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Radical Functions

Describing Transformations of Radical Functions

The common transformations can be applied to radical functions as usual.

Rule

Translation

By adding some number to every function value, a function's graph is translated vertically.

Translate graph upward

A graph is translated horizontally by subtracting a number from the input of the function rule. Note that the number, is subtracted and not added. This is so that a positive leads to a translation to the right, which is the positive -direction.

Translate graph to the right

Rule

Reflection

A function is reflected in the -axis by changing the sign of all function values: Graphically, all points on the graph move to the opposite side of the -axis, while maintaining their distance to the -axis.

Reflect graph in -axis

A graph is instead reflected in the -axis by moving all points on the graph to the opposite side of the -axis. This occurs by changing the sign of the input of the function. Notice that the -intercept is preserved.

Reflect graph in -axis

Rule

Stretch and Shrink

A function graph is vertically stretched or shrunk by multiplying the function rule by some constant : All vertical distances from the graph to the -axis are changed by the factor Thus, preserving any -intercepts.

Stretch graph vertically

By instead multiplying the input of a function rule by some constant its graph will be horizontally stretched or shrunk by the factor Since the -value of -intercepts is they are not affected by this transformation.

Stretch graph horizontally

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Exercise

The graphs of four radical functions are shown in the image.

Compare the graphs with each other or their parent function to match each with its corresponding function rule:

Show Solution
Solution

The radicals in the function rules are either square roots or cube roots. Thus, the parent function of each is either the square root of or the cube root of Let's start with Graph I.

Example

Graph I

From the image, we can see that the function corresponding to Graph I isn't defined for all real numbers. Thus, it must be one of the square root functions. Comparing it to the graph of can give us more information about its function rule.

It looks as though Graph I is a translation of both to the left and downward. Thus, its function rule is similar to but with some number added to the input and some number subtracted from the output. Among the available choices, there is a match,

Example

Graph II

Graph II is very similar to Graph I, so let's compare them.

Here, II looks to be a reflection of I in the -axis. If this is the case, one of the options must be equal to

We can now identify that the function is a reflection of in the -axis. Thus, it corresponds to Graph II.

Example

Graph III

Graph III has to be the graph of one of the square root functions, as it's not defined for all real numbers. There is only one square root function remaining, which must be its match. Looking at the graph, we can confirm that this is the case. Comparing with the sign of the input has been reversed. Thus is a reflection of in the -axis. This is exactly what can be seen in the graphs when comparing I and III.

Example

Graph IV

With only one rule remaining, Graph IV must correspond to To confirm this, can be viewed as a translation of unit to the left and units downward. This can also be seen in its graph.

Thus, we have matched all graphs and function rules.

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Exercise


The rules of and are given such that is a transformation of Express as a function of Then, graph both functions in the same coordinate plane and state the transformation(s) underwent to become

Show Solution
Solution
To write as a function of we can first find an expression for then one for This is done by replacing with in the function rule of Now that we have an expression for we can find by multiplying the expression above by This gives the rule for
The function is now written in terms of without containing the function We'll graph and

Example

Graphing

To graph let's calculate some function values in a table. Since is a square root function, it's not defined when the expression under the radical sign is negative. Thus, a good starting value is

The -values and the function values can now be plotted as points in a coordinate plane. We'll connect the points with a smooth curve. Note that the function is not defined for

Example

Graphing

We'll use the same procedure as we did for graphing The function is also a square root function and is not defined for -values less than Therefore, we will start at for our table values.

The points can now be plotted in the same coordinate plane as again connecting the points with a smooth curve.

Example

Transformations from to

To determine the transformations underwent to become we'll study the function rule of This function can be seen as adding to the input of and multiplying the function value by Therefore, the first transformation is a translation unit to the left.

Furthermore, the second transformation is a reflection in the -axis.

Therefore, has undergone both a horizontal translation and a reflection in the -axis to become

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