x y 4 ( 2 ) Growth x y 100 (. 07 ) Decay G RAPHS OF - PowerPoint PPT Presentation

D AY 109 - E XPLORING E XPONENTIAL M ODELS

W HAT IS AN EXPONENTIAL EQUATION ? An exponential equation has the general form y = ab x    where a 0 , b 0 and b 1

G ROWTH F ACTOR , D ECAY F ACTOR Given the general form y = ab x  When 0 < b < 1, b is the  When b > 1, b is the decay factor growth factor

G ROWTH OR D ECAY ???  x 10 ( 1 . 2 ) y Growth  x y 5 (. 9 ) Decay  x Growth y 50 ( 1 . 54 )  x y 5 . 2 (. 70 ) Decay  x y 4 ( 2 ) Growth  x y 100 (. 07 ) Decay

G RAPHS OF E XPONENTIAL F UNCTIONS  x 10 ( 2 ) y

W HAT IS AN ASYMPTOTE ?  x 10 ( 2 ) y “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.

G RAPHING E XPONENTIAL F UNCTIONS  x y 100 (. 5 ) .5 x Y=100(.5) x X -3 Complete the table using the -2 integers -3 through 3 for x. -1 0 1 2 3

L ET ’ S GRAPH ONE TOGETHER  x y 100 (. 5 ) .5 x Y=100(.5) x X .5 -3 -3 800 .5 -2 -2 400 -1 .5 -1 200 .5 0 0 100 .5 1 1 50 .5 2 2 25 3 .5 3 12.5

L ET ’ S TRY ONE X .5 x y=2(.5) x .5 -3 -3 16  x .5 -2 -2 8 y 2 (. 5 ) .5 -1 -1 4 .5 0 0 2 Complete the table using the .5 1 1 1 integers -3 through 3 for x. .5 2 2 0.5 Then graph the function. .5 3 3 0.25

L ET ’ S TRY ONE  x y 2 (. 5 ) X .5 x y=2(.5) x .5 -3 -3 16 .5 -2 -2 8 .5 -1 -1 4 .5 0 0 2 .5 1 1 1 .5 2 2 0.5 .5 3 3 0.25

L ET ’ S TRY ONE 10 x y=5(10) x X -3  x -2 y 5 ( 10 ) -1 0 Complete the table using the 1 integers -3 through 3 for x. 2 Then graph the function. 3

L ET ’ S TRY ONE  x y 5 ( 10 ) 10 x y=5(10) x X 10 -3 -3 0.005 10 -2 -2 0.05 10 -1 -1 0.5 10 0 0 5 10 1 1 50 10 2 2 500 10 3 3 5000

W RITING E XPONENTIAL E QUATIONS Find the exponential equation passing through the points (3,20) and (1,5). y  x ab Start with the general form.  3 20 ab Choose a point. Substitute for x and y using (3, 20) 20 b  a Solve for a 3 2 20 b   Substitute x and y using (1, 5) and a using a 1 5 b 3 3 b   1 3 Division property of exponents 5 20 b

W RITING E XPONENTIAL E QUATIONS  Find the exponential equation passing through the points (3,20) and (1,5).   2 5 20 b 20 Simplify  5 2 b 20   2 b 4 5 Solve for b  b 2 Go back to the equation for a; substitute 20 20 20 5  b    a in b and solve for a 3 3 2 8 2

W RITING E XPONENTIAL E QUATIONS  Find the exponential equation passing through the points (3,20) and (1,5). y  x ab Going back to the general form, substitute in a and b 5  x y ( 2 ) 2 The exponential equation passing through the points (3,20) and (1,5) is 5  x y ( 2 ) 2

L ET ’ S T RY O NE  Find the exponential equation passing through the points (2,4) and (3,16). y  x ab Start with the general form.  2 4 ab Choose a point. Substitute for x and y using (2, 4) 4  a Solve for a 2 b 4 4   a Substitute x and y using (3, 16) and a using 3 16 b 2 b 2 b   b 3 2 16 4 Division property of exponents

W RITING E XPONENTIAL E QUATIONS  1 16 4 b Simplify  b 4 Solve for b y  x ab Go back to the equation for a; substitute in 4 1 b and solve for a    a 0 . 25 2 4 4  x Going back to the general form, y 0 . 25 ( 4 ) substitute in a and b The exponential equation passing through  x the points (2,4) and (3,16) is y 0 . 25 ( 4 )

P UTTING IT ALL TOGETHER . . .  Find the equation of the exponential function that goes through (1,6) and (0,2). Graph the function.

  x y ab ( 1 , 6 )  1 6 ab 6   6 2 a  a 3 1 b  x y ab   x y ab ( 0 , 2 )  x y 2 ( 3 ) 6  0 2 ( ) b 1 b 6 6     2 b 3 1 b 2

 x 2 ( 3 ) y x 3 x y=2(3) x -3 3 -3 0.074 -2 3 -2 0.22 -1 3 -1 0.66 0 3 0 2 1 3 1 6 2 3 2 18 3 3 3 54

M ODELING G ROWTH WITH AN E XPONENTIAL E QUATION  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.

 EX- a guy puts $1000 into a simple 3% interest account. What is the exponential equation? y  x r = rate 3% (write as 0.03) ab b = 1 + r = 1.03 x = time  x y 1000 ( 1 . 03 ) a = amount put into the account ($1,000)

 EX – a colony of 1000 bacteria cells doubles every hour. What is the exponential equation? r = 1 (why not 2?) y  x ab b = r + 1 = 2 x = time (in hours)  x y 1000 ( 2 ) a = the original number in the colony (1,000 bacteria ) b = r + 1 , where r is the amount of increase. We are increasing by 100% each time something doubles, so r = 1

 EX- a $15000 car depreciates at 10% a year. What is the exponential equation? r = - 10% (the car is worth 10% y  x ab less each year) b = 1 - r = 1 – 0.1 = 0.9  x y 15000 ( 0 . 9 ) x = time (in years) a = amount put into the account ($15,000)