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Rational Functions

Describing Transformations of Rational Functions

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

Rule

Translation

By adding some number to every function value,
g(x)=f(x)+k,
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.
g(x)=f(xh)
Note that the number, h, is subtracted and not added. This is so that a positive h leads to a translation to the right, which is the positive x-direction. This transformation affects the vertical asymptote, but not the horizontal.
Translate graph to the right

Rule

Reflection

A function is reflected in the x-axis by changing the sign of all function values:
g(x)=-f(x).
Graphically, all points on the graph move to the opposite side of the x-axis, while maintaining their distance to the x-axis. Thus, x-intercepts and vertical asymptotes are preserved.
Reflect graph in x-axis

A graph is instead reflected in the y-axis by moving all points on the graph to the opposite side of the y-axis. This occurs by changing the sign of the input of the function.
g(x)=f(-x)
Notice that the y-intercept and horizontal asymptote are preserved.
Reflect graph in y-axis

Rule

Stretch and Shrink

A function's graph is vertically stretched or shrunk by multiplying the function rule by some constant a>0:
g(x)=af(x).
All vertical distances from the graph to the x-axis are changed by the factor a. Thus, preserving any x-intercepts.
Stretch graph vertically

By instead multiplying the input of a function rule by some constant a>0,
g(x)=f(ax),
its graph will be horizontally stretched or shrunk by the factor Since the x-value of y-intercepts is 0, they are not affected by this transformation.
Stretch graph horizontally

Example

Graph the transformed rational function

fullscreen
Describe how the function
is a transformation of its parent function, Then, add the graph of g to the coordinate plane below.
Show Solution expand_more
To transform f into g, 2 has to be added to the input, followed by adding 1 to the function value:
This transformation corresponds with a translation 2 units to the left and 1 unit upward. Thus, we can simply translate the graph of f accordingly to sketch g. This can be done by first choosing some points on the graph of f and plotting their counterpart for g.

Now, g can be sketched by connecting the points with two curves, mimicking the appearance of f. Notice that the asymptotes have been translated 2 units leftward and 1 unit upward as well.

Example

Write the rule of the transformed rational function

fullscreen

Suppose the rational function is stretched vertically by a factor of 5, translated 2 units to the right, and 1 unit downward, resulting in g. Write the rule of g and find its asymptotes.

Show Solution expand_more

Example

The rule of g

To find the rule of g, we must know how the transformations affect the rule of f algebraically. First, it is stretched vertically by a factor of 5. When a function is stretched vertically, its rule is multiplied by the factor it's stretched, in this case 5.
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 2 units to the right, it corresponds to subtracting 2 from the input.
Note that we apply the transformations in the order they are listed, as this will lead to g. The last transformation is a vertical translation downward. Thus, corresponding to subtracting a number from every function value. As the translation is 1 unit, 1 will be subtracted.
All of the transformations have now been applied to f. Thus, the rule of g can be written as

Example

Find the asymptotes of g

Notice that g is a simple rational function. Thus, we can compare the rule of g with the general form of simple rational functions to identify its asymptotes.
Its asymptotes can be identified from the values of h and k. A simple rational function has a vertical asymptote at x=h. Hence, g has a vertical asymptote at x=2. Further, it has a horizontal asymptote at y=k, which in our case is at y=-1.
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