Domain: All real numbers except x=7 Range: All real numbers except y=- 3
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 y= 1x, x≠ 0
Vertical Translations
Translation up k units, k>0
y=1/x+ k
Translation down k units, k>0
y=1/x- k
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=5/x- 7- 3
The given function is a combination of transformations.
Horizontal translation 7 units right
Vertical stretch by a factor of 5
Vertical translation down 3 units
Let's apply these transformations one at a time. We will start by translating the parent function, y= 1x, 7 units right.
Now, let's apply a stretch by a factor of 5.
The last transformation is a vertical translation 3 units down.
Finally, let's look at the graph of the given function and its asymptotes alone.
We can see that the vertical asymtpote is the line x=7, and the equation of the horizontal asymptote is y=- 3. Using this information, we can state the domain and range of the function.
Domain:& All real numbers except x=7
Range:& All real numbers except y=- 3
