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Lesson 3.1, page 376 Exponential Functions
Objective: To graph exponentials equations and functions, and solve
applied problems involving exponential functions and their
graphs.
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.
Exponential Function
The function f(x) = bx, where x is a real number, b > 0 and b 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.)
See Example 1, page 377. Check Point 1: Use the function
f(x) = 13.49 (0.967) x – 1to 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. Compute function values and list the results in a table.
2. Plot the points and connect them with a smooth curve. Be sure to plot enough points to determine how steeply the curve rises.
Check Point 2 -- Graph the exponential function y = f(x) = 3x.
(3,1/27)1/273
(2, 1/9)1/92
(1, 1/3)1/31
(3, 27)273
9
3
1
y = f(x) = 3x
(2, 9)2
(1, 3)1
(0, 1)0
(x, y)x
Check Point 3: Graph the exponential function
(3,1/27)1/273
(2, 1/9)1/92
(1, 1/3)1/31
(3, 27)273
9
3
1
(2, 9)2
(1, 3)1
(0, 1)0
(x, y)x 1( )
3
x
y f x
1( )
3
x
y f x
Characteristics of Exponential Functions, f(x) = bx, 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<b<1, then graph goes down to the right
and is decreasing. Graph is one-to-one and has an inverse. Graph approaches but does not touch x-axis.
Example -- Graph y = 3x +
2.
The graph is that of y = 3x shifted left 2 units.
2433
812
271
90
3
1
1/3
y= 3 x+2
1
2
3
x
Example: Graph y = 4 3x
3.963
3.882
3.671
30
1
5
23
y
1
2
3
x
The graph is a reflection of the graph of y = 3x across the y-axis, followed by a reflection across the x-axis and then a shift up of 4 units.
The number e (page 381) The number e is an irrational
number. Value of e 2.71828 Note: Base e exponential functions
are useful for graphing continuous growth or decay.
Graphing calculator has a key for ex.
Practice with the Number e
Find each value of ex, to four decimal places, using the ex key on a calculator.a) e4 b) e0.25
c) e2 d) e1
Answers:a) 54.5982 b) 0.7788c) 7.3891 d) 0.3679
Compound Interest Formula
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
1 nt
rA P
n
Compound Interest Formula for Continuous Compounding
A = amount in account after t years
P = principal amount of money invested
R = interest rate (decimal form) T = time in years
rtA Pe