Exponential Functions(指数函数)
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B
Transformations
• Horizontal translation: g(x)=bx+c • Shifts the graph to the left if c > 0 • Shifts the graph to the right if c < 0
y 2x
y 2( x 3) y 2( x 4)
n
The Number e - Definition
The table shows 1 A ( 1 ) the values of n as n gets increasingly large.
n
nபைடு நூலகம்
n
1 1 n
n
1 2
2 2.25
5
10 100
2.48832
Compound Interest 复利
• The formula for compound interest:
r A(t ) P 1 n
nt
Where n is the number of times per year interest is being compounded and r is the annual rate.
2.59374246 2.704813829
As n , the approximate value of e (to 9 decimal places) is ≈ 2.718281827
1000
10,000 100,000 1,000,000
2.716923932
2.718145927 2.718268237 2.718280469
y a x (a 1)
y a x (0 a 1)
Characteristics
• • • • The domain of f(x) = ax is R. The range of f(x) = ax consists of all positive real numbers (0, ). The graphs of all exponential functions pass through the point (0,1). This is because f(o) = a0 = 1 (ao). The graph of f(x) = ax approaches but does not cross the x-axis. The x-axis is a horizontal asymptote(水平渐近线). 1 x are symmetric about y-axis x x-axis y a and y ( )
a
Characteristics
Graphing Exponential Functions
When a>b>1 (1)When x>0,the graph of f(x) = ax is above the graph of f(x) = bx . (2)When x<0, the graph of f(x) = ax below which of f(x) = bx .
You Do
• Graph the function f(x) = 2(x-3) +2 • Where is the horizontal asymptote? y=2
You Do
• Graph the function f(x) = 4(x+5) - 3 • Where is the horizontal asymptote? y=-3
Exponential Functions
Definition of Exponential Functions
• The exponential function f with a base a is defined by f(x) =a x where a is a positive constant other than 1 (a > 0, and a≠ 1) and x is any real number.
A(t ) Pe
rt
where P is the initial principle (initial amount)
A(t ) Pe
rt
• If you invest $1.00 at a 7% annual rate that is compounded continuously, how much will you have in 4 years?
1 y 7
x
y 7x
1 y 2
x
y 2x
Graphing Exponential Functions
• So, when a> 1, f(x) has a graph that goes up to the right and is an increasing function(增函数). • When 0 < a < 1, f(x) has a graph that goes down to the right and is a decreasing function(减函数).
y a x (a 0 and a 1) dom ain: R
Graphing Exponential Functions
1 x Sketch the graphs of y 2 and y ( ) 2
x
y2
x
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
x
1 x y( ) 2
y= 3x
y = ex
y = 2x y =e
Natural Base 自然底数 e
• The irrational number e, is called the natural base. • The function f(x) = ex is called the natural exponential function.
• Reflecting • g(x) = -bx reflects the graph about the xaxis. • g(x) = b-x reflects the graph about the yaxis.
y 2x
y 2 x
y 2x
Steps for Multiple Transformations
When 0<b<a<1 (1)When x>0, the graph of f(x) = ax is below the graph of f(x) = bx (2)When x<0, the graph of f(x) = ax above which of f(x) = bx .
If 1 x 0, which one below is true ? A. 5 x 5 x 0.5 x C. 5 x 5 x 0.5 x B. 5 x 0.5 x 5 x D. 0.5 x 5 x 5 x
• Use the following order to graph a function involving more than one transformations. 1. Horizontal Translation 2. Stretching or Shrinking 3. Reflecting 4. Vertical Translation
– A: – B:
0.075 11 12
365(4)
12(4)
1.3486
$1.35
0.072 11 365
1.3337
$1.34
Interest Compounded Continuously
• If interest is compounded “all the time” (MUST use the word continuously), we use the formula
The Number e - Definition
• An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. It models a variety of situations in which a quantity grows or decays continuously: drugs in the body, probabilities, population studies, atmospheric pressure, optics, a bank account producing interest, or a population increasing as its members reproduce, and even spreading rumors! • The number e is defined as the value that n 1 approaches as n gets larger and larger. 1
• Vertical stretching or shrinking, f(x)=cbx: • Stretches the graph if c>1 • Shrinks the graph if 0 <c<1
y 2x
y 4(2x )
1 x y (2 ) 4
Transformations
• Horizontal stretching or shrinking, f(x)=bcx: • Shinks the graph if c >1 • Stretches the graph if 0<c<1
y 2x
y 4(2x )
1 x y (2 ) 4
Transformations
x -2 -1.5 -1 -0.5 0 0.5 1 1.5 y
y
0.25 0.35 0.5 0.71 1 1.41 2 2.83 4
2
Graphing Exponential Functions
• Four exponential functions have been graphed. • Compare the graphs of functions where a>1 to those where a<1
Compound Interest - Example
• Which plan yields the most interest?
– Plan A: A $1.00 investment with a 7.5% annual rate compounded monthly for 4 years – Plan B: A $1.00 investment with a 7.2% annual rate compounded daily for 4 years
The Number e
• The number e is known as Euler’s number. Leonard Euler (1700’s) discovered it’s importance.
• e ≈ 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 62772 40766 30353 54759 45713 82178 52516 64274
Transformations
• Vertical translation f(x) = bx + c • Shifts the graph up if c>0 • Shifts the graph down if c < 0
y 2x
y 2x 3
y 2x 4
Transformations
n
1,000,000,000 2.718281827
1 As n , 1 e n
The Number e - Definition
• Since 2 < e < 3, the graph of y = ex is between the graphs of y = 2x and y = 3x