Start by dividing the numerator by the denominator.
Function: g(x)=25/x-3+7 Graph:
Graph of g as a transformation of the graph of f(x)= ax: translation 3 units right and 7 units up of the graph of f(x)= 25x.
Practice makes perfect
We will start by rewriting 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 - 3 & |l 7x+4
7x/x= 7
r 7 r x-3 & |l 7x+4
Multiply term by divisor
r 7 rl x-3 & |l 7x+4 & 7x-21
Subtract down
r 7 r x-3 & |l 25
The quotient is 7 with a remainder of 25. Let's rewrite the function using this information.
g(x)=7x+4/x-3 ⇔ g(x)=25/x-3+ 7
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)=25/x- 3+ 7
We see above that a = 25, h = 3, and k = 7. Therefore, the graph of g is a translation 3 units right and 7 units up of the graph of f(x)= 25x. We will use this information to draw the graph.
