1. Chapter 8: Exploring Exponential Models

2. What is an exponential equation? An exponential equation has the general form y=ab x

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

4. Growth or Decay??? Growth Decay Growth Decay Growth Decay

5. Graphs of Exponential Functions

6. 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.

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

8. Let’s graph one together X.5 x Y=100(.5) x -3.5 -3 800 -2.5 -2 400.5 -1 200 0.5 0 100 1.5 1 50 2.5 2 25 3.5 3 12.5

9. Let’s try one X.5 x Y=2(.5) x -3 -2 0 1 2 3 Complete the table using the integers -3 through 3 for x. Then graph the function.

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

11. Let’s try one X10 x Y=5(10) x -3 -2 0 1 2 3 Complete the table using the integers -3 through 3 for x. Then graph the function.

12. Let’s try one X10 x Y=5(10) x -3 -2 0 1 2 3

13. Writing Exponential Equations Find the exponential equation passing through the points (3,20) and (1,5). Step 1: Start with the equation in general form. Choose a point. Substitute for x and y using one of the points. In this example: (3, 20) Solve for a

14. Writing Exponential Equations Find the exponential equation passing through the points (3,20) and (1,5). Step 2: Substitute x and y using (1, 5) and replace “a” with the ratio found from step 1: Start with another equation in general form. Simplify. Start by using the division property of exponents

15. Writing Exponential Equations Find the exponential equation passing through the points (3,20) and (1,5). Step 2 continued: Simplifying and solving for b

16. Find the exponential equation passing through the points (3,20) and (1,5). Step 3: Go back to the equation from step 1 and solve for “a” by substituting in the value of “b”.

17. Writing Exponential Equations Find the exponential equation passing through the points (3,20) and (1,5). Going back to the general form, and write the new equation using the found values of “a” and “b”. The exponential equation passing through the points (3,20) and (1,5) is

18. 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

19. 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

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

23. 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.

24. 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)

25. 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

26. 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)