g(x)=-0.5(1/x-1)-2
⇕
g(x)=-0.5(1/x- 1)+( -2)
We can see that a=-0.5, h= 1, and k= -2. From here, we can determine the transformations.
A vertical compression by a factor of 0.5.
A reflection across the x-axis.
A horizontal translation 1 unit to the right.
A vertical translation 2 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 compression, reflection across the x-axis, and vertical translation affect only the y-coordinates.
Object
f(x)=1/x
g(x)=-0.5(1/x-1)+(-2)
Vertical asymptote
x=0
x=0+ 1 ⇕ x=1
Horizontal asymptote
y=0
y=0+( -2) ⇕ y=- 2
Reference point
(-1,-1)
(-1+ 1,-0.5(- 1)+( -2)) ⇕ (0,-1.5)
Reference point
(1,1)
(1+ 1,-0.5(1)+( -2)) ⇕ (2,-2.5)
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=-2 is the horizontal asymptote, we will exclude them from the domain and range, respectively.
