Start by dividing the numerator by the denominator.
Function: g(x)=-3/x-2+4 Graph:
Graph of g as a transformation of the graph of f(x)= ax: translation 2 units right and 4 units up of the graph of f(x)= -3x.
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 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 - 2 & |l 4x-11
4x/x= 4
r 4 r x-2 & |l 4x-11
Multiply term by divisor
r 4 rl x-2 & |l 4x-11 & 4x-8
Subtract down
r 4 r x-2 & |l -3
The quotient is 4 with a remainder of -3. Let's rewrite the function using this information.
g(x)=4x-11/x-2 ⇔ g(x)=-3/x-2+ 4
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)=-3/x- 2+ 4
We see above that a = -3, h = 2, and k = 4. Therefore, the graph of g is a translation 2 units right and 4 units up of the graph of f(x)= -3x. We will use this information to draw the graph.
