Exponential Functions

1
Exponential Functions

• 4.3

2
Laws of Exponents

• aras a(rs)
• (ar)s ars
• (ab)s asbs
• 1s 1
• a-s 1/(as)(1/a)s
• a0 1

3
Exponential Function

• A function of the form
• F(x) ax
• Where a is a positive real number (agt0) and a ?
  1. The domain of f is the set of all real
  numbers.

4
NOTE

• ax is an exponential function
• Like 2x or 5x
• X2 or x5 is a power function

5
Property of Exponential Functions

• Given that f is an exponential function. If x is
  any real number, then
• f(x1)/f(x) a

6
Examples

• 12,22

7
Properties of f(x) ax, agt1

• Domain
• All real numbers
• Range
• Positive real numbers
• X-intercepts
• None
• Y-intercepts
• (0,1)

8
Properties of f(x) ax, agt1

• Asymptote
• No vertical
• Horizontal at y0 as x approaches -8
• Increasing or Decreasing
• Increasing
• One-to-One?
• yes

9
Properties of f(x) ax, agt1

• Contains the points
• (0,1), (1,a), and (-1,1/a)
• The graph of f is smooth and continuous with no
  corners or gaps.

10
Properties of f(x) ax, 0ltalt1

• Domain
• All real numbers
• Range
• Positive real numbers
• X-intercepts
• None
• Y-intercepts
• (0,1)

11
Properties of f(x) ax, 0ltalt1

• Asymptote
• No vertical
• Horizontal at y0 as x approaches 8
• Increasing or Decreasing
• decreasing
• One-to-One?
• yes

12
Properties of f(x) ax, 0ltalt1

• Contains the points
• (0,1), (1,a), and (-1,1/a)
• The graph of f is smooth and continuous with no
  corners or gaps.

13
Transformation of ax

• F(x)c ax c
• 2x 5
• Moves up
• F(x)-c ax - c
• 2x - 5
• Moves down

14
Transformation of ax

• F(x)c ax c
• F(x)-c ax - c
• Range is all y greater than c.
• Horizontal asymptote is y c.
• Check x and y intercepts.

15
Transformation of ax

• F(xc) a(x c)
• 2(x 5)
• Moves left
• F(x-c) a(x c)
• 2(x 5)
• Moves right

16
Transformation of ax

• F(bx) a(bx) bgt1
• 2(3x)
• Horizontal shrink
• F(bx) a(bx) 0ltblt1
• 2(x/3)
• Horizontal stretch

17
Transformation of ax

• bF(x) ba(x) bgt1
• 32(x)
• Vertical stretch
• bF(x) ba(x) 0ltblt1
• (1/3)2(x)
• Vertical shrink

18
Transformation of ax

• -F(x) -a(x)
• -2(x)
• Flips over the x-axis
• F(-x) a(-x)
• 2(-x)
• Flips over the y axis

19
Transformation of ax

• -F(x) -a(x)
• Flips over the x-axis
• Changes the range from all y greater than zero to
  all y less than zero.

20
Examples

• 38,40,42,44

21
What is the following?

• (1(1/n))n as n gets large??

22
e

• 2.71828
• Notice that the calculator has an e key.

23
e

• ex graphs the same as other exponential funtions.

24
Solving Exponential Equations

• Rewrite numbers as a number to a power in order
  to try to get the same base on each side to a
  different power.
• Rewrite each side of the equation and simplify so
  that each side has the same base to different
  powers.
• Solve each by setting the powers equal.

25
Examples

• 5(1-2x) 1/5
• 4(x2) 2x
• (1/2)(1-x) 4
• (e4)x(e(x2))e12