Exponential Functions Revision Notes for Grade 10 NSC Matric Mathematics

Exponential Functions

What are exponential functions?

Exponential functions are mathematical functions where the variable appears as an exponent. These functions model rapid growth or decay in real-world situations like population growth, radioactive decay, and compound interest.

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Two Main Forms of Exponential Functions

The exponential function can be expressed in two primary forms:

Basic form: $y = b^x$ where $b > 0$ and $b \neq 1$
General form: $y = ab^x + q$ where $a$ , $b$ , and $q$ are constants

Basic exponential functions ( $y = b^x$ )

The basic exponential function has the form $y = b^x$ , where $b$ is called the base. The behaviour of the function depends on the value of the base:

When $b > 1$ : The function shows exponential growth
When $0 < b < 1$ : The function shows exponential decay

The graph above shows three exponential growth functions. Notice that all functions pass through the point $(0, 1)$ because any number raised to the power of 0 equals 1.

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Key Properties of Basic Exponential Functions

Domain: $\{x : x \in \mathbb{R}\}$ (all real numbers)
Range: $\{y : y > 0\}$ (all positive real numbers)
Y-intercept: $(0, 1)$
X-intercept: None (the graph never crosses the x-axis)
Horizontal asymptote: $y = 0$ (the x-axis)

Exponential decay functions

When the base is between 0 and 1, we get exponential decay. These functions can be written as $y = \left(\frac{1}{k}\right)^x$ where $k > 1$ .

Notice how these decay functions are reflections of the growth functions about the y-axis.

General form: $y = ab^x + q$

The general form introduces three parameters that transform the basic exponential function:

$a$ : Affects the steepness and orientation of the curve
$b$ : Determines whether the function grows ( $b > 1$ ) or decays ( $0 < b < 1$ )
$q$ : Shifts the graph vertically

Effect of parameter $q$ (vertical shift)

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

When $q > 0$ : The graph shifts upwards by $q$ units
When $q < 0$ : The graph shifts downwards by $q$ units
The horizontal asymptote becomes $y = q$

Effect of parameter $a$ (orientation and steepness)

The parameter $a$ affects both the orientation and steepness of the curve:

For $b > 1$ :

When $a > 0$ : The graph curves upwards (normal exponential growth)
When $a < 0$ : The graph curves downwards (reflected about the horizontal asymptote)

For $0 < b < 1$ :

When $a > 0$ : The graph curves downwards (normal exponential decay)
When $a < 0$ : The graph curves upwards (reflected about the horizontal asymptote)

Domain and range

Domain

For all exponential functions $y = ab^x + q$ , the domain is $\{x : x \in \mathbb{R}\}$ because you can substitute any real number for $x$ .

Range

The range depends on the sign of parameter $a$ :

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Range Rules Based on Parameter $a$

When $a > 0$ :

Since $b^x > 0$ for all real $x$ , we have $ab^x > 0$
Therefore: $ab^x + q > q$
Range: $\{f(x) : f(x) > q\}$

When $a < 0$ :

Since $b^x > 0$ for all real $x$ , we have $ab^x < 0$
Therefore: $ab^x + q < q$
Range: $\{f(x) : f(x) < q\}$

Intercepts

Y-intercept

To find the y-intercept, substitute $x = 0$ : $y = ab^0 + q = a(1) + q = a + q$

The y-intercept is the point $(0, a + q)$ .

X-intercept

To find the x-intercept, set $y = 0$ : $0 = ab^x + q$ $ab^x = -q$ $b^x = -\frac{q}{a}$

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X-intercept Existence

An x-intercept exists only when $-\frac{q}{a} > 0$ (since $b^x$ is always positive).

Asymptotes

All exponential functions of the form $y = ab^x + q$ have a single horizontal asymptote at $y = q$ .

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Why Horizontal Asymptotes Occur

This occurs because:

As $x \to \infty$ or $x \to -\infty$ , the term $ab^x$ either approaches 0 or becomes very large
The function approaches the horizontal line $y = q$

Sketching exponential functions

To sketch an exponential function $y = ab^x + q$ , determine these four characteristics:

Sign of $a$ (determines curve orientation)
Y-intercept at $(0, a + q)$
X-intercept (if it exists)
Horizontal asymptote at $y = q$

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Worked Example 1: Sketching $y = 3 \times 2^x + 2$

Step 1: Examine the equation

$a = 3 > 0$ : Graph curves upwards
$q = 2 > 0$ : Graph is shifted up by 2 units
$b = 2 > 1$ : Exponential growth

Step 2: Find the y-intercept

Let $x = 0$ : $y = 3 \times 2^0 + 2 = 3 \times 1 + 2 = 5$
Y-intercept: $(0, 5)$

Step 3: Find the x-intercept

Set $y = 0$ : $0 = 3 \times 2^x + 2$
$3 \times 2^x = -2$
$2^x = -\frac{2}{3}$
Since $2^x > 0$ always, there is no real solution, so no x-intercept exists.

Step 4: Identify the asymptote

Horizontal asymptote: $y = 2$

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Worked Example 2: Sketching $y = -2 \times 3^x + 6$

Step 1: Examine the equation

$a = -2 < 0$ : Graph curves downwards
$q = 6 > 0$ : Graph is shifted up by 6 units
$b = 3 > 1$ : Exponential function with growth base, but reflected

Step 2: Find the y-intercept

Let $x = 0$ : $y = -2 \times 3^0 + 6 = -2 \times 1 + 6 = 4$
Y-intercept: $(0, 4)$

Step 3: Find the x-intercept

Set $y = 0$ : $0 = -2 \times 3^x + 6$
$2 \times 3^x = 6$
$3^x = 3$
$x = 1$
X-intercept: $(1, 0)$

Step 4: Identify the asymptote

Horizontal asymptote: $y = 6$

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Worked Example 3: Domain and Range of $g(x) = 5 \times 2^x + 1$

Step 1: Find the domain The domain of any exponential function is all real numbers. Domain: $\{x : x \in \mathbb{R}\}$

Step 2: Find the range Since $2^x > 0$ for all real $x$ :

$5 \times 2^x > 0$
$5 \times 2^x + 1 > 1$

Range: $\{g(x) : g(x) > 1\}$

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Exam Tips

Always check the sign of parameter $a$ to determine curve orientation
Remember that exponential functions never touch the x-axis (horizontal asymptote)
When $b > 1$ , the function grows; when $0 < b < 1$ , the function decays
The y-intercept is always $(0, a + q)$
Use key points and asymptotes to sketch accurate graphs

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Key Points to Remember

Exponential functions have the form $y = ab^x + q$ where the variable $x$ appears in the exponent
The domain is always all real numbers, but the range depends on the sign of $a$
The horizontal asymptote is the line $y = q$
Parameter $a$ affects orientation, $b$ determines growth/decay, and $q$ causes vertical shifts
These functions model real-world phenomena like population growth, radioactive decay, and compound interest

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