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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 exponent
2、ial functions have been graphed.Compare the graphs of functions where a 1 to 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 function(减函数
3、).,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 asym
4、ptote(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 approaches but does not cross the x-axis.The x-axis
5、 is a horizontal asymptote(水平渐近线).are symmetric about y-axis,Characteristics,Graphing Exponential FunctionsWhen 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(2)When x0,the graph of f(x)=ax above which of f(x)=bx.,B,Tran
6、sformations,Horizontal translation:g(x)=bx+cShifts the graph to the left if c 0Shifts the graph to the right if c 0,Transformations,Vertical translation f(x)=bx+cShifts the graph up if c 0Shifts the graph down if c 0,Transformations,Horizontal stretching or shrinking,f(x)=bcx:Shinks the graph if c 1
7、Stretches the graph if 0 c 1,Transformations,Vertical stretching or shrinking,f(x)=cbx:Stretches the graph if c 1Shrinks the graph if 0 c 1,Transformations,Reflectingg(x)=-bx reflects the graph about the x-axis.g(x)=b-x reflects the graph about the y-axis.,Steps for Multiple Transformations,Use the
8、following order to graph a function involving more than one transformations.Horizontal TranslationStretching or Shrinking ReflectingVertical Translation,You Do,Graph the function f(x)=2(x-3)+2Where is the horizontal asymptote?,y=2,You Do,Graph the function f(x)=4(x+5)-3Where is the horizontal asympt
9、ote?,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 40766 30353 54759 45713 82178 52516 64274,The Number e-Definition,An irrational number,symbolized by the lette
10、r 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
11、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 gets increasingly large.,As,the approximate value of e(to 9 decimal places)is 2.718281827,The Number e-Definition,Since 2 e
12、 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 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
13、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 investment with a 7.5%annual rate compounded monthly for 4 yearsPlan B:A$1.00 investment with a 7.2%annual rate compounded daily for 4 yearsA:B:,$1.35,$1.34,Int
14、erest Compounded Continuously,If interest is compounded“all the time”(MUST use the word continuously),we use the formula where P is the initial 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:,
