More Basic Functions and Graphs: Background You’ll Need 3

Lumen Learning

More Basic Functions and Graphs: Background You’ll Need 3

Explain the difference between exponential growth and decay

Identify Exponential Growth and Decay

In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth (or decay), we may choose to model the given scenario using the following function:

$y={A}_{0}{b}^{x}$

where ${A}_{0}$ is equal to the value at $x=0$ , $b$ is the base, and $x$ is the exponent. Note that the variable is in the exponent which makes the function exponential.

exponential function

For any real number $x$ , an exponential function is a function with the form

y = A_{0} b^{x}

$y={A}_{0}{b}^{x}$

where

$a$ is a non-zero real number called the initial value and
$b$ is any positive real number such that $b≠1$ .
The domain is $\left(-\infty , \infty \right)$ , or all real numbers
The range is all positive real numbers if $a > 0$
The range is all negative real numbers if $a < 0$
The y-intercept is $\left(0,{A}_{0}\right)$ , and the horizontal asymptote is $y=0$

An exponential function models exponential growth when $b > 1$ and exponential decay when $b < 1$ .

When $b>1$ , the exponential function represents exponential growth. Common applications of exponential growth include doubling time, the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time.

When $b<1$ , the exponential function represents exponential decay. One common application of exponential decay includes calculating half-life, or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes.

Exponential growth and decay graphs have a distinctive shape, as we can see in the graphs below. It is important to remember that, although parts of each of the two graphs seem to lie on the $x$ -axis, they are really a tiny distance above the $x$ -axis.

Graph of y=2e^(3x) with the labeled points (-1/3, 2/e), (0, 2), and (1/3, 2e) and with the asymptote at y=0. — A graph showing exponential growth. The equation is $y=2{e}^{3x}$ .

Graph of y=3e^(-2x) with the labeled points (-1/2, 3e), (0, 3), and (1/2, 3/e) and with the asymptote at y=0. — A graph showing exponential decay. The equation is $y=3{e}^{-2x}$ .