7.2 Alg 2 exploring exponential models

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7.2 Alg 2 exploring exponential models

  1. 1. 7.2 Exploring Exponential Models
  2. 2. What is an exponential equation? An exponential equation has the general form y=abx 1band0b,0wherea
  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??? x )2.1(10y x )9(.5y x )54.1(50y x )70(.2.5y x y )2(4 x )07(.100y Growth Decay Growth Decay Growth Decay
  5. 5. Graphs of Exponential Functions x )2(10y
  6. 6. What is an asymptote? x )2(10y“Walking halfway to the wall” An Asymptote is a line that a graph approaches as x or y increases in absolute value. In this example, the asymptote is the x axis.
  7. 7. Graphing Exponential Functions x )5(.100y X .5x Y=100(.5)x -3 -2 -1 0 1 2 3 Complete the table using the integers -3 through 3 for x.
  8. 8. Let’s graph one together x )5(.100y X .5x Y=100(.5)x -3 .5-3 800 -2 .5-2 400 -1 .5-1 200 0 .50 100 1 .51 50 2 .52 25 3 .53 12.5
  9. 9. Let’s try one x y )5(.2 X .5x Y=2(.5)x -3 -2 -1 0 1 2 3 Complete the table using the integers -3 through 3 for x. Then graph the function.
  10. 10. Let’s try one x y )5(.2 X .5x Y=2(.5)x -3 -2 -1 0 1 2 3
  11. 11. Let’s try one x y )10(5 X 10x Y=5(10)x -3 -2 -1 0 1 2 3 Complete the table using the integers -3 through 3 for x. Then graph the function.
  12. 12. Let’s try one x y )10(5 X 10x Y=5(10)x -3 -2 -1 0 1 2 3
  13. 13. Writing Exponential Equations  Find the exponential equation passing through the points (3,20) and (1,5). x aby a b3 20 1 3 20 5 b b 31 205 b 3 20 ab Start with the general form. Choose a point. Substitute for x and y using (3, 20) Solve for a Substitute x and y using (1, 5) and a using a b3 2 Division property of exponents
  14. 14. Writing Exponential Equations  Find the exponential equation passing through the points (3,20) and (1,5). 2 4 5 20 20 5 205 2 2 2 b b b b Simplify Solve for b 2 5 8 20 2 2020 33 b a Go back to the equation for a; substitute in b and solve for a
  15. 15. Writing Exponential Equations  Find the exponential equation passing through the points (3,20) and (1,5). x aby x y )2( 2 5 Going back to the general form, substitute in a and b The exponential equation passing through the points (3,20) and (1,5) is x y )2( 2 5
  16. 16. Let’s Try One  Find the exponential equation passing through the points (2,4) and (3,16). x aby a b2 4 3 2 4 16 b b 23 416 b 2 4 ab 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 a b2 4 Division property of exponents
  17. 17. Writing Exponential Equations 1 416 b 4b Simplify Solve for b 25.0 4 1 4 4 2 a Go back to the equation for a; substitute in b and solve for a x aby x y )4(25.0 Going back to the general form, substitute in a and b The exponential equation passing through the points (2,4) and (3,16) is x y )4(25.0
  18. 18. Putting it all together . . .  Find the equation of the exponential function that goes through (1,6) and (0,2). Graph the function.
  19. 19. 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.
  20. 20.  EX- a guy puts $1000 into a simple 3% interest account. What is the exponential equation? x y )03.1(1000 r = rate 3% (write as 0.03) b = 1 + r = 1.03 x = time a = amount put into the account ($1,000) x aby
  21. 21.  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 ) x y )2(1000 x aby b = r + 1, where r is the amount of increase. We are increasing by 100% each time something doubles, so r = 1
  22. 22.  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) x y )9.0(15000 x aby

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