Graph of g as a Transformation of the Graph of f(x)= ax: Translation 6 units right and 1 unit up of the graph of f(x)= 24x.
Practice makes perfect
We will rewrite the function, describe the graph of g as a transformation of the graph of f(x)= ax, and draw the graph of g.
Rewriting the Function
We want to rewrite the given rational function so that it is in the form g(x)= ax-h+k. To do so, we will start by dividing the numerator of the function by the denominator.
l r x - 6 & |l x+18
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Divide
x/x= 1
r 1 r x-6 & |l x+18
Multiply term by divisor
r 1 rl x-6 & |l x+18 & x-6
Subtract down
r 1 r x-6 & |l 24
The quotient is 1 with a remainder of 24. Let's rewrite the function using this information.
g(x)=x+18/x-6 ⇔ g(x)=24/x-6+ 1
Describing the Graph as a Transformation of f(x)= ax and Graphing
Let's start by recalling two possible transformations of the function f(x)= ax.
Function
Transformation of the Graph of f(x)= ax
g(x)=a/x- h
Horizontal translation by h units. If h>0, the translation is to the right. If h<0, the translation is to the left.
g(x)=a/x+ k
Vertical translation by k units. If k>0, the translation is up. If k<0, the translation is down.
Let's now consider the obtained equation.
g(x)=24/x- 6+ 1
We can see that a= 24, h= 6, and k= 1. Therefore, the graph of g is a translation 6 units right and 1 unit up of the graph of f(x)= 24x. We will use this information to draw the graph.
