Let's start by recalling possible transformations of the parent function f(x)=x1.
Function | Transformation of the Graph of f(x)=x1 |
---|---|
g(x)=x−h1 | 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)=x1+k | Vertical translation by k units. If k>0, the translation is up. If k<0, the translation is down. |
g(x)=b(x)1 | Horizontal stretch or compression by a factor of b. If 0<b<1, it is a horizontal stretch. If 1<b, it is a horizontal compression. |
g(x)=-x1 | Reflection across the y-axis. |
Now, let's consider the given function. g(x)=-0.5(x−3)1+1⇕g(x)=-0.5(x−3)1+1 We can see that b=-0.5, h=3, and k=1. Also, there is a negative sign in the denominator. From here, we can determine the transformations.
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)=x1 | g(x)=(-0.5(x−3)1)+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).