Exponential functions

Exponential Functions

Exponential functions are a type of mathematical function that can be expressed in the form f(x) = a × b^x, where:

• a is a constant (the initial value or y-intercept),
• b is the base of the exponential function (a positive number, not equal to 1),
• x is the exponent (the variable).

Exponential functions are used to model many real-world situations, such as population growth, radioactive decay, and compound interest.

Characteristics of Exponential Functions

Exponential functions have several important characteristics:

• They have a constant rate of growth or decay. If b > 1, the function grows; if 0 < b < 1, the function decays.
• The graph of an exponential function is always above the x-axis, meaning it never touches or crosses the x-axis.
• As x approaches positive infinity, f(x) approaches positive infinity (the function increases without bound).
• As x approaches negative infinity, f(x) approaches 0 (the function gets closer to the x-axis but never touches it).

Graphing Exponential Functions

To graph an exponential function, you can follow these steps:

1. Identify the values of a and b.
2. Create a table of values by choosing several values for x and calculating f(x).
3. Plot the points on a Cartesian plane.
4. Draw a smooth curve through the points.

For example, consider the function f(x) = 2 × 3^x. We can create a table of values:

Now we can plot the points (-2, 0.22), (-1, 0.67), (0, 2), (1, 6), and (2, 18) on the Cartesian plane. The resulting graph will show an increasing curve starting close to the x-axis and rising steeply as x increases.

Transformations of Exponential Functions

Exponential functions can be transformed in various ways. Common transformations include:

• Vertical shifts: Adding or subtracting a constant to the function shifts the graph up or down. For example, f(x) = 2 × 3^x + 1 shifts the graph of f(x) = 2 × 3^x up by 1 unit.
• Horizontal shifts: Adding or subtracting a constant to the variable shifts the graph left or right. For example, f(x) = 2 × 3^{(x - 1)} shifts the graph right by 1 unit.
• Reflections: Multiplying the function by -1 reflects the graph over the x-axis. For example, f(x) = -2 × 3^x reflects the graph of f(x) = 2 × 3^x over the x-axis.
• Stretching and compressing: Multiplying the function by a constant greater than 1 stretches the graph vertically, while multiplying by a constant between 0 and 1 compresses it. For example, f(x) = 3 × 2^x stretches the graph of f(x) = 2^x vertically.

Exponential Growth and Decay

Exponential growth occurs when the base b is greater than 1. This can be seen in situations like population growth or investment returns. The general formula for exponential growth is:

f(t) = a × (1 + r)^t

Where:

• a is the initial amount,
• r is the growth rate (as a decimal),
• t is the time period.

Exponential decay occurs when the base b is between 0 and 1. This is often used to model radioactive decay or depreciation of assets. The general formula for exponential decay is:

f(t) = a × (1 - r)^t

Where the variables have similar meanings as in the growth formula.

Example of Exponential Growth

Suppose a population of bacteria doubles every hour. If we start with 100 bacteria, the population after t hours can be expressed as:

f(t) = 100 × 2^t

To find the population after 5 hours, substitute t with 5:

f(5) = 100 × 2^5 = 100 × 32 = 3200

Example of Exponential Decay

Suppose a radioactive substance has a half-life of 3 years. If we start with 80 grams, the amount remaining after t years can be expressed as:

f(t) = 80 × (1/2)^{t/3}

To find the amount remaining after 9 years, substitute t with 9:

f(9) = 80 × (1/2)^{9/3} = 80 × (1/2)^3 = 80 × (1/8) = 10

Check your understanding

• 1. Write the equation of an exponential function with a base of 4 and an initial value of 3.
• 2. Graph the function f(x) = 5 × 2^x and label the y-intercept.
• 3. If a population of 500 grows at a rate of 10% per year, what will the population be after 3 years?
• 4. A car depreciates in value by 15% each year. If its initial value is R200 000, what will its value be after 2 years?