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Exponential Growth Section 8.1
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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
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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
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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
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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
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Y = 0 asymptote D= all reals R= all reals>0
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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.
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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)
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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
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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(1 + 0.14) t P(t) = 350(1.14) t Substitute 350 for a and 0.14 for r. Simplify.
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It will take about 31 years for the population to reach 20,000.
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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.
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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
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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 ≈ $1082.43 A=1000(1+.08/365) 365x1 ≈1000(1.000219) 365 ≈ $1083.28
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Graph y= 2·3 x-2 +1
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