Start by drawing the graph of the parent function, f(x)= 1x.
Graph:
Domain: All real numbers except x=- 5 Range: All real numbers except y=- 6
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
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
Reflection
In the x-axis
y=- 1/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=- 8/x+ 5- 6The given function is a combination of transformations.
Horizontal translation 5 units left
Vertical stretch by a factor of 8
Reflection in the x-axis
Vertical translation down 6 units
Let's apply these transformations one at a time. We will start by translating the parent function, f(x)= 1x, 5 units left.
Now, let's apply a stretch by a factor of 8.
The third transformation is a reflection in the x-axis.
The last transformation is a vertical translation 6 units down.
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=- 5, and the equation of the horizontal asymptote is y=- 6. Using this information, we can state the domain and range of the function.
Domain:& All real numbers except x=- 5
Range:& All real numbers except y=- 6
