Transcript Ch. 9 Polynomials 9-1 Multiplying Monomials
Lesson 3.1, page 376 Exponential Functions
Objective: To graph exponentials equations and functions, and solve applied problems involving exponential functions and their graphs.
Look at the following…
4
x
2 3
x
1 Polynomial 4
x
3 Exponential
Real World Connection
Exponential functions are used to model numerous real-world applications such as population growth and decay, compound interest, economics (exponential growth and decay) and more.
REVIEW
Remember: x 0 = 1 Translation – slides a figure without changing size or shape
Exponential Function The function
f
(
x
) =
b x
, where
x
real number,
b
> 0 and
b
is a 1, is called the exponential function , base
b
.
(The base needs to be positive in order to avoid the complex numbers that would occur by taking even roots of negative numbers.)
Examples of Exponential Functions, pg. 376
x
3
x
(4.23)
x
See Example 1, page 377.
Check Point 1: Use the function f(x) = 13.49 (0.967) x – 1 to find the number of О-rings expected to fail at a temperature of 60° F. Round to the nearest whole number.
Graphing Exponential Functions 1.
2.
Compute function values and list the results in a table.
Plot the points and connect them with a smooth curve. Be sure to plot enough points to determine how steeply the curve rises.
2 3 1 2 3
x
0 1 Check Point 2 -- Graph the exponential function
y
=
f
(
x
) = 3
x .
y
=
f
(
x
) = 3
x
1 3 9 27 1/3 1/9 1/27 (
x, y
) (0, 1) (1, 3) (2, 9) (3, 27) ( 1, 1/3) ( 2, 1/9) ( 3,1/27)
x 0 1 2 3 1 2 3 Check Point 3: Graph the exponential function
y
y
1 3 9 27 1/3 1/9 1/27
x
( x, y ) (0, 1) ( 1, 3) ( 2, 9) ( 3, 27) (1, 1/3) (2, 1/9) (3,1/27)
x
Characteristics of Exponential Functions,
f(x) = b
x , pg. 379 Domain = (-∞,∞) Range = (0, ∞) Passes through the point (0,1) If b>1 , then graph goes up to the right and is increasing.
If 0 Graph is one-to-one and has an inverse. Graph approaches but does not touch x-axis. Example -- x + 2 The graph is that of y = 3 x shifted left 2 units . 1 2 3 x 3 2 1 0 y= 3 x+2 1/3 1 3 9 27 81 243 Example: x x 3 2 1 0 1 2 3 The graph is a reflection of the graph of y = 3 x across the y-axis, followed by a reflection across the x -axis and then a shift up of 4 units. y 23 5 1 3 3.67 3.88 3.96 The number e is an irrational number. Value of e 2.71828 Note: Base or decay. e exponential functions are useful for graphing continuous growth Graphing calculator has a key for e x . Practice with the Number e Find each value of e x , to four decimal places, using the e x key on a calculator. a) e 4 b) e 0.25 d) e 1 c) e 2 Answers: a) 54.5982 c) 7.3891 b) 0.7788 d) 0.3679 Natural Exponential Function x Remember e is a number e lies between 2 and 3 Compound Interest Formula A P 1 r n nt A = amount in account after t years P = principal amount of money invested R = interest rate (decimal form) N = number of times per year interest is compounded T = time in years Compound Interest Formula for Continuous Compounding A Pe rt A = amount in account after t years P = principal amount of money invested R = interest rate (decimal form) T = time in years See Example 7, page 384. Compound Interest Example Check Point 7: A sum of $10,000 is invested at an annual rate of 8%. Find the balance in that account after 5 years subject to a) quarterly compounding and b) continuous compounding.Observing Relationships
Connecting the Concepts
Graph y = 3
.
Graph y = 4
3
The number
e (page 381)