1、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.,Graphing Exponential Functions,Sketch the graphs of,Graphing Exponential Functions,Four expo
2、nential functions have been graphed. Compare the graphs of functions wherea 1to those where a 1,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 fun
3、ction(减函数).,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
4、 x-axis is a horizontal asymptote(水平渐近线).are symmetric about x-axis,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 appro
5、aches but does not cross the x-axis. The x-axis is a horizontal asymptote(水平渐近线).are symmetric about y-axis,Characteristics,Graphing Exponential Functions When ab1 (1)When x0,the graph of f(x) = ax is above the graph of f(x) = bx . (2)When x0, the graph of f(x) = ax is below the graph of f(x) = bx (
6、2)When x0, the graph of f(x) = ax above which of f(x) = bx .,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,Transformations,Vertical translation f(x) = bx + c Shifts the graph up if c 0 Shifts the graph down if c 0,Transfo
7、rmations,Horizontal stretching or shrinking, f(x)=bcx: Shinks the graph if c 1 Stretches the graph if 0 c 1,Transformations,Vertical stretching or shrinking, f(x)=cbx: Stretches the graph if c 1 Shrinks the graph if 0 c 1,Transformations,Reflecting g(x) = -bx reflects the graph about the x-axis. g(x
8、) = b-x reflects the graph about the y-axis.,Steps for Multiple Transformations,Use the following order to graph a function involving more than one transformations. Horizontal Translation Stretching or Shrinking Reflecting Vertical Translation,You Do,Graph the function f(x) = 2(x-3) +2 Where is the
9、horizontal asymptote?,y = 2,You Do,Graph the function f(x) = 4(x+5) - 3 Where is the horizontal asymptote?,y = - 3,The Number e,The number e is known as Eulers number. Leonard Euler (1700s) discovered its importance.e 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 62772 40
10、766 30353 54759 45713 82178 52516 64274,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, populat
11、ion 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 approaches as n gets larger and larger.,The Number e - Definition,The table shows the values of as n
12、 gets increasingly large.,As , the approximate value of e (to 9 decimal places) is 2.718281827,The Number e - Definition,Since 2 e 3, the graph of y = ex is between the graphs of y = 2x and y = 3x,y =e,y = 2x,y = 3x,y = ex,Natural Base 自然底数 e,The irrational number e, is called the natural base. The
13、function f(x) = ex is called the natural exponential function.,Compound Interest 复利,The formula for compound interest:,Where n is the number of times per year interest is being compounded and r is the annual rate.,Compound Interest - Example,Which plan yields the most interest? Plan A: A $1.00 inves
14、tment 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 yearsA: B:,$1.35,$1.34,Interest Compounded Continuously,If interest is compounded “all the time” (MUST use the word continuously), we use the formulawhere P is the i
15、nitial principle (initial amount),If you invest $1.00 at a 7% annual rate that is compounded continuously, how much will you have in 4 years?You will have a whopping $1.32 in 4 years!,You Do,You decide to invest $8000 for 6 years and have a choice between 2 accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment?,You Do Answer,1st Plan: 2nd Plan:,
