Lesson 108: Identifying and Graphing Exponential Functions

New Concept

An exponential function is a function of the form $f(x) = ab^x$, where $a$ and $b$ are nonzero constants and $b$ is a positive number not equal to 1.

What’s next

Next, you'll practice evaluating, identifying, and graphing these powerful functions to see how they behave in different situations.

Property

An exponential function is a function of the form $f(x) = ab^x$, where $a$ and $b$ are nonzero constants and $b$ is a positive number not equal to 1.

Explanation

Think of an exponential function as a 'super-multiplier' machine! The variable $x$ is in the exponent, which means as $x$ changes, the $y$ values are repeatedly multiplied by the base $b$. This creates either rapid growth or decay, making the function shoot up or dive down incredibly fast.

Examples

• Evaluate $f(x) = 3^x$ for $x = -2$. Solution: $f(-2) = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.
• Evaluate $f(x) = 4(2)^x$ for $x = 3$. Solution: $f(3) = 4(2)^3 = 4(8) = 32$.
• The population of a town is modeled by $P(t) = 5000(1.03)^t$, where $t$ is the number of years.

Property

For a set of ordered pairs, if the x-values change by a constant amount, an exponential function exists if the y-values change by a constant factor (a common ratio).

Explanation

To check if a set of points is exponential, look for a consistent multiplier. If each y-value is the result of multiplying the previous one by the same number, you've found an exponential pattern. This constant factor is the base of the function. It’s all about multiplication, not addition!

Examples

• The set \{(1, 4), (2, 12), (3, 36)\} is exponential because the y-values multiply by 3 for each +1 change in x.
• The set \{(1, 5), (2, 10), (3, 15)\} is not exponential because it increases by adding 5, which is a linear pattern.
• For \{(0, 2), (1, 8), (2, 32)\}, the ratios are $\frac{8}{2}=4$ and $\frac{32}{8}=4$. This is exponential with a base of 4.

Let's see if we can find a consistent growth pattern hiding in this set of points. This example shows how to use the core definition of an exponential function to test a set of points.

Example Problem

Determine if the set of ordered pairs satisfies an exponential function:
$\{ (0, 5), (-2, \frac{5}{4}), (1, 10), (-1, \frac{5}{2}) \}$

Step-by-Step

1. First, we arrange the ordered pairs so that the x-values are in increasing order.

2. We can see that the x-values increase by a constant amount of 1.
3. Now, we check if there is a common ratio between the y-values. We do this by dividing each y-value by the one before it.

4. Because the ratio between consecutive y-values is constant, the set of ordered pairs does satisfy an exponential function. The common ratio, or base, is $b=2$.

Property

To graph an exponential function, make a table of ordered pairs for different x-values and plot the points. The graph will always be a smooth curve that approaches, but never touches, the x-axis.

Explanation

Graphing reveals the function's signature 'swoop' curve. Pick a few x-values (include negatives and zero!), calculate the y-values, and connect the dots. The graph will either shoot up for growth ($b>1$) or dive down toward zero for decay ($0<b<1$), always staying on one side of the x-axis.

Examples

• For $y = 2(3)^x$, points like $(-1, \frac{2}{3})$, $(0, 2)$, and $(1, 6)$ form a growth curve rising to the right.
• For $y = 10(\frac{1}{2})^x$, points like $(-1, 20)$, $(0, 10)$, and $(1, 5)$ form a decay curve falling to the right.
• The graph of $y = -(4)^x$ is a reflection of $y=4^x$ across the x-axis, showing a downward-facing curve.