3.13.1

Algebraic Functions =

polynomial and rational functions.

Transcendental Functions = exponential and logarithmic functions.

Algebraic vs. Transcendental

The exponential function f with

base a is denoted by f(x) = ax where a > 0, a ≠ 1, and where x is any real number. Sometimes you will have irrational exponents.

Definition of Exponential Function

Use a calculator to evaluate each function

at the indicated value of x.A)f(x) = 2x

B)f(x) = 2-x

C)f(x) = .6x

Example 1: Evaluating Exponential Functions

In the same coordinate plane, sketch the

graph of each function by hand.A)f(x) = 2x

B)g(x) = 4x

Example 2: Graphs of y = ax

In the same coordinate plane, sketch the

graph of each function by hand.

A)f(x) = 2-x

B)g(x) = 4-x

Example 3: Graphs of y = a-x

1. ax∙ay = ax+y

2. ax / ay = ax-y

3. a-x = 1 / ax

4. a0 = 15. (ab)x = ax ∙bx

6. (ax)y = axy

7. (a / b)x = ax / bx

8. |a2| = |a|2 = a2

Properties of Exponents

Each of the following graphs is a

transformation of the graph of f(x) = 3x.

f(x) = 3x+1 one unit to the leftf(x) = 3x-1 one unit to the rightf(x) = 3x + 1 one unit upf(x) = 3x -1 one unit downf(x) = -3x reflect about x-axisf(x) = 3-x reflect about y-axis

Transformations of Graphs of Exponential Functions

e ≈ 2.718281828 ← natural base

The function f(x) = ex is called the natural exponential function and the graph is similar to that of f(x) = ax. The base e is your constant and x is the variable. The number e can be approximated by the expression [1 + 1 / x] x.

The Natural Base e

Use a calculator to evaluate the function

f(x) = ex at each indicated value of x.

A)x = -2

B)x = .25

C)x = -.4

Example 4: Evaluating the Natural Exponential

Function

Sketch the graph of each natural

exponential function.

A)f(x) = 2e.24x

B)g(x) = 1 / 2e-.58x

Example 5: Graphing Natural Exponential

Functions

After t years, the balance A in an account

with principal P and annual interest rate r (in decimal form) is given by the following formulas:

1.For n compoundings per year: A = P (1 + r / n)nt

2.For continuous compoundings: a = Pert.

Formulas for Compound Interest

A) A total of $12,000 is invested at an annual

interest rate of 4% compounded annually. Find the balance in the account after 1 year.

B) A total of $12,000 is invested at an annual interest rate of 3%. Find the balance after 4 years if the interest is compounded quarterly.

Example 6: Finding the Balance for Compound

Interest