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 - 1 & |l 3x-5
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Divide
3x/x= 3
r 3 r x-1 & |l 3x-5
Multiply term by divisor
r 3 rl x-1 & |l 3x-5 & 3x-3
Subtract down
r 3 r x-1 & |l -2
The quotient is 3 with a remainder of -2. Let's rewrite the function using this information.
g(x)=3x-5/x-1 ⇔ g(x)=-2(1/x-1)+ 3
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)=-2(1/x- 1)+ 3
We can see that a=2, h= 1, and k= 3. Also, there is a negative sign. From here, we can determine the transformations.
A vertical stretch by a factor of 2.
A reflection across the x-axis.
A horizontal translation 1 unit to the right.
A vertical translation 3 units up.
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, the reflection across the x-axis, and the vertical translation affect only the y-coordinates.
Feature
f(x)=1/x
g(x)=-2(1/x-1)+3
Vertical asymptote
x=0
x=0+ 1 ⇕ x=1
Horizontal asymptote
y=0
y=0+ 3 ⇕ y=3
Reference point
(-1,-1)
(- 1+ 1,-2(- 1)+ 3) ⇕ (0,5)
Reference point
(1,1)
(1+ 1,-2(1)+ 3) ⇕ (2,1)
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=1 is the vertical asymptote and y=3 is the horizontal asymptote, we will exclude them from the domain and range, respectively.
