Graphs of Reciprocal Functions (HSC SSCE Mathematics Advanced): Revision Notes

Graphs of Reciprocal Functions

What is a reciprocal function?

A reciprocal function has the form:

$f(x) = \frac{k}{x}$

where $k$ is a constant and $k \neq 0$.

This type of function produces a graph called a rectangular hyperbola. The graph consists of two smooth curves that never touch the axes.

The simplest form is $f(x) = \frac{1}{x}$ (where $k = 1$). This creates one curve in the first quadrant (where $x > 0$ and $f(x) > 0$) and another curve in the third quadrant (where $x < 0$ and $f(x) < 0$).

Components of the reciprocal function

In the function $f(x) = \frac{k}{x}$:

• $k$ is the constant that scales the hyperbola. This value must not equal zero ($k \neq 0$).
• $x$ is the independent variable. The function is undefined when $x = 0$.

Understanding asymptotes

An asymptote is a straight line (or curve) that a graph approaches as $x$ tends towards positive infinity ($+\infty$), negative infinity ($-\infty$), or some particular value.

For example:

• The curve $f(x) = 2^x$ has an asymptote at $f(x) = 0$ as $x$ tends to $-\infty$
• The curve $f(x) = \frac{1}{x}$ has asymptotes at $x = 0$ and $f(x) = 0$

Key features of reciprocal function graphs

Vertical asymptote

The function $f(x) = \frac{k}{x}$ has important characteristics based on how fractions work:

The denominator cannot be zero because division by zero is undefined. In this case, the denominator is $x$, so the function is undefined at $x = 0$.

This creates a vertical asymptote at $x = 0$. As $x$ approaches $0$ from the positive side ($x \to 0^+$):

• If $k > 0$, then $f(x) \to \infty$
• If $k < 0$, then $f(x) \to -\infty$

As $x$ approaches $0$ from the negative side ($x \to 0^-$):

• If $k > 0$, then $f(x) \to -\infty$
• If $k < 0$, then $f(x) \to \infty$

Horizontal asymptote

A fraction equals zero only if the numerator is zero. Here, the numerator is $k$, and we know that $k \neq 0$. Therefore, $f(x)$ can never equal zero.

This creates a horizontal asymptote at $f(x) = 0$.

Rectangular shape

The asymptotes intersect at right angles (perpendicular to each other), which gives the graph its "rectangular" hyperbola name.

The graph above illustrates $f(x) = \frac{k}{x}$ for $k = 1$, $k = 3$, and $k = -2$, showing the hyperbolic shape and asymptotes.

Effect of the sign of k

The sign of $k$ determines which quadrants contain the curves:

• If $k > 0$: curves lie in quadrants 1 and 3
• If $k < 0$: curves lie in quadrants 2 and 4

Effect of the magnitude of k

The magnitude (absolute value) of $k$ scales the hyperbola. Larger $|k|$ values stretch the curves away from the origin, while smaller $|k|$ values bring the curves closer to the origin.

In other words:

• A larger $|k|$ makes the curves "wider" and further from the axes
• A smaller $|k|$ makes the curves "tighter" and closer to the axes

End behaviour

The behaviour of $f(x) = \frac{k}{x}$ as $x$ approaches infinity or negative infinity describes how $f(x)$ approaches the horizontal asymptote.

As x approaches positive infinity

As $x \to \infty$, the denominator becomes very large, making $\frac{k}{x}$ approach $0$:

• If $k > 0$, then $f(x) \to 0$ from the positive side
• If $k < 0$, then $f(x) \to 0$ from the negative side

As x approaches negative infinity

As $x \to -\infty$, the denominator becomes a large negative number:

• If $k > 0$, then $f(x) \to 0$ from the negative side
• If $k < 0$, then $f(x) \to 0$ from the positive side

Worked example 1: Creating a table and plotting the graph

Worked example 2: Describing end behaviour

Remember!