1. Aim: What is an exponential function?
Do Now:
Solve/evaluate: 64 = 23n + 1
Poll: n = 5/3
x = 6561
Rewrite in exponential form

2. Properties of Exponents
Product of Powers Property
am • an = am+n
Power of Power Property
(am)n = am•n
Power of Product Property
(ab)m = ambm
Negative Power Property
a-n = 1/an, a 0
Zero Power Property
a0 = 1
Quotients of Powers Property
Power of Quotient Property

3. beware! -23/2 is not the same as (-2)3/2
Types of exponents
Positive Integer Exponent
an = a • a • a • • • • a
n factors
Zero Exponent
a0 = 1
Rational Exponent 1/n
Rational Exponent m/n
Negative Exponent - m/n
beware! /2 is not the same as (-2)3/2

4. Exponential Function - Outbreak
Disease Control - “Outbreak”
y = 2x

5. y = a • bx y = 1 • 2x y = 2x Exponential Function
. . have variables as exponents, and
where a  0, base b > 0, and also b  1.
When a = 1, graph always goes through (0,1)
(0,1)
The domain (x) is the set of real numbers: (-, )
The range (y) is the set of positive real numbers: (0, )
The x-axis is a horizontal asymptote: ax  0 as x  -
If b > 1, the graph is increasing and continuous
What happens as b increases in value?
y = 1 • 2x
y = 2x
As b increases in value from 1, the slope of
the graph gets steeper

6. y = a • bx The b Affect y = (1/2)x y = 2x y = 1 (0,1)
If b > 1, the graph is increasing - Growth
If 0 < b < 1
the graph is decreasing - Decay
If b is a positive number other than 1,
the graphs of y = bx and y = (1/b)x
are reflections through the y-axis of each other
What if b = 1?
horizontal line: y = 1

7. if b < 0, no longer the exponential function
The b Affect: b < 0
y = a • bx
What if b < 0?
Let b = (-2)?
y = 1 • (-2)x
if b < 0, no longer the exponential function
graph:
table: x = .1
table: x = 1

8. y = a • bx The a Affect y = 2x y = (2)2x y = (4)2x
Graph: y = 2x; y = (2)2x; y = (4)2x
a = 4
(0,4)
(0,2)
a = 2
(0,1)
a = 1

9. The a Affect
y = a • bx
(0,-1)
Graph f(x) = -2x = -1 • 2x
Graph f(x) = -(1/2)x = -1 • (1/2)x
Graph f(x) = -6x = -1 • (6)x

10. Graph the exponential functions
The a affect
Graph the exponential functions

11. substitute & solve for a
Model Problem
Write an exponential function y = abx for a graph that includes (2, 2) and (3, 4)
general form
substitute (2,2) for x & y
solve for a
substitute for a
substitute (3,4) for x & y
b = 2
simplify & solve for b
a = 1/2
substitute & solve for a
substitute for a and b

12. y  2.7183 is asymptotic to f(x).
Where’d e Come From?
Graph
y 
Poll for asymptote: Use graphing calculator to find to nearest thousandth
y  is asymptotic to f(x).
x = 1
x = 100
x = 10000

13. Natural Exponential Function
f(x) = ex
f(x) = ( )x
Evaluate e2
e-1
e0.48 = = =
Graph f(x) = 2e0.24x
Graph f(x) = (1/2)e-0.58x

14. Transforming Functions
If k and h are positive numbers and
f(x) is a function, then
f(x) + k shifts f(x) up k units
f(x) – k shifts f(x) down k units
f(x + h) shifts f(x) left h units
f(x – h) shifts f(x) right h units
f(x) = (x + h)2 + k - parabolic
f(x) = |x + h| + k absolute value
ex. f(x) = (x – 4)2 + 4 is the image of g(x) = x2
after a shift of 4 units to the right and four
units up or a translation of T4,4.
NOTE: k is not the y-intercept.
what is the y-intercept for f(x) = (x – 4)2 + 4?

15. Transforming the Exponential Function
Graph the exponential functions y = 4x, y = 4x + 2, y = 4x – 3

16. Transforming the Exponential Function
Graph the exponential functions y = 2x+3, y = 2x-2, y = 2x–1 – 2
(1, 1)
(2, 1)
(-3, 1)
(0, 1)

17. Transforming the Exponential Function
Write the equation and graph the exponential function f(x) = 2x after a reflection in both the x- & y-axes, and a vertical translation of 4.
Graph y = 2x
Graph y = (1/2)x
Activity Center: send graph and equation
Reflect thru x-axis: y = -(1/2)x
Shift up 4 units y = -(1/2)x + 4

18. Transforming the Exponential Function
Write the equation and graph the exponential function after a reflection in the x-axis, vertically stretched by a factor of 2, a vertical translation of 2 and a horizontal of 2.
Graph y = 2x
Reflect thru x-axis: y = -2x
Activity Center: send equation and graph
Shift right 2 units y = -(1/2)x –2
Shift up 2 units y = -(1/2)x –2 + 2

19. Transforming the Exponential Function
The graph of y = (1/4)x is translated 5 units upward, 8 units to the right and then reflected in the x-axis. What’s the equation?
Activity Center: send equation and graph
f(x) = -f(x)

20. Transforming the Exponential Function
The graph of y = (1/4)x is translated 5 units upward, 8 units to the right and then reflected in the x-axis. What’s the equation?
Activity Center: send equation and graph
f(x) = -f(x)