Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

Exponential Functions and Their Graphs

The exponential function f with base is defined by f(x) = ^x where > 0, ≠ 1, and x is any real number.

For instance, f(x) = 4^x and g(x) = 0.5^x are exponential functions.

Rules for Exponential Functions:

1. Product Rule

2. Quotient Rule

3. Power Rule

4. Negative Exponent Rule

The value of f(x) = 4^x when x = 2 is f(2) = 4² = 16

The value of f(x) = 4^x when x = -2 is f(-2) = 4^-2 = 1⁄16

The value of g(x) = 1.2^x when x = 2 is g(2) = 1.2² = 1.44

Example #1: Sketch the graph of f(x) = 4^x

x	f(x)	(x, f(x))
-2		(-2, 1⁄16)
-1		(-1, 1⁄4)
0	4⁰ = 1	(0, 1)
1	4¹ = 4	(1, 4)
2	4² = 16	(2, 16)

Example #2: Sketch the graph of g(x) = 4^x - 1. State teh domain and range.

x	g(x)	(x, g(x)-1)
-2		(-2, -15⁄16)
-1		(-1, -3⁄4)
0	4⁰ = 1	(0, 0)
1	4¹ = 4	(1, 3)
2	4² = 16	(2, 15)
3	4³ = 64	(3, 64)

The graph of this function is a vertical translation of the graph of f(x) = 4^x down one unit.

Domain: (-∞, ∞)

Range: (-1, ∞)

Example #3: Sketch the graph of g(x) = 4^-x. State teh domain and range.

x	g(x)	(x, g(x))
-2	4^-(-2) = 4² = 16	(-2, 16)
-1	4^-(-1) = 4¹ = 4	(-1, 4)
0	4⁰ = 1	(0, 1)
1	4^-1 = 1⁄4	(1, 1⁄4)
2	4^-2 = 1⁄16	(2, 1⁄16)
3	4^-3 = 1⁄64	(3, 1⁄64)

The graph of this function is a vertical translation of the graph of f(x) = 4^x in the y-axis.

Domain: (-∞, ∞)

Range: (0, ∞)

In Examples 2 and 3 above, the graphs have a horizontal asymptote of y = -1 and y = 0 and are transformed with respect to the original function f(x) = 4^x.

Rules for Transformations of Graphs of Exponential Functions:

f(x) + a is f(x) shifted upward "a" units
f(x) - a is f(x) shifted downward "a" units
f(x + a) is f(x) shifted left "a" units
f(x - a) is f(x) shifted right "a" units
-f(x) is f(x) reflected across the x-axis
f(-x) is f(x) reflected across the y-axis

The Natural Exponential Function

The graph of f(x) = e^x

x	f(x)
-2	0.14
-1	0.38
0	1
1	2.72
2	7.39

The irrational number e, where e ≈ 2.718281828... is used in applications involving growth and decay. This number is called the natural base.

The function f(x) = e^x is called the natural exponential function

A common scientific application of exponential functions is radioactive decay.

Example #4: After t years, teh remaining mass y (in grams) of 23 grams of a radioactive element whose half-life is 45 years is given by:

How much of the initial mass remains after 150 years?

Solution:

Formulas for Compound Interest: After t years, the balance A in an account with principal P and annual interest r (in decimal form) is given by the following formulas.

Example #5: Determine the balance A for P dollars invested at rate r for t years, compounded n times per year. Principal $400, rate (r = 7%) and time (t = 20 years).