Definition: An exponential function is a function in the form:
where x is the independent variable, b > 0, b ≠ 1, and a ≠ 0.
Unlike linear functions, the rate of change is not constant for exponential functions.
Exponential functions can be used to represent growth and decay, where:
a = initial amount/y-intercept (0, a)
b = growth/decay factor,
Exponential growth: b > 1
Exponential decay: 0 < b < 1
x = time
Example: Tell whether the function represents growth or decay, then graph:
Step 1: Use the value of the base, b, to decide if the function shows growth or decay. Since b = 1.5 is greater than 1, we know this is an exponential growth function.
Step 2: Graph the function by using a table of values.
Choosing several values for x (here we’ll use -2, 1, 2, 4), plug the values in to the function to complete the table:
For example, when x = 2:
f(2) = 1.5² = 2.25
Step 3: Using each row of the table as an ordered pair, plot the points to graph. For example, the first row gives us (-2, 0.4). Connect to points in a curve to from the graph of the function:
Notice that as the x-values decrease, the function gets closer to the x-axis. Because the value of f(x) cannot equal 0, the x-axis is an asymptote, or a line that the function approaches but never reaches.
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