8 1 Exploring Exponential Models What is an

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 l When b > 1, b is the growth factor l When 0 < b < 1, b is the decay factor

Growth or Decay? ? ? Growth Decay

Graphs of Exponential Functions

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 X . 5 x Y=100(. 5)x -3 -2 -1 0 1 2 3 Complete the table using the integers -3 through 3 for x.

Let’s graph one together X . 5 x 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

Let’s try one X . 5 x 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.

Let’s try one X -3 -2 -1 0 1 2 3 . 5 x Y=2(. 5)x

Let’s try one X 10 x 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.

Let’s try one X -3 -2 -1 0 1 2 3 10 x Y=5(10)x

Writing Exponential Equations l 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 l 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 l 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 l 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. . . l Find the equation of the exponential function that goes through (1, 6) and (0, 2). Graph the function.

Modeling Growth with an Exponential Equation l 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.

l 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)

l 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

l 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)