g(x)=1/-0.5(x-3)+1
⇕
g(x)=1/- 12(x- 3)+ 1
We can see that b=2, h= 3, and k= 1. Also, there is a negative sign in the denominator. From here, we can determine the transformations.
A horizontal stretch by a factor of 2.
A reflection across the y-axis.
A horizontal translation 3 units to the right.
A vertical translation 1 unit up.
Using these transformations, we can find the asymptotes and the reference points of the graph of g(x). Note that the horizontal stretch, the reflection across the y-axis, and the horizontal translation affect only the x-coordinates, while the vertical translation affects only the y-coordinates.
Feature
f(x)=1/x
g(x)=(1/- 12(x-3))+1
Vertical asymptote
x=0
x=0+ 3 ⇕ x=3
Horizontal asymptote
y=0
y=0+ 1 ⇕ y=1
Reference point
(-1,-1)
(-2(- 1)+ 3,- 1+ 1) ⇕ (5,0)
Reference point
(1,1)
(-2(1)+ 3,1+ 1) ⇕ (1,2)
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=3 is the vertical asymptote and y=1 is the horizontal asymptote, we will exclude them from the domain and range, respectively.
