g(x)=1/2(x+2)+3
⇕
g(x)=1/11/2(x-( -2))+ 3
We can see that b=12, h= -2, and k= 3. From here, we can determine the transformations.
A horizontal compression by a factor of 12.
A horizontal translation 2 units to the left.
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 compression 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)=1/11/2(x-(-2))+3
Vertical asymptote
x=0
x=0+( -2) ⇕ x=-2
Horizontal asymptote
y=0
y=0+ 3 ⇕ y=3
Reference point
(-1,-1)
(1/2(- 1)+( -2),- 1+ 3) ⇕ (-2.5,2)
Reference point
(1,1)
(1/2(1)+( -2),1+ 3) ⇕ (-1.5,4)
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=3 is the horizontal asymptote, we will exclude them from the domain and range, respectively.
