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

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

g(x)=f(x−h)

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

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

g(x)=f(-x)

Notice that the y-intercept and horizontal asymptote are preserved.
Reflect graph in y-axis

g(x)=a⋅f(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

g(x)=f(a⋅x),

its graph will be horizontally stretched or shrunk by the factor $a1 .$ Since the x-value of y-intercepts is 0, they are not affected by this transformation.
Stretch graph horizontally

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.

Suppose the rational function $f(x)=x1 $ 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*

$Generalx−ha Form+k Fu nctiong(x)x−25 −1 $

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. {{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!

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