02Dtruncus

toc = Truncus =

Power Functions
A power function has the form y = x n.
 * 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
math . \qquad y = x^{-2} \text{ can be written as } y=\dfrac{1}{x^2} \qquad. math

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 → ¥, y → 0 so y = 0 is an asymptote.

For the same function, if we consider x-values between 0 and 1, we get:

Looking at the table from right to left, we 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 → 0, y → ¥ x = 0 is also an asymptote.

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.

Graph of Standard Truncus
Now plot these points and sketch the graph. Clearly draw in and label the asymptotes. math . \qquad y=\dfrac{1}{x^2} \qquad. math The asymptotes are along the axis, so the equations of the asymptotes are: x = 0 and y = 0

The domain is all Real values of x, except 0. Domain: x Î R\{0} Range: y Î 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)

We can apply the standard transformations such as dilations, translations and reflections when sketching a truncus. media type="custom" key="7960152" media type="custom" key="7960164" ... media type="custom" key="7960168"

media type="custom" key="7960172"

Notice that as with the hyperbola: Go to top of page : flat
 * 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)