Graph of g as a Transformation of the Graph of f(x)= ax: Translation 5 units right and 2 units up of the graph of f(x)= 6x.
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 - 5 & |l 2x-4
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Divide
2x/x= 2
r 2 r x-5 & |l 2x-4
Multiply term by divisor
r 2 rl x-5 & |l 2x-4 & 2x-10
Subtract down
r 2 r x-5 & |l 6
The quotient is 2 with a remainder of 6. Let's rewrite the function using this information.
g(x)=2x-4/x-5 ⇔ g(x)=6/x-5+ 2
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)=6/x- 5+ 2
We can see that a= 6, h= 5, and k= 2. Therefore, the graph of g is a translation 5 unit right and 2 units up of the graph of f(x)= 6x. We will use this information to draw the graph.
