02.22truncus

Truncus

Power Functions

A power function has the form y = xn.

• For n = 1, we get a linear graph,
• For n = 2 we get a parabola,
• For n = 3 we get a cubic.
• For n = –1, we get a hyperbola.
• For n = –2, we get a graph called a truncus.

The Standard Truncus

… … $y = x^{-2} \text{ can be written as } y=\dfrac{1}{x^2}$

If we consider a table of values for the function

We can see that as the value of x increases, the value of y approaches zero but never reaches zero.

• Notice that the values get small faster than they do for a hyperbola.
• In symbols, as $x \rightarrow +\infty, \; y \rightarrow 0^+$
• In symbols, as $x \rightarrow -\infty, \; y \rightarrow 0^+$

We can also see that as the value of x decreases from 1 to 0, the value of y increases to infinity.

• At x = 0, y is undefined.
• Again, compared to a hyperbola, the values get larger much more quickly.
• In symbols, as $x \rightarrow 0^+, \; y \rightarrow +\infty$
• In symbols, as $x \rightarrow 0^-, \; y \rightarrow +\infty$

Notice that squaring a negative number results in a positive answer. So taking the reciprocal of a squared negative number also gives a positive answer.

That means the negative half of a truncus also has positive y-values

Graph of Standard Truncus

Now plot these points and sketch the graph. Clearly draw in and label the asymptotes.

… … $y=\dfrac{1}{x^2}$

… … The asymptotes are along the axis, so the equations of the asymptotes
… … are: $x = 0 \text{ and } y = 0$

… … The domain is all Real values of x, except 0.

… … Domain: $x \in R \backslash \{ 0 \}$

… … Range: $y \in R^+$

… … (or) Range: $\{ y: y > 0 \}$

… … There are no stationary points and no intercepts.

… … The graph passes through the points (1, 1) and (–1, 1)

Transformations on the Truncus

We can apply the standard transformations such as dilations, translations and reflections when sketching a truncus.

Notice that as with the hyperbola:

• we can find the vertical asymptote by setting the denominator to zero and solving for x
• we can find the horizontal asymptote by replacing the fraction with zero and solving the result for y (provided the fraction has been simplifed)

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