Exponential Growth & Decay

In Algebra 1, one of the functions studied was the exponential function. An exponential function is defined as equation image indicator $LaTeX: y=a\cdot b^{\left(x-h\right)}+k$ , where $LaTeX: y=a\cdot b^x$ is the parent function. "a" is a real number and "b" is greater than 0. "h" and "k" are real numbers.

h is the horizontal shift. Since the formula says x - h, a number subtracted from x gives you a positive (right) shift. You are subtracting a positive number from x. Likewise, x + h will give you a negative (left) shift since you are subtracting a negative h value. A simple way to remember this is that the horizontal shift will always be the opposite of what you see in the equation. x - 5 will be a shift right 5, while x + 5 will shift the graph left 5.

"b" is the base of the exponent, and if its value is greater than 1, then exponential growth is present. "b" is the base of the exponent, and if its value is between 0 and 1, then exponential decay is present.

The input (or domain) of the exponential function is the variable in the exponent, and the output (or range) is the number obtained after the computations.

The asymptote (or boundary line) has the equation of y = k.

In exponential growth functions when a is greater than 0, the ends of the graph behave like the following: as x goes to the left forever (negative infinity), the y (or f(x)) approaches the asymptote (y = k) from above ; and as x goes to the right forever (positive infinity), the y (or f(x)) goes up forever (to positive infinity). When a is less than 0, the ends of the graph behave like the following: as x goes to the left forever (negative infinity), the y (or f(x)) goes down forever (to negative infinity); and as x goes to the right forever (positive infinity), the y (or f(x)) approaches the asymptote (y = k) from below.

Note that when we are graphing an exponential function "by hand" that it is best to create a table of values, and the easiest numbers to use are (-2, -1, 0, 1, 2). Once we are more familiar with the behavior of exponential growth and decay function graphs, then using 1 and 0 should be sufficient (along with the asymptote) to help draw a "quick graph."

image of 4 graphs comparing end behaviors

In this lesson, we will explore the graphs and applications of exponential growth functions and exponential decay functions.

Let's start by watching the following video that introduces us to exponential growth functions and exponential decay functions.

Graphing Exponential Growth and Decay Functions

Now that we have a good understanding of what exponential growth functions and exponential decay functions are, let's watch some videos that explore the graphs of exponential growth functions and exponential decay functions, as well as their various characteristics.

It's now time for us to explore and practice working with the graphs of exponential growth and exponential decay functions.

Graphs of Exponential Growth & Decay Presentation and Practice

Applications of Exponential Growth and Decay Functions

We can apply what we have learned from the graphs of exponential growth and decay functions to real-world problems. To introduce us to some of these applications, watch the following videos that explore the applications of exponential growth and decay functions in real-world problems.

IMAGES CREATED BY GAVS