The Truncus (VCE SSCE Mathematical Methods): Revision Notes

The Truncus

Understanding the truncus function

The truncus is a distinctive U-shaped curve that appears when we graph certain reciprocal functions. The most basic form of a truncus comes from the function:

$$y = \frac{1}{x^2} = x^{-2} \text{ for } x \neq 0$$

This function creates a symmetrical curve that has some fascinating properties. Unlike many functions you've encountered, this one behaves quite uniquely around the origin and at extreme values.

Building the graph from values

To understand how this graph takes shape, let's examine some specific coordinate points. We can calculate y-values for various x-values between -4 and 4:

Notice the pattern in these values:

• When $x = \pm 1$, we get $y = 1$
• When $x = \pm 2$, we get $y = \frac{1}{4}$
• When $x = \pm \frac{1}{2}$, we get $y = 4$

When we plot these points and connect them with a smooth curve, we get the characteristic truncus shape:

Key features of the truncus

What makes the truncus special

The truncus has several important characteristics that distinguish it from other function graphs. Understanding these features will help you sketch and analyse truncus graphs confidently.

First, the function is undefined at x = 0. We cannot divide by zero, so there is no y-value when x equals zero. This creates a gap in the graph at the origin.

Second, the function only produces positive values. Since we're squaring x in the denominator, the result is always positive (regardless of whether x itself is positive or negative). This means the graph exists entirely above the x-axis.

Asymptotes explained

Behaviour at extreme values

Understanding how the function behaves as x approaches different values helps us sketch accurate graphs. Here's what happens:

• As $x \to \infty$, $y \to 0^+$ (y approaches zero from above as x gets very large and positive)
• As $x \to -\infty$, $y \to 0^+$ (y approaches zero from above as x gets very large and negative)
• As $x \to 0^+$, $y \to \infty$ (y increases without bound as x approaches zero from the right)
• As $x \to 0^-$, $y \to \infty$ (y increases without bound as x approaches zero from the left)

This behaviour creates the characteristic "double peak" appearance of the truncus, with the curve rising steeply on both sides of the y-axis.

Transformations of the truncus

Just like other functions you've studied, the truncus can be transformed through translations, dilations, and reflections. The transformation rules you've learned apply here as well.

All graphs of the form:

$$y = \frac{a}{(x-h)^2} + k$$

will maintain the same basic truncus shape. Each parameter affects the graph differently:

• The value a controls vertical stretching, compression, and reflection. If $a$ is negative, the graph is reflected in the horizontal asymptote (it opens downward instead of upward).
• The value h shifts the graph horizontally. The graph moves $h$ units to the right if $h$ is positive, or $|h|$ units to the left if $h$ is negative.
• The value k shifts the graph vertically, moving it up by $k$ units if $k$ is positive, or down by $|k|$ units if $k$ is negative.

Finding asymptotes in transformed graphs

Worked example

Remember!