Start by drawing the graph of the parent function, f(x)= 1x.
Graph:
Domain: {x| x≠ 0} Range: {f(x)| f(x)≠ 2}
Practice makes perfect
We want to graph 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
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.
f(x)= - 12/x+ 2
The given function is a combination of transformations.
Vertical stretch by a factor of 12
Reflection in the x-axis
Vertical translation up 2 units
Let's apply these transformations one at a time. We will start by stretching the parent function, f(x)= 1x, by a factor of 12.
The second transformation is a reflection in the x-axis.
The last transformation is a vertical translation 2 units up.
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=0, and the equation of the horizontal asymptote is y=2. Using this information, we can state the domain and range of the function.
Domain:& {x| x≠ 0} Range:& {f(x)| f(x)≠ 2}
