6D - Truncus function

Learning intentions:

In this section we will examine:

    • The truncus graph and its transformations (dilations, reflections and translations).
    • How to graph the truncus function from a given equation.

The truncus graph

The truncus function has a general form:

    • All truncus graphs have the same shape, they are just transformed (dilated, reflected and translated) according to the values of the parameters a, h and k.

Introduction to the truncus graph

Consider the following table of values with numbers and the square of their reciprocal values:

Table 1 - Numbers and the square of their reciprocal values.

If we plot these point on a set of axes we can see the shape of the truncus graph:

Draw asymptotes as a dotted line on the graph. Make sure you use a ruler!

Figure 1 - The truncus graph with asymptotes y = 0 and x = 0.

Asymptotes

An asymptote is a line (or curve) where the distance between the curve and the asymptote approaches zero as they tend toward infinity.

This means that the curve of the truncus will approach the vertical and horizontal asymptotes but never actually reach them.

Consider the truncus with the general equation:

The horizontal asymptote occurs at y = k.

    • As x approaches ∞, y approaches k.
    • As x approaches -∞, y approaches k.

The vertical asymptote occurs at x = h.

    • As x approaches h (from left), y approaches ∞.
    • As x approaches h (from right), y approaches ∞.

When the parameters, h and k, are changed the asymptotes undergo translation.

Looking ahead: Domain of a function

In the near future we will examine the domain of a function. The domain is the set of all of the x-coordinates (or first elements of an order pair). When we consider domains of functions they can be maximal/implied or they can be restricted.

    • For the truncus function, the maximal domain is restricted to expression in the denominator not equalling zero. This is because you cannot divide by zero - it is undefined.

Looking ahead: Range of a function

In the near future we will examine the range of a function. The range is the set of all of the y-coordinates (or second elements of an order pair). When we consider the range of a functions they are influenced by domain and the values a function (rule) returns.

    • For the truncus function, the range is restricted because there is no value of x which gives y = k because the term
    • cannot equal zero (given that a ≠ 0). The range is also restricted due to the squared term, giving all positive numbers.

Transformations of the truncus graph

When considering transformations of the truncus graph, it is easiest to have the equation in the following form:

We can consider the effects of each parameter (a, h and k) on the hyperbola graph.

    • a will cause a vertical dilation by a factor of a from the x-axis.
      • Where a < 0 the graph is reflected in the x-axis.
    • h will cause a horizontal translation of h units parallel to the x-axis.
    • k will cause a vertical translation of k units parallel to the y-axis.

Examining the combined effects of a, h and k

The dynamic GeoGebra worksheet illustrates the combined effect of a, h and k on the truncus graph.

  • Please use the sliders to adjust the parameters and observe the transformations.

Coming soon!

Graphing the truncus function

When graphing the truncus function you must label:

    • Any axial intercepts (if they exist) with their coordinates.
    • Vertical and horizontal asymptotes with their equations.

Before graphing the square root function, always make sure it is in the following form:

To find the x-intercept let y = 0 and solve for x.

To find the y-intercept let x = 0 and solve for y.

The asymptotes exist at x = h and y = k.

6D - VIDEO EXAMPLE 1:

Graph the following truncus and state the maximal domain and range:

6D - VIDEO EXAMPLE 2:

Graph the following truncus and state the maximal domain and range:

6D - VIDEO EXAMPLE 3:

Graph the following truncus and state the maximal domain and range:

Success criteria:

You will be successful if you can:

    1. Identify and describe transformations of the truncus graph.
    2. Sketch the truncus graph from a given equation.
    3. Label axial intercepts with their coordinates and asymptotes with their equations.