1YTruncus


 * __Truncus__ **

The truncus is introduced in the VCE Maths Methods course (here)

math \text{If we consider the function } y=\dfrac{1}{x^2} math

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 as they approach 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 → ¥ so x = 0 is also an asymptote.

Notice that squaring a negative number results in a positive answer. So the function is __always__ postive.


 * Sketch Graph of the Standard Truncus **

If we plot the standard parabola, y = x 2 (in blue), and then consider the reciprocal of each point, we get the standard truncus (in red).


 * A vertical asymptote occurs where y = 0.
 * The graphs intercept at y = 1
 * When x < 1, y becomes large (y > 1)
 * When x > 1, y becomes small (y < 1)
 * The standard truncus is always positive

math y=\dfrac{1}{x^2} math

The equations of the asymptotes are: x = 0 and y = 0

Domain: x Î **R\**{0} Range: y Î **R** + (or) Range: {y: y > 0}

There are no stationary points and no intercepts.

Example 1

Sketch the parabola y = (x + 2) 2 (in blue) and take the reciprocal to obtain the sketch of the truncus (in red) math y = \dfrac{1}{(x+2)^2} math


 * A vertical asymptote occurs where y = 0.
 * The graphs intercept at y = 1
 * When x < 1, y becomes large (y > 1)
 * When x > 1, y becomes small (y < 1)

The equations of the asymptotes are: x = –2 and y = 0

Domain: x Î **R \ {**–2} Range: y Î **R** + (or) Range: {y: y > 0}

There are no stationary points.

There is a y-intercept at y = ¼

There are no x-intercepts


 * This is the __standard truncus__ translated to the left by 2 units**

Return to Summary of Coordinate Geometry .