6.3: Exponential Functions

Exploring Exponential Functions: Contrasting with Linear Functions

For those familiar with linear functions, delving into exponential functions introduces a new realm of mathematical relationships. While linear functions grow steadily by a constant rate, exponential functions exhibit rapid growth or decay, governed by a constant factor. Let's explore the key differences between exponential and linear functions:

Exponential Functions

Exponential functions are characterized by a base raised to a variable exponent. They follow the form: $f(x) = a \times b^x$, where:

• $a$ is the initial value or y-intercept,
• $b$ is the base of the exponential function,
• $x$ is the variable or input value.

Linear Functions

Linear functions, on the other hand, grow steadily by a constant rate. They follow the form: $f(x) = mx + b$, where:

• $m$ is the slope or rate of change,
• $b$ is the y-intercept.

Common Ratios vs. Common Differences

Understanding the concept of common ratio and common difference is key to grasping the fundamental differences between exponential and linear functions.

Common Ratio in Exponential Functions

Exponential functions exhibit a common ratio, denoted by the base of the exponent. This common ratio represents the factor by which the function grows or decays with each successive input. Let's consider an example exponential function:

$f(x) = 2^x$

For this function, the base $2$ serves as the common ratio. As $x$ increases by $1$, the function's value doubles, demonstrating exponential growth. Here's a table illustrating the function's values:

$x$	$f(x)$
0	1
1	2
2	4
3	8

In this example, the common ratio $2$ indicates that each successive value is twice the previous one.

Common Difference in Linear Functions

Linear functions, on the other hand, exhibit a common difference, which represents the rate of change or slope of the function. This common difference dictates the consistent increment or decrement between consecutive outputs as the input variable increases by a fixed amount. Let's consider an example linear function:

$g(x) = 3x + 2$

In this function, the coefficient of $x$, which is $3$, serves as the common difference. As $x$ increases by $1$, the function's value increases by $3$. Here's a table illustrating the function's values:

$x$	$g(x)$
0	2
1	5
2	8
3	11

In this example, the common difference $3$ indicates that each successive value is $3$ more than the previous one.

Contrast and Comparison

• Exponential Functions: Have a common ratio, indicating rapid growth or decay with each successive input.
• Linear Functions: Have a common difference, indicating steady incremental change with each successive input.

By understanding these concepts, you'll gain insight into how exponential and linear functions behave differently and how their patterns can be represented mathematically.

Graphing Exponential Functions

Graphing exponential functions allows us to visualize their behavior and understand how they grow or decay. Here are the steps to graph an exponential function:

1. Make a Table of Values

The first step is to create a table of values by choosing a range of input values for $x$ and calculating the corresponding output values for $f(x)$. Choose a range of values that will give you a clear understanding of the function's behavior. For example, you can select both positive and negative values of $x$ to see how the function behaves in different regions.

2. Plot the Ordered Pairs

Once you have calculated the output values for each input value, plot the ordered pairs $(x, f(x))$ on the Cartesian plane. Use a graphing calculator or software if needed, or simply plot the points manually.

3. Draw a Smooth Curve Through the Points

After plotting the ordered pairs, connect them with a smooth curve that represents the overall behavior of the function. Since exponential functions exhibit rapid growth or decay, the curve will typically be steep. Ensure that the curve passes through or near all the plotted points to accurately represent the function.

Example

Let's graph the exponential function $f(x) = 2^x$ over the interval $-2 \leq x \leq 2$.

Table of Values

$x$	$f(x) = 2^x$
-2	$\frac{1}{4}$
-1	$\frac{1}{2}$
0	1
1	2
2	4

Plotting the Ordered Pairs

Now, plot the ordered pairs on the grid: $(-2, \frac{1}{4}), (-1, \frac{1}{2}), (0, 1), (1, 2), (2, 4)$.

Drawing the Smooth Curve

Finally, connect the points with a smooth curve that passes through or near all the plotted points. Since the function is $f(x) = 2^x$, the curve will exhibit exponential growth, rising steeply as $x$ increases.

By following these steps, you can accurately graph exponential functions and gain insight into their behavior and characteristics.