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# 8.2 Exploring exponential models

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### 8.2 Exploring exponential models

1. 1. 8.2 Writing Exponential Models
2. 2. What is an exponential equation? An exponential equation has the general form y=abx where a ≠ 0, b〉 0 and b ≠ 1
3. 3. Growth Factor, Decay Factor Given the general form y=abx  When b > 1, b is the growth factor  When 0 < b < 1, b is the decay factor
4. 4. Growth or Decay??? y = 10(1.2) y = 5(.9) x Growth x Decay y = 50(1.54) y = 5.2(.70) y = 4(2) x y = 100(.07) x x Growth Decay Growth x Decay
5. 5. Writing Exponential Equations  Find the exponential equation passing through the points (3,20) and (1,5). y = ab 20 = ab 20 =a 3 b x 3 Start with the general form. Choose a point. Substitute for x and y using (3, 20) Solve for a 20 1 5= 3 b b 1−3 5 = 20b Substitute x and y using (1, 5) and a using Division property of exponents 2 =a b3
6. 6. Writing Exponential Equations  Find the exponential equation passing through the points (3,20) and (1,5). 5 = 20b −2 Simplify 20 5= 2 b 20 2 b = =4 Solve for b 5 b=2 20 20 20 5 a= 3 = 3 = = b 2 8 2 Go back to the equation for a; substitute in b and solve for a
7. 7. Writing Exponential Equations  Find the exponential equation passing through the points (3,20) and (1,5). y = ab x 5 x y = (2) 2 Going back to the general form, substitute in a and b The exponential equation passing through 5 the points (3,20) and (1,5) is y = ( 2) x 2
8. 8. Let’s Try One  Find the exponential equation passing through the points (2,4) and (3,16). y = ab x 4 = ab 4 =a 2 b 2 4 3 16 = 2 b b 16 = 4b 3− 2 Start with the general form. Choose a point. Substitute for x and y using (2, 4) Solve for a Substitute x and y using (3, 16) and a using Division property of exponents 4 =a b2
9. 9. Writing Exponential Equations 16 = 4b 1 Simplify b=4 y = ab Solve for b x 4 1 a = 2 = = 0.25 4 4 y = 0.25(4) x Go back to the equation for a; substitute in b and solve for a Going back to the general form, substitute in a and b The exponential equation passing through x the points (2,4) and (3,16) is y = 0.25(4)
10. 10. Putting it all together . . .  Find the equation of the exponential function that goes through (1,6) and (0,2). Graph the function.
11. 11. Modeling Growth with an Exponential Equation  The growth factor can be found in word problems using b=1+r where r = rate or amount of increase. You can substitute your new b into your general equation to find the exponential function.
12. 12.  EX- a guy puts \$1000 into a simple 3% interest account. What is the exponential equation? r = rate 3% (write as 0.03) b = 1 + r = 1.03 x = time a = amount put into the account (\$1,000) y = ab x y = 1000 (1.03) x
13. 13.  EX – a colony of 1000 bacteria cells doubles every hour. What is the exponential equation? r = 1 (why not 2?) b=r+1=2 x = time (in hours) a = the original number in the colony (1,000 bacteria ) y = ab x y = 1000 (2) x b = r + 1, where r is the amount of increase. We are increasing by 100% each time something doubles, so r = 1
14. 14.  EX- a \$15000 car depreciates at 10% a year. What is the exponential equation? r = - 10% (the car is worth 10% less each year) b = 1 - r = 1 – 0.1 = 0.9 x = time (in years) a = amount put into the account (\$15,000) y = ab x y = 15000 (0.9) x
15. 15. Compound Interest  The formula for compound interest: nt  r A(t ) = P 1 + ÷  n Where n is the number of times per year interest is being compounded and r is the annual rate.
16. 16. 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 years  Plan B: A \$1.00 investment with a 7.2% annual rate compounded daily for 4 years  12(4) A:11 + 0.075  ≈1.3486  ÷ 12   \$1.35 365(4)  B:  0.072  11 + ≈1.3337 ÷ 365    \$1.34