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This lesson will discuss how to apply different transformations to radical functions. It will also show how to identify these transformations in the graphs of different radical 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 parent function and the graph of the radical function are drawn on the same coordinate plane.

Find the values of and
Explore

## Translating a Graph

The graph of the function is shown in the coordinate plane. By changing the values of and observe how the graph is horizontally and vertically translated.

Discussion

A translation of a function is a transformation that shifts a graph vertically or horizontally. As with the graph of any other function, a vertical translation of the graph of a radical function is achieved by adding some number to every output value of the function rule. Consider the parent 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. This transformation can also be shown on a coordinate plane.
Example

Ignacio has just started to learn about transformations of radical functions. In order to get some extra practice, he goes online to to find a worksheet for the topic. He found a pretty good exercise, which is divided into parts A, B, and C.
Help Ignacio solve the exercise!

a Equation:

Graph:

b Equation:

Graph:

c Equation:

Graph:

### Hint

a To translate the graph of to the right units, find an equation for
b To translate the graph of up units, find an equation for
c To translate the graph of down units and to the left units, find an equation for

### Solution

a To translate the graph of a function units to the right, the number must be subtracted from the input of the function.
The right-hand side of the equation can be simplified by distributing the in the radicand.
Now both functions can be drawn on the same coordinate plane.
b To translate the graph of a function units up, the number must be added to the output of the function.
The right-hand side of the equation can be simplified by adding the terms.
Now both functions can be drawn on the same coordinate plane.
c To translate the graph of a function units down and units to the left, must be subtracted from the output and must be added to the input.
The right-hand side of the equation can be simplified by subtracting the terms.
Now both functions can be drawn on the same coordinate plane.
Pop Quiz

## Stating the Translation

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

Explore

## Stretching and Shrinking a Graph

The graph of the radical function is shown in the coordinate plane. By changing the values of and observe how the graph is vertically and horizontally stretched and shrunk.

Discussion

## Stretch and Shrink of Radical Functions

A function graph is vertically stretched or shrunk by multiplying the output of the function rule by some positive constant Consider the radical function
If is greater than the graph 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 neither stretch nor shrink. All vertical distances from the graph to the axis are changed by the factor
Similarly, the graph of a radical function is horizontally stretched or shrunk by multiplying the input of the function rule by some positive constant Consider this time the parent function
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.

### Extra

Finding and

Suppose that a function is horizontally or vertically stretched/shrunk, and that the graphs of the transformed and the original function are both drawn on the same coordinate plane. Then, the values of or can be found by following these procedures.

 Finding Select two points with the same coordinate, one point on the parent function and the other point on the transformed function. The value of is the quotient of the coordinate of and the coordinate of Select two points with the same coordinate, one point on the parent function and the other point on the transformed function. The value of is the quotient of the coordinate of and the coordinate of
Example

## Stretching and Shrinking Radical Functions

After mastering vertical and horizontal translations of radical functions, Ignacio is having a hard time understanding vertical and horizontal stretches and shrinks of this type of function. He asked for some help from his very good friend Jordan.

In order to help him, Jordan asks Ignacio to consider the following function.
She then tells him to complete the next two exercises.
a Write the equation and draw the graph of a function whose graph is a vertical stretch of the graph of by a factor of
b Write the equation and draw the graph of a function whose graph is a horizontal shrink of the graph of by a factor of

a Equation:

Graph:

b Equation:

Graph:

### Hint

a To vertically stretch the graph of by a factor of multiply the output by
b To horizontally shrink the graph of by a factor of multiply the input by

### Solution

a To vertically stretch the graph of by a factor of the output of the function must be multiplied by
The right-hand side of the function can be simplified by distributing the
Now both functions can be drawn on the same coordinate plane.
b To horizontally shrink the graph of by a factor of the input of the function must be multiplied by
Now both functions can be drawn on the same coordinate plane.
Pop Quiz

## Stating the Factor of Stretch or Shrink

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

Discussion

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. This means changing the sign of the coordinate of every point on the graph of a function. Consider the radical function
This reflection can be shown on a coordinate plane.
A reflection in the axis is instead achieved by changing the sign of every input value. In this case, note that the domain of the radical function needs to be modified from all non-negative numbers to all non-positive numbers. Consider this time the parent function
This transformation can also be shown on a coordinate plane.
Example

Ignacio is now feeling pretty confident about transformations of radical functions again. Now he turns his attention to reflections.

Ignacio considers the following radical function.
a Write the equation and draw the graph of a function whose graph is a reflection in the axis of the graph of
b Write the equation and draw the graph of a function whose graph is a reflection in the axis of the graph of

a Equation:

Graph:

b Equation:

Graph:

### Hint

a To reflect the graph of a function in the axis, multiply the output by
b To reflect the graph of a function in the axis, multiply the input by

### Solution

a To reflect the graph of the given function in the axis, the output needs to be multiplied by
The right-hand side of the function can be simplified by distributing the
Now both functions can be graphed on the same coordinate plane.
b To reflect the graph of the given function in the axis, the input needs to be multiplied by
Now, both functions can be graphed on the same coordinate plane.
Example

## Combining Transformations

Ignacio confidently states that he can now solve any exercise about transformations of radical functions. Jordan skeptically challenges her friend to solve an exercise that combines transformations.

Help Ignacio solve Jordan's challenge!

Equation:
Graph:

### Hint

Start by multiplying the output by to find the function that represents the vertical stretch. Then, multiply the input by to reflect the graph in the axis. Finally, subtract units from the input.

### Solution

To help Ignacio, the transformations will be applied one at a time.

### Vertical Stretch by a Factor of

First, the equation of the vertical stretch by a factor of will be found. To do so, multiply the output of the function by
Simplify the right-hand side of the above equation by distributing the
The graph of the function is a vertical stretch by a factor of of the graph of Let be this function.

### Reflection in the axis

To reflect the graph of a function in the axis, multiply the input by
The graph of this function is a reflection in the axis of the graph of Let be this function.

### Translation Units to the Right

Finally, to horizontally translate the graph of a function units to the right, subtract from the input.
Simplify the right-hand side of the equation by distributing the and adding the terms in the radicand.
The graph of this final function is a vertical stretch by a factor of followed by a reflection in the axis, and a translation units to the right of the graph of

### Graph

The transformations can be shown on a coordinate plane.
Closure

## Transforming the Graph of a Function

With the topics learned in this lesson, the challenge presented at the beginning can be solved. The graphs of the radical function and its corresponding parent function are given.

Find the values of and

### Hint

Start by considering a vertical stretch. Then, consider vertical and horizontal translations.

### Solution

Disregard translations for a moment. In the graph of it can be understood as, after moving unit to the right of the starting point the graph increases vertically unit. Conversely, in the graph of after moving unit to the right of the starting point the graph increased vertically units.

This means that the graph of is a vertical stretch of the graph of by a factor of Therefore, the value of is
Next, pay close attention to the starting point of each curve. It can be concluded that the graph of the parent function needs to be translated units to the left to match the other graph. Therefore, the value of is
Finally, note that the coordinate of the initial point of both graphs is Therefore, there is no vertical translation. This means that
It has been found that and The obtained function can be simplified.
The transformations can be shown on a graph.