Start by drawing the graph of the parent function, f(x)= 1x.
Graph:
Domain: All real numbers except x=- 3 Range: All real numbers except y= 0
Practice makes perfect
We want to draw the asymptotes and the graph of the given function. We will start by considering some possible transformations.
Transformations of f(x)= 1x, x≠0
Horizontal Translations
Translation right h units, h>0
y=1/x- h
Translation left h units, h>0
y=1/x+ h
Vertical Stretch or Shrink
Vertical stretch, a>1
y=a/x
Vertical shrink, 0< a< 1
y=a/x
Note that if the graph of the function is translated, the asymptotes are also translated in the same distance and direction. Consider the function.
y=2/x+ 3
The given function is a combination of transformations.
Horizontal translation 3 units left
Vertical stretch by a factor of 2
Let's apply these transformations one at a time. We will start by translating the parent function, f(x)= 1x, 3 units left.
Now, let's apply a stretch by a factor of 2.
Finally, let's look at the graph of the given function and its asymptotes alone.
We can see that the vertical asymptote is the line x=- 3, and the equation of the horizontal asymptote is y=0. Using this information, we can state the domain and range of the function.
Domain:& All real numbers except x=- 3
Range:& All real numbers except y=0
