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Describing Transformations of Rational Functions

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

Rule

Translation

By adding some number to every function value, a function graph is translated vertically. Notice that the horizontal asymptote changes, while the vertical one is unaffected.

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. This transformation affects the vertical asymptote, but not the horizontal.

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. Thus, -intercepts and vertical asymptotes are preserved.

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 and horizontal asymptote are preserved.

Reflect graph in -axis

Rule

Stretch and Shrink

A function's 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

Describe how the function is a transformation of its parent function, Then, add the graph of to the coordinate plane below.

Show Solution
Solution

To transform into has to be added to the input, followed by adding to the function value: This transformation corresponds with a translation units to the left and unit upward. Thus, we can simply translate the graph of accordingly to sketch This can be done by first choosing some points on the graph of and plotting their counterpart for

Now, can be sketched by connecting the points with two curves, mimicking the appearance of Notice that the asymptotes have been translated units leftward and unit upward as well.

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Exercise

Suppose the rational function is stretched vertically by a factor of translated units to the right, and unit downward, resulting in Write the rule of and find its asymptotes.

Show Solution
Solution
Example

The rule of

To find the rule of we must know how the transformations affect the rule of algebraically. First, it is stretched vertically by a factor of When a function is stretched vertically, its rule is multiplied by the factor it's stretched, in this case The second transformation is a horizontal translation. When a function is moved horizontally, a number is subtracted from the input of the function rule. Hence, since the function is translated units to the right, it corresponds to subtracting from the input. Note that we apply the transformations in the order they are listed, as this will lead to The last transformation is a vertical translation downward. Thus, corresponding to subtracting a number from every function value. As the translation is unit, will be subtracted. All of the transformations have now been applied to Thus, the rule of can be written as

Example

Find the asymptotes of

Notice that is a simple rational function. Thus, we can compare the rule of with the general form of simple rational functions to identify its asymptotes. Its asymptotes can be identified from the values of and A simple rational function has a vertical asymptote at Hence, has a vertical asymptote at Further, it has a horizontal asymptote at which in our case is at

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