– Evaluate exponential functions.
– Graph exponential functions.
– Evaluate functions with base e.
– Use compound interest formulas.
• The exponential function is very important in math because it is used to model many real life situations.
• For example: population growth and decay, compound interest, economics, and much more.
• How is this different from functions that we worked with previously?
• Some DID have exponents, but NOW, the variable is found in the exponent.
– (Example: is NOT an exponential function)
• To graph an exponential function, follow the steps listed:
1. Compute some 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.
Graph of an Exponential Function
• Graph
1

• Subtract 3 from x-values
(move 3 units left)
• Subtract 4 from y-values
(move 4 units down)
Note: Point (0,1) has now been moved to (-3,-3)
1
2
3
−
1
−
2
−
3
Example
Graph the exponential function y = f(x) = 3 x .
x y = f ( x ) = 3 x ( x, y )
0 1 (0, 1)
3
9
27
1/3
1/9
1/27
(1, 3)
(2, 9)
(3, 27)
(
−
1,
1/3)
(
−
2,
1/9)
(
−
3,
1/27)
Graph y = 3 x + 2
1
2
3 x
−
3
−
2
−
1
0 y
1/3
1
3
9
27
81
243
.
Example
= 3 xxxx
Example
Graph the exponential function x
0
−
1
−
2
−
3
1
2
3
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)
Characteristics of _________
• The domain consists of all real numbers: (∞, ∞)
• The range consists of all positive real numbers: (0, ∞)
• The y-intercept is the point (0,1) (a non-zero base raised to a zero exponent = 1).
• If the base b is greater than 1, the graph extends up as you go right of zero, and gets very close to zero as you go left. (Is an increasing function)
• If the base b lies between 0 & 1, the graph extends up as you go left of zero, and gets very close to zero as you go right. (Is a decreasing function)
2

Other Characteristics of _________
• This function is one-to-one and has an inverse that is a function.
• The graph approaches, but does not touch, the xaxis. The x-axis (y=0) is a horizontal asymptote.
• Transformations of the exponential function are treated as transformation of polynomials. (Follow order of operations, x’s do the opposite of what you think.)
• e is known as the natural base
(Most important base for exponential functions.)
• e is an irrational number
(can’t write its exact value)
• We approximate e
Remember
• e is a number
• e lies between 2 and 3
Graphs of Exponential Functions, Base e
Graph f(x) = e x x f ( x )
0.135
−
2
−
1
0
0.368
1
1
2
2.718
7.389
Graph f(x) = e x + 2 .
x f ( x )
−
4
−
3
−
2
−
1
0
1
0.135
0.368
1
2.718
7.389
20.086
Example
Graph f (x) = 2
− e
−
Example
3x .
1
2 x
−
2
−
1
0 f ( x )
−
401.43
−
18.09
1
1.95
1.99
3

• Exponential growth (compound interest!)
• Exponential decay (decomposition of radioactive substances)
• A father sets up a savings account for his daughter.
He puts $1000 in an account that is compounded quarterly at an annual interest rate of 8%.
How much money will be in the account at the end of 10 years? (Assume no other deposits were made after the original one.)
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