aim: exponential functions course: alg. 2 & trig. aim: what is an exponential function? do now:...

Aim: Exponential Functions Course: Alg. 2 & Trig.

Aim: What is an exponential function?

Do Now:Solve/evaluate: 64 = 23n + 1

3 7 25 3r s t

734 x3

4 5

Rewrite in exponential form


Zero Power Property

Properties of Exponents

Product of Powers Property

a0 = 1

Power of Power Property

Power of Product Property

Negative Power Property

Quotients of Powers Property

Power of Quotient Property

am • an = am+n

(am)n = am•n

a-n = 1/an, a 0

(ab)m = ambm

am

an am n , a 0

(a

b)m

am

bm , b 0


Types of exponents

Positive Integer Exponent an = a • a • a • • • • a

n factors

Zero Exponent a0 = 1

Rational Exponent 1/n

a1 n an

Rational Exponent m/n

am n ( an )m

Negative Exponent - m/n

a m n 1

am n

If a 0, am n ( an )m

results in a nonreal number

ex . ( 2)3 2 imaginary number

beware! -23/2 is not the same as (-2)3/2


Exponential Function - Outbreak

Disease Control - “Outbreak”

y = 2x


Exponential Function

y = a • bx

. . have variables as exponents, andwhere a 0, base b > 0, and also b 1.

The x-axis is a horizontal asymptote: ax 0 as x -

The domain (x) is the set of real numbers: (-, )The range (y) is the set of positive real numbers: (0, )

If b > 1, the graph is increasing and continuous

As b increases in value from 1, the slope ofthe graph gets steeper

y = 1 • 2x

y = 2x

When a = 1, graph always goes through (0,1)

(0,1)

What happens as b increases in value?


4

2

-2

-4

-5 5

The b Affect

y = 2xy = (1/2)x

(0,1)

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

If b > 1, the graph is increasing - Growth

What if b = 1? horizontal line: y = 1

y = 1

y = a • bx

If 0 < b < 1


4

2

-2

-4

-5 5

f x = 2x

The b Affect: b < 0

y = a • bx

Let b = (-2)?

What if b < 0?

if b < 0, no longer the exponential function

y = 1 • (-2)x

graph: table: x = 1table: x = .1


y = a • bx

The a Affect

(0,1) a = 1

(0,2) a = 2

(0,4)a = 4

y = 2x

y = (2)2x

y = (4)2x

Graph: y = 2x; y = (2)2x; y = (4)2x


4

2

-2

-4

-5 5

f x = 2x

4

2

-2

-4

-5 5

f x = -2x

The a Affect

Graph f(x) = -(1/2)x = -1 • (1/2)x

Graph f(x) = -6x = -1 • (6)x

4

2

-2

-4

-5 5

g x = -1

2

x

4

2

-2

-4

-5 5g x = -6x (0,-1)

y = a • bx

Graph f(x) = -2x = -1 • 2x


The a affect

Graph the exponential functions

11,

5 552

1,6

xx x

yy y

10

5

-5

-10

-10 10

h x = 1

5

x10

5

-5

-10

-10 10

h x = 61

5

x

10

5

-5

-10

-10 10

h x = -21

5

x


Model Problem

Write an exponential function y = abx for a graph that includes (2, 2) and (3, 4)

xy ab 22 ab 2

2a

b

2

2 xy bb

32

24 b

b

3 3 22

24 2 2b b b

b

b = 2

2

2

2a a = 1/2 1

22

xy

general form substitute (2,2) for x & ysolve for a

substitute for a substitute (3,4) for x & y

simplify & solve for b

substitute & solve for a substitute for a and b


6

4

2

-2

-5 5

r x = 1+1

x x

Where’d e Come From?

Graph x

f xx

11

6

4

2

-2

-5 5

y 2.7183

y 2.7183 is asymptotic to f(x).

x = 1 x = 100 x = 10000


4

3

2

1

-1

-2 2

t x = ex

Natural Exponential Function

f(x) = ex

f(x) = (2.71828. . .)x

Evaluate e2

e-1

e0.48

= 7.389056099

= 0.3678794412

= 1.616074402

10

8

6

4

2

-2

-5 5 10

q1 x = 2e0.24x

Graph f(x) = 2e0.24x Graph f(x) = (1/2)e-0.58x

10

8

6

4

2

-2

-5 5 10

r1 x = 1

2e-0.58x


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?


10

5

-5

-10

-10 10

h x = 4x

Transforming the Exponential Function

Graph the exponential functions y = 4x, y = 4x + 2, y = 4x – 3

10

5

-5

-10

-10 10

h x = 4x10

5

-5

-10

-10 10

q x = 4x-3

h x = 4x+2


6

4

2

-2

-4

-6

-5 5

u x = 2x

(0, 1)

6

4

2

-2

-4

-6

-5 5

u x = 2x+3

(-3, 1)


Graph the exponential functions y = 2x+3, y = 2x-2, y = 2x–1 – 2

6

4

2

-2

-4

-6

-5 5

v x = 2x-2

(2, 1)

6

4

2

-2

-4

-6

-5 5

w x = 2x-1-2

(1, 1)



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 = (1/2)x

Reflect thru x-axis: y = -(1/2)x

Shift up 4 units y = -(1/2)x + 4

4

2

-2

-4

-5 5

f x = 2x

f x = 1

2

x

f x = -1

2

x

f x = -1

2

x+4

Graph y = 2x



Graph y = 2x

Reflect thru x-axis: y = -2x

Shift right 2 units y = -(1/2)x –2

6

4

2

-2

-4

-6

-5 5

h1 x = 2x

6

4

2

-2

-4

-6

-5 5

h1 x = -2x

6

4

2

-2

-4

-6

-5 5

h1 x = -2x-2

Shift up 2 units y = -(1/2)x –2 + 2

6

4

2

-2

-4

-6

-5 5

h1 x = -2x-2+2

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.



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?10

8

6

4

2

-2

-4

-6

-8

5 10 15

f x = 0.25x

g x = 0.25x+5

h x = 0.25x-8+5

r x = - 0.25x-8+5

f(x) = -f(x)

aim: exponential functions course: alg. 2 & trig. aim: what is an exponential function? do now:...

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