Hyperbolas and truncuses

Hyperbola and truncus functions are classified as power functions due to their distinct forms, characterised by negative exponents. Unlike polynomials, this form results in unique graphs that feature an additional key characteristic: asymptotes. Asymptotes are lines that describe the value or direction a graph approaches but never reaches as \(x\) tends to infinity or as the graph approaches undefined points.

The basic form of a hyperbola is:

The basic form of a truncus is:

Both functions have asymptotes at \(x = 0\) and \(y = 0\) for their basic forms.

Use this page to revise the following concepts of hyperbolas and truncuses:

• Graphing Hyperbola and Truncus Functions
• Solving Hyperbola and Truncus Equations
• Transformations of Hyperbola and Truncus Functions

Graphing Hyperbola and Truncus Functions

Hyperbola and truncus functions have unique graphical representations. Key features of their graphs include:

• Shape: Graph that has symmetry along some axes and approaches two asymptotes
• Asymptotes: Lines that describe the value or direction a graph approaches. Both functions have both \(x\)- and \(y\)- asymptotes
• \(x\)- and \(y\)-intercepts: The points where the graph crosses the axes.
• End points (if applicable): The boundaries of the graph when the domain is restricted. This is the determined similarly for other functions, by substituting the ends of the domains into the function.

Shape

Asymptotes

The asymptotes for the basic forms of both a hyperbola and a truncus are located at \(x = 0\) (vertical asymptote) and \(y = 0\) (horizontal asymptote). These asymptotes can be adjusted for transformed variations by identifying the relevant translation factors:

• The vertical asymptote at \(x = 0\) is shifted by the horizontal translation factor, corresponding to changes in the \(x\)-coordinate.
• The horizontal asymptote at \(y = 0\) is shifted by the vertical translation factor, corresponding to changes in the \(y\)-coordinate.

Axial Intercepts

The axial intercepts are the points where the graphs cross the axes. Since there are two axes, there are two types of axial intercepts:

• The \(x\)-intercept: the point(s) where the graph crosses the \(x\)-axis. These intercepts are also known as the solutions or roots of the equation.
• The \(y\)-intercept: the point where the graph crosses the \(y\)-axis

The \(x\)-axis corresponds to the horizontal line \(y = 0\); and the \(y\)-axis corresponds to the vertical line \(x = 0\). As they correspond to these values, these values can be substituted into the function to determine the \(x\)- and \(y\)- intercepts.

A key difference in the axial intercepts of a hyperbola and a truncus, compared to other functions, is that they may not be defined if the asymptotes coincide with the axes.

Solving Hyperbola and Truncus Equations

Understanding the behaviour of hyperbola and truncus graphs is key to solving these equations. Transformations, particularly vertical translations, determine whether solutions exist. Without a vertical shift, the horizontal asymptote lies on the \(x\)-axis, making the reciprocal function undefined at those values.

Once the graph's transformation is understood, substituting \(f(x) = 0\) into the equation is a straightforward way to find solutions. Alternatively, a "shortcut" formula derived from the general form can be used, provided the conditions for its validity are understood.

Hyperbola

For a hyperbola given in the form:

The solution for x is:

From this equation, we can observe that \( b \neq 0\) and \(d \neq 0\) for the value to defined, and the denominator in the term \(-\frac{a}{bd}\) cannot be 0.

However, we also have the condition that \(a \neq 0\). If \(f\left(x\right) = 0\) and \(a = 0\), then the equation simplifies to \(0 = \frac{0}{b\left(x - c\right)} + d\). This reduces to \(d = 0\) which does not satisfy the original condition where \(d \neq 0\).

Truncus

For a truncus given in the form:

The solution for \(x\) is:

From this equation, we can observe that \(b \neq 0\) and \(d \neq 0\) for the value to be defined, as the denominator in \(\frac{a}{b\left(-d\right)}\) cannot be 0. Similarly, \(a \neq 0\) for the same reasons as defined for the hyperbola.

There is another condition, which is to ensure that \(\frac{a}{b\left(-d\right)} > 0\) for there to be a real solution.
This can be simplified to the two cases:

• Positive truncus, with vertical translation down:
  \[a > 0, d < 0\]
• Negative truncus, with a vertical translation up:
  \[a < 0, d > 0\]

Transformations of Hyperbola and Truncus Functions

Transformations for both hyperbola and truncus functions can only be clearly defined when in the given forms:

• Hyperbola function:

\[f(x) = \frac{a}{b\left(x -c\right)} + d\]

• Truncus function:

\[f(x) = \frac{a}{b\left(x -c\right)^{2}} + d\]

In both functions, the transformation factors correspond to the following characteristics:

• Dilation from the \(x\)-axis and reflection in the \(x\)-axis \((a)\)
  • Determines the vertical dilation by a factor of \(a\)
  • If \(a < 0\), there is also a reflection in the \(x\)-axis
• Dilation from the \(y\)-axis and reflection in the \(y\)-axis \((b)\)
  • Determine the horizontal dilation by a factor of \(\frac{1}{b}\)
  • If \(b < 0\), there is also a reflection in the \(y\)-axis
    • For truncus functions the reflection does not affect the appearance of the graph, as it is symmetrical about the vertical asymptote.
• Horizontal translation \((c)\), if:
  • \(c > 0\), translation of c units in the positive direction of the \(x\)-axis (the graph shifts \(c\) units to the right)
  • \(c < 0\), translation of \(c\) units in the negative direction of the \(x\)-axis (the graph shifts \(c\) units to the left)
• Vertical translation (\(d)\), if:
  • \(d > 0\), translation of \(d\) units in the positive direction of the \(y\)-axis (the graph shifts \(d\) units upwards)
  • \(d < 0\), translation of \(d\) units in the negative direction of the \(y\)-axis (the graph shifts \(d\) units downwards)

Note that the horizontal and vertical translations also define the asymptotes.

• Vertical asymptote:

\[x = c\]

• Horizontal asymptote:

\[y = d\]

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