Outline
Exponential Functions
Natural Base e
College Algebra & Trigonometry I
4.1 - Exponential Functions
Math 1100
North Carolina Central University
Math & C.S. Department
Hicham Qasmi - hqasmi@nccu.edu
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
1
Exponential Functions
Definition
Graphing
Characteristics of f (x) = bx
Transformations
2
Natural Base e
Definition
Example
Compound Interest
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
1
Exponential Functions
Definition
Graphing
Characteristics of f (x) = bx
Transformations
2
Natural Base e
Definition
Example
Compound Interest
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Outline
1
Exponential Functions
Definition
Graphing
Characteristics of f (x) = bx
Transformations
2
Natural Base e
Definition
Example
Compound Interest
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Exponential Functions
Definition
The exponential function f with base b is defined by
f (x) = bx
where b is a positive constant other than 1 (b > 0 with b 6= 1)
and x is any real number.
Example
f (x) = 2x
Here the base b is 2
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Outline
1
Exponential Functions
Definition
Graphing
Characteristics of f (x) = bx
Transformations
2
Natural Base e
Definition
Example
Compound Interest
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Graphing
It is understood what bx means for x being a rational number,
eg: 103 = 10 × 10 × 10 = 1, 000
√
What about irrational power such as b 2 ?
√
In these cases, since 2 ≃ 1.414213 . . ., we can consider the
set of numbers
b1 , b1.4 , b1.41 , b1.414 , . . .
and find the number it approaches.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Graphing
It is understood what bx means for x being a rational number,
eg: 103 = 10 × 10 × 10 = 1, 000
√
What about irrational power such as b 2 ?
√
In these cases, since 2 ≃ 1.414213 . . ., we can consider the
set of numbers
b1 , b1.4 , b1.41 , b1.414 , . . .
and find the number it approaches.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Graphing
It is understood what bx means for x being a rational number,
eg: 103 = 10 × 10 × 10 = 1, 000
√
What about irrational power such as b 2 ?
√
In these cases, since 2 ≃ 1.414213 . . ., we can consider the
set of numbers
b1 , b1.4 , b1.41 , b1.414 , . . .
and find the number it approaches.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Graphing
It is understood what bx means for x being a rational number,
eg: 103 = 10 × 10 × 10 = 1, 000
√
What about irrational power such as b 2 ?
√
In these cases, since 2 ≃ 1.414213 . . ., we can consider the
set of numbers
b1 , b1.4 , b1.41 , b1.414 , . . .
and find the number it approaches.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Example of Exponential Function
Graph f (x) = 2x
Math 1100
College Algebra & Trigonometry I
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Outline
Exponential Functions
Natural Base e
Example of Exponential Function
Graph f (x) = 2x
5
4
3
2
1
0
−5 −4 −3 −2 −1 0
−1
1
2
3
4
5
−2
−3
−4
−5
Math 1100
College Algebra & Trigonometry I
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Outline
Exponential Functions
Natural Base e
Example of Exponential Function
Graph f (x) = 2x
5
4
•
3
2
1•
•
•
•
0
−5 −4 −3 −2 −1 0
−1
•
1
2
3
4
5
−2
−3
−4
−5
Math 1100
College Algebra & Trigonometry I
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Outline
Exponential Functions
Natural Base e
Example of Exponential Function
Graph f (x) = 2x
5
4
•
3
2
1•
•
•
•
0
−5 −4 −3 −2 −1 0
−1
•
1
2
3
4
5
−2
−3
−4
−5
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Example of Exponential Function
Graph f (x) =
x
1
2
Math 1100
College Algebra & Trigonometry I
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Outline
Exponential Functions
Natural Base e
Example of Exponential Function
Graph f (x) =
x
1
2
5
4
3
2
1
0
−5 −4 −3 −2 −1 0
−1
1
2
3
4
5
−2
−3
−4
−5
Math 1100
College Algebra & Trigonometry I
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Outline
Exponential Functions
Natural Base e
Example of Exponential Function
Graph f (x) =
x
1
2
5
4
•
3
• 2
1•
•
0
−5 −4 −3 −2 −1 0
−1
1
•
2
•
3
4
5
−2
−3
−4
−5
Math 1100
College Algebra & Trigonometry I
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Outline
Exponential Functions
Natural Base e
Example of Exponential Function
Graph f (x) =
x
1
2
5
4
•
3
• 2
1•
•
0
−5 −4 −3 −2 −1 0
−1
1
•
2
•
3
4
5
−2
−3
−4
−5
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Outline
1
Exponential Functions
Definition
Graphing
Characteristics of f (x) = bx
Transformations
2
Natural Base e
Definition
Example
Compound Interest
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Characteristics of f (x) = bx
1
The domain of f (x) = bx is (−∞, ∞).
The range is (0, ∞).
2
The graph of f (x) = bx always passes through point (0, 1).
3
If b > 1, then f (x) = bx goes up to the right and is an increasing
function.
The greater the value of b, the steeper the increase.
4
If 0 < b < 1, then f (x) = bx goes down to the right and is a
decreasing function.
The smaller the value of b, the steeper the decrease.
5
f (x) = bx is a one-to-one and has an inverse function
6
The x-axis is a horizontal asymptote. The graph approaches but does
not cross y = 0.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Characteristics of f (x) = bx
1
The domain of f (x) = bx is (−∞, ∞).
The range is (0, ∞).
2
The graph of f (x) = bx always passes through point (0, 1).
3
If b > 1, then f (x) = bx goes up to the right and is an increasing
function.
The greater the value of b, the steeper the increase.
4
If 0 < b < 1, then f (x) = bx goes down to the right and is a
decreasing function.
The smaller the value of b, the steeper the decrease.
5
f (x) = bx is a one-to-one and has an inverse function
6
The x-axis is a horizontal asymptote. The graph approaches but does
not cross y = 0.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Characteristics of f (x) = bx
1
The domain of f (x) = bx is (−∞, ∞).
The range is (0, ∞).
2
The graph of f (x) = bx always passes through point (0, 1).
3
If b > 1, then f (x) = bx goes up to the right and is an increasing
function.
The greater the value of b, the steeper the increase.
4
If 0 < b < 1, then f (x) = bx goes down to the right and is a
decreasing function.
The smaller the value of b, the steeper the decrease.
5
f (x) = bx is a one-to-one and has an inverse function
6
The x-axis is a horizontal asymptote. The graph approaches but does
not cross y = 0.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Characteristics of f (x) = bx
1
The domain of f (x) = bx is (−∞, ∞).
The range is (0, ∞).
2
The graph of f (x) = bx always passes through point (0, 1).
3
If b > 1, then f (x) = bx goes up to the right and is an increasing
function.
The greater the value of b, the steeper the increase.
4
If 0 < b < 1, then f (x) = bx goes down to the right and is a
decreasing function.
The smaller the value of b, the steeper the decrease.
5
f (x) = bx is a one-to-one and has an inverse function
6
The x-axis is a horizontal asymptote. The graph approaches but does
not cross y = 0.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Characteristics of f (x) = bx
1
The domain of f (x) = bx is (−∞, ∞).
The range is (0, ∞).
2
The graph of f (x) = bx always passes through point (0, 1).
3
If b > 1, then f (x) = bx goes up to the right and is an increasing
function.
The greater the value of b, the steeper the increase.
4
If 0 < b < 1, then f (x) = bx goes down to the right and is a
decreasing function.
The smaller the value of b, the steeper the decrease.
5
f (x) = bx is a one-to-one and has an inverse function
6
The x-axis is a horizontal asymptote. The graph approaches but does
not cross y = 0.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Characteristics of f (x) = bx
1
The domain of f (x) = bx is (−∞, ∞).
The range is (0, ∞).
2
The graph of f (x) = bx always passes through point (0, 1).
3
If b > 1, then f (x) = bx goes up to the right and is an increasing
function.
The greater the value of b, the steeper the increase.
4
If 0 < b < 1, then f (x) = bx goes down to the right and is a
decreasing function.
The smaller the value of b, the steeper the decrease.
5
f (x) = bx is a one-to-one and has an inverse function
6
The x-axis is a horizontal asymptote. The graph approaches but does
not cross y = 0.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Characteristics of f (x) = bx
1
The domain of f (x) = bx is (−∞, ∞).
The range is (0, ∞).
2
The graph of f (x) = bx always passes through point (0, 1).
3
If b > 1, then f (x) = bx goes up to the right and is an increasing
function.
The greater the value of b, the steeper the increase.
4
If 0 < b < 1, then f (x) = bx goes down to the right and is a
decreasing function.
The smaller the value of b, the steeper the decrease.
5
f (x) = bx is a one-to-one and has an inverse function
6
The x-axis is a horizontal asymptote. The graph approaches but does
not cross y = 0.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Characteristics of f (x) = bx
1
The domain of f (x) = bx is (−∞, ∞).
The range is (0, ∞).
2
The graph of f (x) = bx always passes through point (0, 1).
3
If b > 1, then f (x) = bx goes up to the right and is an increasing
function.
The greater the value of b, the steeper the increase.
4
If 0 < b < 1, then f (x) = bx goes down to the right and is a
decreasing function.
The smaller the value of b, the steeper the decrease.
5
f (x) = bx is a one-to-one and has an inverse function
6
The x-axis is a horizontal asymptote. The graph approaches but does
not cross y = 0.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Characteristics of f (x) = bx
1
The domain of f (x) = bx is (−∞, ∞).
The range is (0, ∞).
2
The graph of f (x) = bx always passes through point (0, 1).
3
If b > 1, then f (x) = bx goes up to the right and is an increasing
function.
The greater the value of b, the steeper the increase.
4
If 0 < b < 1, then f (x) = bx goes down to the right and is a
decreasing function.
The smaller the value of b, the steeper the decrease.
5
f (x) = bx is a one-to-one and has an inverse function
6
The x-axis is a horizontal asymptote. The graph approaches but does
not cross y = 0.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Characteristics of f (x) = bx
1
The domain of f (x) = bx is (−∞, ∞).
The range is (0, ∞).
2
The graph of f (x) = bx always passes through point (0, 1).
3
If b > 1, then f (x) = bx goes up to the right and is an increasing
function.
The greater the value of b, the steeper the increase.
4
If 0 < b < 1, then f (x) = bx goes down to the right and is a
decreasing function.
The smaller the value of b, the steeper the decrease.
5
f (x) = bx is a one-to-one and has an inverse function
6
The x-axis is a horizontal asymptote. The graph approaches but does
not cross y = 0.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Outline
1
Exponential Functions
Definition
Graphing
Characteristics of f (x) = bx
Transformations
2
Natural Base e
Definition
Example
Compound Interest
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Transformations
Transformation
Vertical Shift
Horizontal Shift
Reflection
Vertical Stretch
Vertical Shrinking
Horizontal Stretch
Horizontal Shrinking
upward
downward
leftward
rightward
about y-axis
about x-axis
c>1
0<c<1
c>1
0<c<1
Math 1100
Equation
g(x) = bx + c
g(x) = bx − c
g(x) = bx+c
g(x) = bx−c
g(x) = −bx
g(x) = b−x
g(x) = cbx
g(x) = cbx
g(x) = bcx
g(x) = bcx
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Transformations
Transformation
Vertical Shift
Horizontal Shift
Reflection
Vertical Stretch
Vertical Shrinking
Horizontal Stretch
Horizontal Shrinking
upward
downward
leftward
rightward
about y-axis
about x-axis
c>1
0<c<1
c>1
0<c<1
Math 1100
Equation
g(x) = bx + c
g(x) = bx − c
g(x) = bx+c
g(x) = bx−c
g(x) = −bx
g(x) = b−x
g(x) = cbx
g(x) = cbx
g(x) = bcx
g(x) = bcx
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Transformations
Transformation
Vertical Shift
Horizontal Shift
Reflection
Vertical Stretch
Vertical Shrinking
Horizontal Stretch
Horizontal Shrinking
upward
downward
leftward
rightward
about y-axis
about x-axis
c>1
0<c<1
c>1
0<c<1
Math 1100
Equation
g(x) = bx + c
g(x) = bx − c
g(x) = bx+c
g(x) = bx−c
g(x) = −bx
g(x) = b−x
g(x) = cbx
g(x) = cbx
g(x) = bcx
g(x) = bcx
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Transformations
Transformation
Vertical Shift
Horizontal Shift
Reflection
Vertical Stretch
Vertical Shrinking
Horizontal Stretch
Horizontal Shrinking
upward
downward
leftward
rightward
about y-axis
about x-axis
c>1
0<c<1
c>1
0<c<1
Math 1100
Equation
g(x) = bx + c
g(x) = bx − c
g(x) = bx+c
g(x) = bx−c
g(x) = −bx
g(x) = b−x
g(x) = cbx
g(x) = cbx
g(x) = bcx
g(x) = bcx
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Transformations
Transformation
Vertical Shift
Horizontal Shift
Reflection
Vertical Stretch
Vertical Shrinking
Horizontal Stretch
Horizontal Shrinking
upward
downward
leftward
rightward
about y-axis
about x-axis
c>1
0<c<1
c>1
0<c<1
Math 1100
Equation
g(x) = bx + c
g(x) = bx − c
g(x) = bx+c
g(x) = bx−c
g(x) = −bx
g(x) = b−x
g(x) = cbx
g(x) = cbx
g(x) = bcx
g(x) = bcx
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Transformations
Transformation
Vertical Shift
Horizontal Shift
Reflection
Vertical Stretch
Vertical Shrinking
Horizontal Stretch
Horizontal Shrinking
upward
downward
leftward
rightward
about y-axis
about x-axis
c>1
0<c<1
c>1
0<c<1
Math 1100
Equation
g(x) = bx + c
g(x) = bx − c
g(x) = bx+c
g(x) = bx−c
g(x) = −bx
g(x) = b−x
g(x) = cbx
g(x) = cbx
g(x) = bcx
g(x) = bcx
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Transformations
Transformation
Vertical Shift
Horizontal Shift
Reflection
Vertical Stretch
Vertical Shrinking
Horizontal Stretch
Horizontal Shrinking
upward
downward
leftward
rightward
about y-axis
about x-axis
c>1
0<c<1
c>1
0<c<1
Math 1100
Equation
g(x) = bx + c
g(x) = bx − c
g(x) = bx+c
g(x) = bx−c
g(x) = −bx
g(x) = b−x
g(x) = cbx
g(x) = cbx
g(x) = bcx
g(x) = bcx
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Transformations
Transformation
Vertical Shift
Horizontal Shift
Reflection
Vertical Stretch
Vertical Shrinking
Horizontal Stretch
Horizontal Shrinking
upward
downward
leftward
rightward
about y-axis
about x-axis
c>1
0<c<1
c>1
0<c<1
Math 1100
Equation
g(x) = bx + c
g(x) = bx − c
g(x) = bx+c
g(x) = bx−c
g(x) = −bx
g(x) = b−x
g(x) = cbx
g(x) = cbx
g(x) = bcx
g(x) = bcx
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Transformations
Transformation
Vertical Shift
Horizontal Shift
Reflection
Vertical Stretch
Vertical Shrinking
Horizontal Stretch
Horizontal Shrinking
upward
downward
leftward
rightward
about y-axis
about x-axis
c>1
0<c<1
c>1
0<c<1
Math 1100
Equation
g(x) = bx + c
g(x) = bx − c
g(x) = bx+c
g(x) = bx−c
g(x) = −bx
g(x) = b−x
g(x) = cbx
g(x) = cbx
g(x) = bcx
g(x) = bcx
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Transformations
Transformation
Vertical Shift
Horizontal Shift
Reflection
Vertical Stretch
Vertical Shrinking
Horizontal Stretch
Horizontal Shrinking
upward
downward
leftward
rightward
about y-axis
about x-axis
c>1
0<c<1
c>1
0<c<1
Math 1100
Equation
g(x) = bx + c
g(x) = bx − c
g(x) = bx+c
g(x) = bx−c
g(x) = −bx
g(x) = b−x
g(x) = cbx
g(x) = cbx
g(x) = bcx
g(x) = bcx
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Graphing
Characteristics of f (x) = bx
Transformations
Transformations
Transformation
Vertical Shift
Horizontal Shift
Reflection
Vertical Stretch
Vertical Shrinking
Horizontal Stretch
Horizontal Shrinking
upward
downward
leftward
rightward
about y-axis
about x-axis
c>1
0<c<1
c>1
0<c<1
Math 1100
Equation
g(x) = bx + c
g(x) = bx − c
g(x) = bx+c
g(x) = bx−c
g(x) = −bx
g(x) = b−x
g(x) = cbx
g(x) = cbx
g(x) = bcx
g(x) = bcx
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Outline
1
Exponential Functions
Definition
Graphing
Characteristics of f (x) = bx
Transformations
2
Natural Base e
Definition
Example
Compound Interest
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Natural Base e
Consider the number
1+
1 n
n
for various values of n.
As n −→ ∞ , this approaches the irrational number e:
1+
1 n
−→ e
n n→∞
The approximation for e is
e ≃ 2.718281827 . . .
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Natural Base e
Consider the number
1+
1 n
n
for various values of n.
As n −→ ∞ , this approaches the irrational number e:
1+
1 n
−→ e
n n→∞
The approximation for e is
e ≃ 2.718281827 . . .
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Natural Base e
Consider the number
1+
1 n
n
for various values of n.
As n −→ ∞ , this approaches the irrational number e:
1+
1 n
−→ e
n n→∞
The approximation for e is
e ≃ 2.718281827 . . .
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Natural Base e
Consider the number
1+
1 n
n
for various values of n.
As n −→ ∞ , this approaches the irrational number e:
1+
1 n
−→ e
n n→∞
The approximation for e is
e ≃ 2.718281827 . . .
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Natural Base e
The number e is called the natural base when used in
exponential expressions.
The corresponding exponential function is
f (x) = ex
It is called the natural exponential function.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Natural Base e
The number e is called the natural base when used in
exponential expressions.
The corresponding exponential function is
f (x) = ex
It is called the natural exponential function.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Natural Base e
The number e is called the natural base when used in
exponential expressions.
The corresponding exponential function is
f (x) = ex
It is called the natural exponential function.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Outline
1
Exponential Functions
Definition
Graphing
Characteristics of f (x) = bx
Transformations
2
Natural Base e
Definition
Example
Compound Interest
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Example
The function
f (x) = 3.6 e0.02x
describes world population f (x), in millions after 1969.
Use the function to find the world population in the year 2020.
−→
f (51) = 9.98 billions
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Example
The function
f (x) = 3.6 e0.02x
describes world population f (x), in millions after 1969.
Use the function to find the world population in the year 2020.
−→
f (51) = 9.98 billions
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Outline
1
Exponential Functions
Definition
Graphing
Characteristics of f (x) = bx
Transformations
2
Natural Base e
Definition
Example
Compound Interest
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest
Compound Interest is interest accrued on the original
investment as well as any interest earned.
The initial investment is called the principal amount. It is
governed by the equation
A = P(1 + r )t
where r is the interest rate given as a decimal
and the interest is calculated annually for t years.
A is the amount at the end of t years.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest
Compound Interest is interest accrued on the original
investment as well as any interest earned.
The initial investment is called the principal amount. It is
governed by the equation
A = P(1 + r )t
where r is the interest rate given as a decimal
and the interest is calculated annually for t years.
A is the amount at the end of t years.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest
Compound Interest is interest accrued on the original
investment as well as any interest earned.
The initial investment is called the principal amount. It is
governed by the equation
A = P(1 + r )t
where r is the interest rate given as a decimal
and the interest is calculated annually for t years.
A is the amount at the end of t years.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest
Compound Interest is interest accrued on the original
investment as well as any interest earned.
The initial investment is called the principal amount. It is
governed by the equation
A = P(1 + r )t
where r is the interest rate given as a decimal
and the interest is calculated annually for t years.
A is the amount at the end of t years.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest
Compound Interest is interest accrued on the original
investment as well as any interest earned.
The initial investment is called the principal amount. It is
governed by the equation
A = P(1 + r )t
where r is the interest rate given as a decimal
and the interest is calculated annually for t years.
A is the amount at the end of t years.
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest n times per year
Instead of computing the interest annually, we can compute it
n times a year.
For example, semiannually, quarterly, or monthly.
The equation for n compoundings per year is:
r nt
A=P 1+
n
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest n times per year
Instead of computing the interest annually, we can compute it
n times a year.
For example, semiannually, quarterly, or monthly.
The equation for n compoundings per year is:
r nt
A=P 1+
n
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest n times per year
Instead of computing the interest annually, we can compute it
n times a year.
For example, semiannually, quarterly, or monthly.
The equation for n compoundings per year is:
r nt
A=P 1+
n
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest n times per year
Instead of computing the interest annually, we can compute it
n times a year.
For example, semiannually, quarterly, or monthly.
The equation for n compoundings per year is:
r nt
A=P 1+
n
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest n times per year
If we let n → ∞, we get continuous compounding:
and A si given by the formula
A = P ert
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest n times per year
If we let n → ∞, we get continuous compounding:
and A si given by the formula
A = P ert
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest n times per year
If we let n → ∞, we get continuous compounding:
and A si given by the formula
A = P ert
Math 1100
College Algebra & Trigonometry I
Outline
Exponential Functions
Natural Base e
Definition
Example
Compound Interest
Compound Interest Example
You decide to invest $8, 000 for 6 years and you have a choice
between two accounts.
The first pays 7% per year, compounded monthly.
The second pays 6.85% per year, compounded continuously.
Which is the better investment?
Math 1100
College Algebra & Trigonometry I