8.1 Exploring Exponential Models

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

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

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 -2 -1 1 2 3 1 2 3 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 -2 -1 1 2 3

Let’s try one X 10x Y=5(10)x -3 -2 -1 1 2 3 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 -2 -1 1 2 3

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.

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.

Real-World Situation The population of the United States in 1994 was about 260 million with an average annual rate of increase about 0.7%. Find the growth factor for that year. Suppose the rate of growth continued to be 0.7%. Write a function to model this population growth.

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)

EX – a colony of 1000 bacteria cells doubles every hour 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)