Rectangular Hyperbolas (VCE SSCE Mathematical Methods): Revision Notes

Rectangular Hyperbolas

Introduction to rectangular hyperbolas

A rectangular hyperbola is a special type of curve that follows the rule:

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

Table of values

To understand the shape of this curve, we can create a table showing how $y$ changes as $x$ varies:

Notice that:

• As $x$ gets larger (positively or negatively), $y$ gets smaller
• When $x$ is positive, $y$ is positive
• When $x$ is negative, $y$ is negative
• There is no value of $x$ that produces $y = 0$

Graph of the basic rectangular hyperbola

When we plot these points and connect them smoothly, we get a distinctive curve with two separate branches:

The graph has two separate parts, one in the first quadrant (top right) and one in the third quadrant (bottom left). These branches never meet and never touch the axes.

Asymptotes

Asymptotes are lines that the curve approaches but never actually touches or crosses. Every rectangular hyperbola has two asymptotes that help define its shape.

Horizontal asymptote

Looking at the graph, we can see that as $x$ gets very large (in either direction), the value of $y$ gets closer and closer to zero. We write this using special notation:

• As $x \to \infty$, $y \to 0^+$ (read as: "as $x$ approaches infinity, $y$ approaches $0$ from the positive side")
• As $x \to -\infty$, $y \to 0^-$ (read as: "as $x$ approaches negative infinity, $y$ approaches $0$ from the negative side")

The curve approaches the $x$-axis (the line $y = 0$) but never crosses it. Therefore, the line $y = 0$ is called a horizontal asymptote.

Vertical asymptote

Similarly, as $x$ approaches zero from either direction, the magnitude of $y$ becomes extremely large. We write this as:

• As $x \to 0^+$, $y \to \infty$ (read as: "as $x$ approaches zero from the positive side, $y$ approaches infinity")
• As $x \to 0^-$, $y \to -\infty$ (read as: "as $x$ approaches zero from the negative side, $y$ approaches negative infinity")

The curve approaches the $y$-axis (the line $x = 0$) but never crosses it. Therefore, the line $x = 0$ is called a vertical asymptote.

Transformations of rectangular hyperbolas

Dilations from an axis

When we multiply the basic function by a constant, we create a dilation (stretch) of the graph. The diagram below shows three related hyperbolas:

The graphs of $y = \frac{2}{x}$ and $y = \frac{1}{3x}$ have the same basic shape and the same asymptotes ($x = 0$ and $y = 0$) as $y = \frac{1}{x}$, but they are stretched vertically by different amounts.

Reflection in the x-axis

When we multiply the function by $-1$, we reflect the graph in the $x$-axis:

The graph of $y = -\frac{1}{x}$ is the reflection of $y = \frac{1}{x}$ in the $x$-axis. Notice that:

• The asymptotes remain the same: $x = 0$ and $y = 0$
• The branches now appear in the second and fourth quadrants instead of the first and third
• Similarly, $y = -\frac{2}{x}$ is the reflection of $y = \frac{2}{x}$ in the $x$-axis

Translations

We can shift (translate) the graph horizontally and vertically by adding constants. Consider the function:

$$y = \frac{1}{x - 1} + 3$$

This represents the basic graph $y = \frac{1}{x}$ translated:

• $1$ unit to the right
• $3$ units upward

Finding the new asymptotes:

When a rectangular hyperbola is translated, its asymptotes also shift:

• The vertical asymptote moves to $x = 1$
• The horizontal asymptote moves to $y = 3$

Finding the intercepts:

Unlike the basic hyperbola, translated hyperbolas often cross the axes. To find these intercepts:

For the $y$-intercept, substitute $x = 0$:

$$y = \frac{1}{0 - 1} + 3 = -1 + 3 = 2$$

So the $y$-intercept is at $(0, 2)$.

For the $x$-intercept, substitute $y = 0$:

$$0 = \frac{1}{x - 1} + 3$$ $$-3 = \frac{1}{x - 1}$$ $$-3(x - 1) = 1$$ $$-3x + 3 = 1$$ $$x = \frac{2}{3}$$

So the $x$-intercept is at $\left(\frac{2}{3}, 0\right)$.

Sketching rectangular hyperbolas

Using dilations, reflections, and translations, we can sketch any rectangular hyperbola of the form:

$$y = \frac{a}{x - h} + k$$

Where:

• $a$ determines the dilation and orientation (positive or negative)
• $h$ determines the horizontal shift
• $k$ determines the vertical shift