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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

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.
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.
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

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.

a Graph:

Equation:

b Graph:

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.
b The reflection of the graph of in the axis is given by the equation of
To obtained equation can be simplified.
Therefore, the graph of is a reflection of the graph of in the 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

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
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
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.

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.
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.
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.

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.

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.
The resulting function can be simplified by squaring the binomial and then distributing the
Simplify right-hand side
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.
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

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.
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
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.

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.

a Equation:

Graph:

b Equation:

Graph:

c Equation:

Graph:

### 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.
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.
It is worth noting that the obtained function can be simplified by squaring the binomial.
Simplify right-hand side