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)=a 1x-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
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Divide
-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(1/x-2)+( -4)
Describing the Graph as a Transformation of f(x)= ax and Graphing
Let's start by recalling possible transformations of the parent function f(x)= 1x.
Function
Transformation of the Graph of f(x)= 1x
g(x)=1/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)=1/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 function.
g(x)=3(1/x- 2)+( -4)
We can see that a=3, h= 2, and k= -4. From here, we can determine the transformations.
A vertical stretch by a factor of 3.
A horizontal translation 2 units to the right.
A vertical translation 4 units down.
Using these transformations, we can find the asymptotes and the reference points of the graph of g(x). Note that the horizontal translation affects only the x-coordinates, while the vertical stretch and the vertical translation affect only the y-coordinates.
Feature
f(x)=1/x
g(x)=3(1/x-2)+(-4)
Vertical asymptote
x=0
x=0+ 2 ⇕ x=2
Horizontal asymptote
y=0
y=0+( -4) ⇕ y=-4
Reference point
(-1,-1)
(- 1+ 2,3(- 1)+( -4)) ⇕ (1,-7)
Reference point
(1,1)
(1+ 2,3(1)+( -4)) ⇕ (3,-1)
Next, we will use the table above to graph f(x) and g(x).
Finally, we will state the domain and range of g(x). Since x=2 is the vertical asymptote and y=-4 is the horizontal asymptote, we will exclude them from the domain and range, respectively.
