Exponential FunctionsAlgebra III, Sec. 3.1

Objective

Recognize, evaluate, and graph exponential functions.

Important Vocabulary

Algebraic functions – linear, polynomial, and rational functions

Transcendental functions – nonalgebraic functions (exponential & logarithmic functions)

Natural base e – 2.718281828…

Continuous compounding – the number of compoundings increase without bound

Exponential Functions

The exponential function f with base a is denoted by ______________ where a > 0, a ≠ 1, and x is any real number.

Example (on your handout)

Use a calculator to evaluate the expression

2.6265

Example 1

Use a calculator to evaluate each function at the indicated value of x.

a.) at x = π

687.2913

Example 1

Use a calculator to evaluate each function at the indicated value of x.

b.) at x = ½

0.3536

Example 1

Use a calculator to evaluate each function at the indicated value of x.

c.) at x = -2.5

1.7469

Graphs of Exponential Fns

For a > 1, the graph of is ___________________ over its domain.

For a > 1, the graph of is ___________________ over its domain.

increasing

decreasing

Graphs of Exponential Fns (cont.)

For the graph of or , a > 1,

the domain is ___________,

the range is __________,

and the y-intercept is __________.

Also both graphs have _________ as a horizontal asymptote.

(-∞, ∞)

(0, ∞)

(0, 1)

y = 0

Example 2In the same coordinate plane, sketch the graph of each function.

a.)

b.) Create a table.

x f(x) g(x)

-2

-1

0

1

2

Plot the points. Sketch the curve.

Example 3In the same coordinate plane, sketch the graph of each function.

a.)

b.) Create a table.

x f(x) g(x)

-2

-1

0

1

2

Plot the points. Sketch the curve.

Example (on your handout)

Sketch the graph of the function.

Create a table. x f(x)

-2

-1

0

1

2

Plot the points. Sketch the curve.

Exponents

x x2 x3 x4 x5

1

2

3

4

5

6

Example 4

Solve using the one-to-one property.

a.) Write as common bases.

Exponents must equal.

Solve for x.

Example 4

Solve using the one-to-one property.

b.)Write as common bases.

Exponents must equal.

Solve for x.

Example 5

Describe the graph as a transformation of the graph of

a.)

b.)

c.)

Horizontal shift: 3 units right

Reflection over x-axis

Reflection over y-axis, Vertical shift: 3 units up

The Natural Base e

The natural exponential function is given by the function _______________.

e = 2.718281828… is the constant x is the variable

Example 6Use a calculator to evaluate the function given by

at each individual value of x to four decimal places.

a.) x = -0.4

b.) x = -7.1

c.) x = 0.72

0.6703

0.0008

2.0544

Example (on your handout)

Use a calculator to evaluate the expression

1.8221

Example 7Sketch the graph of the natural exponential function.

Create a table. x f(x)

-2

-1

0

1

2

Plot the points. Sketch the curve.

ApplicationsAfter t years, the balance A in an account with principle P and annual interest rate r (in decimal form) is given by the formulas:

For n compoundings per year:

For continuous compounding:

Example (on your handout)

Find the amount in an account after 10 years if $6000 is invested at an interest rate of 7%.

a.) compounded monthly

b.) compounded continuously

Example 8On the day of a child’s birth, a deposit of $25,000 is made in a trust fund that pays 8.25% interest. Determine the balance in this account on the child’s 26th birthday if the interest is compounded…

a.) quarterly.

b.) monthly.

c.) continuously.