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{{ printedBook.courseTrack.name }} {{ printedBook.name }} The common transformations can be applied to rational functions as usual.

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. $g(x)=f(x−h)$ 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

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

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

Stretch graph vertically

By instead multiplying the input of a function rule by some constant $a>0,$ $g(x)=f(a⋅x),$ 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

Describe how the function $g(x)=x+21 +1$ is a transformation of its parent function, $f(x)=1/x.$ Then, add the graph of $g$ to the coordinate plane below.

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To transform $f$ into $g,$ $2$ has to be added to the input, followed by adding $1$ to the function value: $f(x+2)+1=x+21 +1=g(x).$ 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.

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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 $5.$ $5⋅f(x)=5⋅x1 =x5 $ 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. $5⋅f(x−2)=x−25 $ 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. $5⋅f(x−2)−1=x−25 −1$ All of the transformations have now been applied to $f.$ Thus, the rule of $g$ can be written as $g(x)=x−25 −1.$

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. $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 $y=-1.$

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