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8.1 Exponential Growth

8.1 Exponential Growth. p. 465. Exponential Function. f(x) = b 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.

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8.1 Exponential Growth

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  1. 8.1 Exponential Growth p. 465

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

  3. 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 2 raised to any power Will NEVER be zero!! y = 0

  4. Lets look at the activity on p. 465 • This shows of y= a * 2x • 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)

  5. These are true of: • y = abx • If a>0 & b>1 ……… • The function is an Exponential Growth Function

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

  7. D+ D= all reals R= all reals>0 y = 0 Always mark asymptote!!

  8. Example 2 y = 0 • Graph y = - (3/2)x • Plot (0, -1) and (1, -3/2) • Connect with a curve • Mark asymptote • D=?? • All reals • R=??? • All reals < 0

  9. To graph a general Exponential Function: • y = a bx-h + k • Sketch y = a bx • 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.

  10. Example 3 Graph y = 3·2x-1-4 • Lightly sketch y=3·2x • 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)

  11. D= all reals R= all reals >-4 y = -4

  12. Now…you try one! • Graph y= 2·3x-2 +1 • State the Domain and Range! • D= all reals • R= all reals >1 y=1

  13. 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

  14. 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) quarterly c) daily A=1000(1+.08/4)4x1 =1000(1.02)4 ≈ $1082.43 A=1000(1+ .08/1)1x1 = 1000(1.08)1 ≈ $1080 A=1000(1+.08/365)365x1 ≈1000(1.000219)365 ≈ $1083.28

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