Upload
pearl
View
58
Download
0
Tags:
Embed Size (px)
DESCRIPTION
Precalculus Lesson 3.1. Check: NONE. Warm-up. Find the Domain and identify any asymptotes. Answers. Objective(3.1). Evaluate and graph exponential functions. Use the natural base e and compound interest formulas. The graph of f ( x ) = a x , a > 1. - PowerPoint PPT Presentation
Citation preview
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