AEF - Graph f(x) = b^x (Lesson)

Graph $f(x) = b^x$

Imagine that you are offered a job that pays $1 on the first day, then $2 on the second day, $4 on the third, and $8 on the fourth. You will continue to get paid in this manner for as long as you hold the job. How can you determine how much money you will have on your 32^nd day of work? Your 100^th day of work? This situation represents an exponential function, and that is what we will learn about in this module.  An exponential function is a nonlinear function in which the independent variable is the exponent.  

 

Let's try graphing one of these functions by making a table: $f\left(x\right)=\left(2^x\right)$ 

Plot these points.

x	-2	-1	0	1	2
f(x)	1/4	1/2	1	2	4

 

Connect the curve. 

 

This graph represents exponential growth because the base of the function is greater than 1.

Let's try graphing a different function by making a table: $f\left(x\right)=\left(\frac{1}{2}\right)^x$

Plot these points.

x	-2	-1	0	1	2
f(x)	4	2	1	1/2	1/4

 

Connect the curve.

 

This graph represents exponential decay because the base of the function is greater than 0 but less than 1.

Both of the functions graphed above have an asymptote. An asymptote is a line that a graph approaches more and more closely but never touches. For both of those functions, the asymptote is the line y = 0.

                     

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Graph $f(x) = b^x$ Practice

Complete the tables for each of the functions below. Given the x values of a table, what are the appropriate y values for the function?

1. f(x) = 3^x

x	-2	-1	0	1	2
y

2. $f(x)=(\frac{1}{3})^x$

x	-2	-1	0	1	2
y

Identify the following functions as exponential growth or exponential decay:

3. f(x) = (0.5)^x

4. $y=(\frac{5}{3})^x$

5. f(x) = 3^x

6. $y=(\frac{3}{4})^x$

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