Exponential Functionsand the number e

Precalculus11-2 and 11-3

p. 704

CalculatorGraph y = bx

b = 0.5b = 0.75

b = 2b = 5

Characteristics of graphs of y = bx

b > 1 0 < b < 1

Domain all real numbers all real numbers

Range

y-inercept

Behavior

Horizontal asymptote

Vertical asymptote

Characteristics of graphs of y = bx

b > 1 0 < b < 1

Domain all real numbers all real numbers

Range {y | y > 0} {y | y > 0}

y-inercept

Behavior

Horizontal asymptote

Vertical asymptote

Characteristics of graphs of y = bx

b > 1 0 < b < 1

Domain all real numbers all real numbers

Range {y | y > 0} {y | y > 0}

y-inercept (0, 1) (0, 1)

Behavior

Horizontal asymptote

Vertical asymptote

Characteristics of graphs of y = bx

b > 1 0 < b < 1

Domain all real numbers all real numbers

Range {y | y > 0} {y | y > 0}

y-inercept (0, 1) (0, 1)

Behavior Continuousone-to-oneincreasing

Continuousone-to-onedecreasing

Horizontal asymptote

Vertical asymptote

Characteristics of graphs of y = bx

b > 1 0 < b < 1

Domain all real numbers all real numbers

Range {y | y > 0} {y | y > 0}

y-inercept (0, 1) (0, 1)

Behavior Continuousone-to-oneincreasing

Continuousone-to-onedecreasing

Horizontal asymptote

negative x-axis positive x-axis

Vertical asymptote

Characteristics of graphs of y = bx

b > 1 0 < b < 1

Domain all real numbers all real numbers

Range {y | y > 0} {y | y > 0}

y-inercept (0, 1) (0, 1)

Behavior • Continuous• one-to-one• increasing

• Continuous• one-to-one• decreasing

Horizontal asymptote

negative x-axis positive x-axis

Vertical asymptote none none

CalculatorGraphy = 4x

y = 4x + 2y = 4x – 3

CompareO All the graphs are continuous, increasing,

and one-to-one.

CompareO All the graphs are continuous, increasing,

and one-to-one.O The y-intercept and horizontal asymptotes

for each graph are different.

CompareO All the graphs are continuous, increasing,

and one-to-one.O The y-intercept and horizontal asymptotes

for each graph are different.O Only one graph has an x-intercept

Exponential Growth or Decay

N is the final amount

Exponential Growth or Decay

N is the final amount

Exponential Growth or Decay

N0 is the initial amount

N is the final amount

Exponential Growth or Decay

N0 is the initial amount

r is the rate of growth or decay per time period

N is the final amount

Exponential Growth or Decay

N0 is the initial amount

r is the rate of growth or decay per time period

t is the number of time periods

Compound Interest

A is the final amount

Compound Interest

A is the final amount

Compound Interest

P is the principal or initial investment

A is the final amount

Compound Interest

P is the principal or initial investment

r is the annual interest rate

A is the final amount

Compound Interest

P is the principal or initial investment

r is the annual interest rate

n is the number of times interest is paid

A is the final amount

Compound Interest

P is the principal or initial investment

r is the annual interest rate

t is the number of years

n is the number of times interest is paid

The Number e

The Number eO e is not a variable.

The Number eO e is not a variable.O e is a constant like .π

The Number eO e is not a variable.O e is a constant like .π

The Number eO e is not a variable.O e is a constant like .π