Start by dividing the numerator by the denominator.
Function: g(x)=1/x+1+2 Graph:
Graph of g as a Transformation of the Graph of f(x)= ax: Translation 1 unit left and 2 units up of the graph of f(x)= 1x.
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 + 1 & |l 2x+3
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Divide
2x/x= 2
r 2 r x+1 & |l 2x+3
Multiply term by divisor
r 2 rl x+1 & |l 2x+3 & 2x+2
Subtract down
r 2 r x+1 & |l 1
The quotient is 2 with a remainder of 1. We can rewrite the function using this information.
g(x)=2x+3/x+1 ⇔ g(x)=1/x+1+ 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.
Now consider the obtained equation.
g(x)=1/x+1+2 ⇔ g(x)=1/x-( - 1)+ 2
We can see that a= 1, h= - 1, and k= 2. Therefore, the graph of g is a translation 1 unit left and 2 units up of the graph of f(x)= 1x. We will use this information to draw the graph.
