PrecalculusLesson 3.1

Check: NONE

WARM-UP

Find the Domain and identify any asymptotes.

2

22

2 5 3 11. ( ) 2. ( )2 3

x xf x g xx x

ANSWERS

1. : ,. .: . .: 2

2. : ,3 3,. .: 3 . .: 0

Dv a None ha y

Dv a x ha y

OBJECTIVE(3.1)

Evaluate and graph exponential functions.

Use the natural base e and compound interest formulas.

The graph of f(x) = ax, a > 1y

x(0, 1)

Domain: (–, )

Range: (0, )

Horizontal Asymptote y = 0

4

4

The graph of f(x) = ax, 0 < a < 1y

x(0, 1)

Domain: (–, )

Range: (0, )Horizontal Asymptote

y = 0

4

4

Exponential Function: An equation in the form f(x) = Cax.

If 0 < a < 1 , the graph represents exponential decay

If a > 1, the graph represents exponential growth

Examples: f(x) = (1/2)x f(x) = 2x

Exponential Decay Exponential Growth

These graphs “shift” according to changes in their equation...

Take a look at how the following graphs compare to the original graph of f(x) = (1/2)x :

f(x) = (1/2)x f(x) = (1/2)x + 1 f(x) = (1/2)x – 3

Vertical Shift: The graphs of f(x) = Cax + k are shifted vertically by k units.

Take a look at how the following graphs compare to the original graph of f(x) = (2)x :

f(x) = (2)x f(x) = (2)x – 3 f(x) = (2)x + 2 – 3

Horizontal Shift: The graphs of f(x) = Cax – h are shifted horizontally by h units.

Notice that f(0) = 1

(0,1)

Notice that this graphis shifted 3 units to theright.

(3,1)

Notice that this graphis shifted 2 units to theleft and 3 units down.

(-2,-2)

Take a look at how the following graphs compare to the original graph of f(x) = (2)x :

f(x) = (2)x f(x) = –(2)x f(x) = –(2)x + 2 – 3

Notice that f(0) = 1

(0,1)

This graphis a reflection of f(x) = (2)x . The graph isreflected over the x-axis.

(0,-1)

Shift the graph of f(x) = (2)x ,2 units to the left. Reflect the graph over the x-axis. Then, shift the graph 3 units down

(-2,-4)

Example: Sketch the graph of f(x) = 2x.

x

x f(x) (x, f(x))-2 ¼ (-2, ¼)-1 ½ (-1, ½)0 1 (0, 1)1 2 (1, 2)2 4 (2, 4)

y

2–2

2

4

Example: Sketch the graph of g(x) = 2x – 1. State the domain and range.

x

yThe graph of this function is a vertical translation of the graph of f(x) = 2x

down one unit .

f(x) = 2x

y = –1 Domain: (–, )

Range: (–1, )

2

4

Example: Sketch the graph of g(x) = 2-x. State the domain and range.

x

yThe graph of this function is a reflection the graph of f(x) = 2x in the y-axis.

f(x) = 2x

Domain: (–, )

Range: (0, ) 2–2

4

Also written as g(x)=(1/2)x

Complete front of notes 3.1

The graph of f(x) = ex

y

x2 –2

2

4

6

x f(x)-2 0.14-1 0.380 11 2.722 7.39

The irrational number e, where

e 2.718281828…

is used in applications involving growth and decay.

Using techniques of calculus, it can be shown that

ne

n

n

as 11

Complete back of notes 3.1

Formula

Continuous Compounding Formula

6.6 The Natural Base, e

A: amount of the investment with interest

P: principal (initial investment)

r: interest rate

t: time in years

A = Pert

Continuous Compounding Formula

Example

6.6 The Natural Base, e

An investment of $7400 at 12% interest is compounded continuously.

How much will the investment be worth in 15 years?

A = Pert = 7400e0.12(15) 44,767.39

Formula

• “n” Compoundings per year

1ntrA P

n

n: # of compoundings per year

Quarterly: semi-annually:

Daily:

“N” Compounding Formula

An investment of $7400 at 12% interest is compounded quarterly.

4 10.127400 14

How much will the investment be worth in 10 years?

1ntrA P

n

$24,139.08

Classwork: Notes hand-out 3.1Practice with formulas p. 225 # 61,

65, 71Homework(3.1) p. 224 #10, 13-16, 22, 24, 28, 38, 42, 60-66 even,72

Assignments

Quiz 3.1-3.2 Friday

Describe the transformation.

CLOSURE

f(x) = 4x

2

2

5

1. ( ) 42. ( ) 4 33. ( ) 4 64. State the domain for each function.5. State the Range for each function.

x

x

x

f xf xf x