Day 109 - Exploring Exponential Models

What is an exponential equation? An exponential equation has the general form y = abx

Growth Factor, Decay Factor When 0 < b < 1, b is the decay factor Given the general form y = abx When b > 1, b is the growth factor

Growth or Decay??? Growth Decay Growth Decay Growth Decay

Graphs of Exponential Functions

In this example, the asymptote is the x axis. What is an asymptote? “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.

Graphing Exponential Functions Y=100(.5)x -3 -2 -1 1 2 3 Complete the table using the integers -3 through 3 for x.

Let’s graph one together X .5x Y=100(.5)x -3 .5-3 800 -2 .5-2 400 -1 .5-1 200 .50 100 1 .51 50 2 .52 25 3 .53 12.5

Let’s try one X .5x y=2(.5)x -3 .5-3 16 -2 .5-2 8 -1 .5-1 4 .50 2 1 .51 .52 0.5 3 .53 0.25 Complete the table using the integers -3 through 3 for x. Then graph the function.

Let’s try one X .5x y=2(.5)x -3 .5-3 16 -2 .5-2 8 -1 .5-1 4 .50 2 1 .51 .52 0.5 3 .53 0.25

Let’s try one X 10x y=5(10)x -3 -2 -1 1 2 3 Complete the table using the integers -3 through 3 for x. Then graph the function.

Let’s try one X 10x y=5(10)x -3 10-3 0.005 -2 10-2 0.05 -1 10-1 0.5 100 5 1 101 50 2 102 500 3 103 5000

Writing Exponential Equations Find the exponential equation passing through the points (3,20) and (1,5). 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 Division property of exponents

Writing Exponential Equations Find the exponential equation passing through the points (3,20) and (1,5). Simplify Solve for b Go back to the equation for a; substitute in b and solve for a

Writing Exponential Equations Find the exponential equation passing through the points (3,20) and (1,5). Going back to the general form, substitute in a and b The exponential equation passing through the points (3,20) and (1,5) is

Let’s Try One Find the exponential equation passing through the points (2,4) and (3,16). 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

Writing Exponential Equations Simplify Solve for b 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 the points (2,4) and (3,16) is

Putting it all together . . . Find the equation of the exponential function that goes through (1,6) and (0,2). Graph the function.

x 3x y=2(3)x -3 3-3 0.074 -2 3-2 0.22 -1 3-1 0.66 30 2 1 31 6 32 18 3 33 54

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.

EX- a guy puts $1000 into a simple 3% interest account 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)

a = the original number in the colony (1,000 bacteria ) 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 ) 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 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)