the translation is upwards. Conversely, if is negative, then the translation is downwards.
This can be seen on the coordinate plane.
translation down
b The graph of is a horizontal translation of the graph of by units. If is positive, the translation is to the right. Conversely, if is negative, then the translation is to the left. Note that since is subtracted from if is positive, then the number is subtracted from On the other hand, if is negative, the number is added to
Here, since the given graph is to be translated units to the right, the value of is Therefore, is subtracted from the variable This can be seen on the coordinate plane.
translation to the left
It is worth noting that the obtained function can be simplified by squaring the binomial.
▼
Simplify right-hand side
ExpandNegPerfectSquare

CommutativePropMult

Commutative Property of Multiplication

CalcPow

Calculate power

Multiply

Multiply

c The graph of is a horizontal translation by units followed by a vertical translation by If is positive, the horizontal translation is to the right, and if it is negative, this translation is to the left. Similarly, if is positive, the vertical translation is upwards. If is negative, this translation is downwards.
Again, special attention must be paid to the sign of This time, since the graph is to be translated units to the left, the value of is Therefore, must be subtracted from This is the same as adding to
combined translations
Again, the obtained function can be simplified by squaring the binomial.
▼
Simplify right-hand side
ExpandPosPerfectSquare

CommutativePropMult

Commutative Property of Multiplication

CalcPow

Calculate power

Multiply

Multiply

SubTerm

Subtract term

Pop Quiz

Stating the Translation

The graph of the quadratic function and a vertical or horizontal translation are shown in the coordinate plane.

Find the value of the constant
Closure

Transforming the Graph of a Function

With the topics learned in this lesson, the challenge presented at the beginning can be solved. The parabola that corresponds to the quadratic function is given.

parabola
By applying transformations to the above graph, draw the graph of

Answer
transformed quadratic graph

Hint

Rewrite the function whose graph is to be drawn to clearly identify the transformations.

Solution
First, the function will be rewritten to identify the transformations.
▼
Rewrite
WriteDiff

Write as a difference

FactorOut

Factor out

NegBaseToPosBase

AddNeg

CommutativePropMult

Commutative Property of Multiplication

The expression in parenthesis corresponds to the right-hand side of the equation of but with a negative sign for the variable Therefore, According to the topics learned in this lesson, the graph of can be explained as a sequence of transformations.
  1. Reflection in the axis.
  2. Vertical stretch by a factor of
  3. Translation units down.
This sequence of transformations can be seen on a coordinate plane.
quadratic graph
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