Hyperbolic Functions Revision Notes for Grade 10 NSC Matric Mathematics

Hyperbolic Functions

What are hyperbolic functions?

Hyperbolic functions are mathematical functions that have the general form:

$y = \frac{a}{x} + q$

where $a$ and $q$ are constants, and $a \neq 0$ .

These functions are called hyperbolic because their graphs form curves called hyperbolas. The simplest hyperbolic function is $y = \frac{1}{x}$ .

The basic hyperbolic function y = 1/x

Let's start by examining the basic hyperbolic function $h(x) = \frac{1}{x}$ .

When we create a table of values for this function, we notice several important features:

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Key observations about $h(x) = \frac{1}{x}$ :

The function is undefined at $x = 0$ (we cannot divide by zero)
For positive x-values, we get positive y-values
For negative x-values, we get negative y-values
As x gets closer to zero, the y-values become very large (positive or negative)

The graph shows that $h(x) = \frac{1}{x}$ has two separate curves (called branches):

One branch lies in the first quadrant (where both x and y are positive)
The other branch lies in the third quadrant (where both x and y are negative)

Key characteristics of hyperbolic functions

Domain and range

For any hyperbolic function $y = \frac{a}{x} + q$ :

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Domain: $\{x : x \in \mathbb{R}, x \neq 0\}$

The function is defined for all real numbers except zero

Range: $\{y : y \in \mathbb{R}, y \neq q\}$

The function can take any real value except the value q

Asymptotes

Hyperbolic functions have two types of asymptotes (lines that the curve approaches but never touches):

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Vertical asymptote: The line $x = 0$ (the y-axis)

As x approaches zero, the function values approach positive or negative infinity

Horizontal asymptote: The line $y = q$

As x approaches positive or negative infinity, the function values approach q

Intercepts

y-intercept: To find where the graph crosses the y-axis, we set $x = 0$ : $y = \frac{a}{0} + q$ This is undefined, so hyperbolic functions have no y-intercept.

x-intercept: To find where the graph crosses the x-axis, we set $y = 0$ : $0 = \frac{a}{x} + q$ $-q = \frac{a}{x}$ $x = \frac{a}{-q} = -\frac{a}{q}$

The x-intercept exists only when $q \neq 0$ , and is located at $\left(-\frac{a}{q}, 0\right)$ .

Axes of symmetry

Hyperbolic functions have two axes of symmetry:

The line $y = x + q$
The line $y = -x + q$

Effects of parameters a and q

Understanding how the parameters $a$ and $q$ affect the shape and position of hyperbolic functions is crucial for sketching them accurately.

Effect of parameter q

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The parameter $q$ causes a vertical shift of the entire graph:

When $q > 0$ , the graph shifts upward by q units
When $q < 0$ , the graph shifts downward by |q| units
When $q = 0$ , there is no vertical shift (basic hyperbola)

The horizontal asymptote is always the line $y = q$ .

Effect of parameter a

The parameter $a$ determines the shape and position of the hyperbola:

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When $a > 0$ :

The hyperbola lies in the first and third quadrants
If $a > 1$ , the branches are further from the axes
If $0 < a < 1$ , the branches are closer to the axes

When $a < 0$ :

The hyperbola lies in the second and fourth quadrants
If $a < -1$ , the branches are further from the axes
If $-1 < a < 0$ , the branches are closer to the axes

Sketching hyperbolic functions

To sketch a hyperbolic function $y = \frac{a}{x} + q$ , follow these steps:

Step-by-step method

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Systematic approach to sketching hyperbolic functions:

Identify the parameters $a$ and $q$ from the equation
Determine the quadrants where the graph lies (based on the sign of $a$ )
Find the asymptotes:
- Vertical: $x = 0$
- Horizontal: $y = q$
Calculate the x-intercept (if it exists): $x = -\frac{a}{q}$
Plot additional points if needed for accuracy
Draw the hyperbola with two smooth curves approaching the asymptotes

Worked examples

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Worked Example 1: Sketching g(x) = 2/x + 2

Let's sketch the graph of $g(x) = \frac{2}{x} + 2$ .

Step 1: Identify parameters

$a = 2 > 0$ , so the graph lies in the first and third quadrants
$q = 2$

Step 2: Find asymptotes

Vertical asymptote: $x = 0$
Horizontal asymptote: $y = 2$

Step 3: Find x-intercept

Set $y = 0$ :
$0 = \frac{2}{x} + 2$
$-2 = \frac{2}{x}$
$x = -1$
So the x-intercept is $(-1, 0)$ .

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Worked Example 2: Sketching y = -4/x + 7

Let's sketch the graph of $y = \frac{-4}{x} + 7$ .

Step 1: Identify parameters

$a = -4 < 0$ , so the graph lies in the second and fourth quadrants
$q = 7$

Step 2: Find asymptotes

Vertical asymptote: $x = 0$
Horizontal asymptote: $y = 7$

Step 3: Find x-intercept

Set $y = 0$ :
$0 = \frac{-4}{x} + 7$
$-7 = \frac{-4}{x}$
$x = \frac{4}{7}$
So the x-intercept is $\left(\frac{4}{7}, 0\right)$ .

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Worked Example 3: Domain and range of g(x) = 2/x + 2

For the function $g(x) = \frac{2}{x} + 2$ :

Domain: The function is undefined when $x = 0$ , so the domain is $\{x : x \in \mathbb{R}, x \neq 0\}$ .

Range: The function cannot equal 2 (the horizontal asymptote), so the range is $\{g(x) : g(x) \in \mathbb{R}, g(x) \neq 2\}$ .

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Key Points to Remember:

Hyperbolic functions have the general form $y = \frac{a}{x} + q$ where $a \neq 0$
The domain is always all real numbers except zero: $\{x : x \in \mathbb{R}, x \neq 0\}$
The range excludes the horizontal asymptote value: $\{y : y \in \mathbb{R}, y \neq q\}$
Asymptotes: Vertical at $x = 0$ , horizontal at $y = q$
Parameter effects: $q$ shifts vertically, $a$ determines shape and quadrants (positive $a$ → quadrants I & III, negative $a$ → quadrants II & IV)

Hyperbolic Functions (Grade 10 NSC Matric Mathematics): Revision Notes