Graph of g as a Transformation of the Graph of f(x)= ax: Translation 7 units left and 9 units up of the graph of f(x)= -66x.
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 + 7 & |l 9x-3
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Divide
9x/x= 9
r 9 r x+7 & |l 9x-3
Multiply term by divisor
r 9 rl x+7 & |l 9x-3 & 9x+63
Subtract down
r 9 r x+7 & |l -66
The quotient is 9 with a remainder of -66. Let's rewrite the function using this information.
g(x)=9x-3/x+7 ⇔ g(x)=-66/x+7+ 9
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)=-66/x+7+9 ⇔ g(x)=-66/x-( -7)+ 9
We can see that a= -66, h= - 7, and k= 9. Therefore, the graph of g is a translation 7 units left and 9 units up of the graph of f(x)= -66x. We will use this information to draw the graph.
