Exponential Growth Section 8.1

Exponential Function  f(x) = ab x where the base b is a positive number other than one.  Graph f(x) = 2 x  Note the end behavior  x→+∞ f(x)→+∞  x→-∞ f(x)→0  y=0 is an asymptote

Asymptote  A line that a graph approaches as you move away from the origin The graph gets closer and closer to the line y = 0 ……. But NEVER reaches it

f(x) = 2 x  Passes thru the point (0,a) (the y intercept is a)  The x-axis is the asymptote of the graph  D is all reals (the Domain)  R is y>0 if a>0 and y<0 if a<0 (the range) These are true of:  y = ab x  If a>0 & b>1 ………  The function is an Exponential Growth Function

Ex  Graph y = ½ 3 x  Plot (0, ½) and (1, 3/2)  Then, from left to right, draw a curve that begins just above the x-axis, passes thru the 2 points, and moves up to the right

Y = 0 asymptote D= all reals R= all reals>0

To graph a general Exponential Function:  y = a b x-h + k  Sketch y = a b x  h= ??? k= ???  Move your 2 points h units left or right …and k units up or down  Then sketch the graph with the 2 new points.

Example 3 Graph y = 3·2 x-1 -4  Lightly sketch y=3·2 x  Passes thru (0,3) & (1,6)  h=1, k=-4  Move your 2 points to the right 1 and down 4  AND your asymptote k units (4 units down in this case)

In the formula, the base of the exponential expression, 1 + r, is called the growth factor. Similarly, 1 – r is the decay factor. You can model growth or decrease with the following formula: decay by a constant percent increase

Example  In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth. Use a graph to predict when the population will reach 20,000  Exponential growth function P(t) = a(1 + r) t P(t) = 350( ) t P(t) = 350(1.14) t Substitute 350 for a and 0.14 for r. Simplify.

 It will take about 31 years for the population to reach 20,000.

 A motor scooter purchased for $1000 depreciates at an annual rate of 15%. Write an exponential function and graph the function. Use the graph to predict when the value will fall below $100.

Compound Interest  A=P(1+r/n) nt  P - Initial principal  r – annual rate expressed as a decimal  n – compounded n times a year  t – number of years  A – amount in account after t years

Compound interest example  You deposit $1000 in an account that pays 8% annual interest.  Find the balance after I year if the interest is compounded with the given frequency.  a) annually b) quarterlyc) daily A=1000(1+.08/1) 1x1 = 1000(1.08) 1 ≈ $1080 A=1000(1+.08/4) 4x1 =1000(1.02) 4 ≈ $ A=1000(1+.08/365) 365x1 ≈1000( ) 365 ≈ $

Graph y= 2·3 x-2 +1