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Unit 3: Exponential and Logarithmic Functions
Section 3.1 Graphs of Exponential Functions including transformations and base โeโ Notes: Exponential Function- function containing a variable exponent f(x) = ๐ ๐ฅ where b is a positive constant other than 1 (b > 0 and b โ 1) think about why b โ 1 Using a half sheet of graph paper, sketch the graph of f(x) = 2 ๐ฅ Then find the following: lim ๐ฅโโโ 2 ๐ฅ lim ๐ฅโโ 2 ๐ฅ =0 = โ
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Sketch the graphs of the following functions on the same set of axes as you graphed f(x) = 2 ๐ฅ .
g(x) = 3 ๐ฅ h(x) = 4 ๐ฅ j(x) = ๐ฅ k(x) = ๐ฅ l(x) = ๐ฅ Then come up with as many properties of exponential functions that you can (there are a total of 7) Domain is (-โ, โ) Range is (0, โ) Horizontal asymptote at y = 0 y-intercept = 1 If b > 1, graph is increasing If 0 < b < 1, graph is decreasing When b > 1, as b increases, the graph gets steeper When 0 < b < 1, as b decreases, the graph gets steeper It is a 1:1 function
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Using the transformation rules we learned in Chapter 1, describe the transformations that are occurring in the examples below. Be sure to list them in the correct order. f(x) = โ2 (โ๐ฅ+4) g(x) = 3ยท 2 (๐ฅโ3) โ 6 h(x) = 1 2 ( โ2 (๐ฅ+1) ) + 5 left 4, reflect over the x and y axis right 3, vertically stretched by a factor of 3, down 6 left 1, vertically shrunk by a factor of ยฝ, reflect over the x-axis, up 5
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Natural Base โeโ The natural base โeโ is an irrational number symbolized by the letter e. It was discovered by a Swiss mathematician Leonard Euler who proved that: lim ๐โโ ๐ ๐ e where e โ โฆ e is called the natural base What does the graph of f(x) = ๐ ๐ฅ look like? Sketch the graph on the same grid as all the others.
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Example In 1969, the world population was 3.6 billion, with a growth rate of 2% per year. The function f(x) = 3.6 ๐ 0.02๐ฅ describes world population, f(x), in billions, x years after Find the world population for 2020. In 2000, the world population was approximately 6 billion but the growth rate had slowed to 1.3%. Find the world population in 2050. f(x) = 6 ๐ 0.013๐ฅ ๐คโ๐๐๐ ๐ฅ=# ๐๐ ๐ฆ๐๐๐๐ ๐ ๐๐๐๐ 2000 9.98 billion 11.49 billion
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Compound Interest Interest computed on original investment as well as any already accumulated interest. ๐ด= ๐(1+ ๐ ๐ ) ๐๐ก Where: A = final balance P = principal (original investment) r = annual percentage rate in decimal form n = number of compounding periods per year t = number of years Be sure to know the following: Annually: n = 1 Semi-annually: n = 2 Quarterly: n = 4
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Some banks use continuous compounding where the number of compounding periods increases infinitely.
as n โโ, (1+ 1 ๐ ) ๐ โ๐ Therefore: A = Pert is the formula for continuous compounding Example: Suppose $10000 is invested at an annual rate of 8%. Find the balance in the account after 5 years when interest is compounded: a) Quarterly b) continuously ๐ด= 10000( ) 4ยท5 A = 10000e.08ยท5 A = $14,918.25 A = $14,859.47
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Homework: Pg /19-24 all, 29, 39, 51 Day 2: Pg. 397/ 53,55,57,61,63,67,73
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