7.6 EXPONENTIAL FUNCTIONS:

1. Function Rule: An equation that describes a function. 7.6 EXPONENTIAL FUNCTIONS: Exponent: A number that shows repeated multiplication.

2. GOAL:

3. Definition: An EXPONENTIAL FUNCTION is a function of the form: Constant Base Exponent Where a ≠ 0, b > o, b ≠ 1, and x is a real number.

4. We must be able to identify exponential functions from given data values. IDENTIFYING: Ex:Does the table represent an exponential function? If so, provide the function rule.

5. To answer the question we must take a look at what is happening in the table. + 1 + 1 + 1 ×3 ×3 ×3 The starting point is -1 when x = 0 The independent variable x increases by 1 The dependent variable y is multiplied by 3

6. Taking the info to consideration, we can see that the equation for the problem is: + 1 + 1 + 1 y=a∙bx ×3 ×3 ×3 Here the difference of ×3 becomes the base. Notice: we begin with -1 when x = 0 or a = -1 y=a∙bx  y = -1∙3x

7. YOU TRY IT: Does the table represent an exponential function? If so, provide the function rule.

8. SOLUTION: Taking the info to consideration, we can see that the equation for the problem is: + 1 + 1 + 1 y=a∙bx ×4 ×4 ×4 Here the difference of ×4becomes the base. Notice: we begin with 2 when x = 1 or a = 1/2  y =½ ∙4x y=a∙bx

9. Summary: Linear Functions:  y = mx + b The difference in the independent variable (y) is in form of addition or subtraction. Exponential Equations:  y = abxThe difference in the independent variable (y) is multiplication

10. We must be able to evaluate exponential functions. EVALUATING: Ex:An investment of $5000 doubles in value every decade. Write a function and provide the worth of the investment after 30 years.

11. To provide the solution we must know the following formula: EVALUATING: A = P∙2x A = total P = Principal (starting amount) 2 = doubles x = time

12. SOLUTION: An investment of $5000 doubles in value every decade. Write a function and provide the worth of the investment after 30 years. Amount: unknown A = P∙2x $5000 Principal: A = 5000∙23 2 Doubles: A = 5000∙(8) 30 yrs(3 decades) Time (x): A = 40,000

13. YOU TRY IT: Suppose 30 flour beetles are left undisturbed in a warehouse bin. The beetle population doubles each week. Provide a function and the population after 56 days.

14. SOLUTION: Suppose 30 flour beetles are left undisturbed in a warehouse bin. The beetle population doubles each week. Provide a function and the population after 56 days. Amount: unknown A = P∙2x 30 Principal: A = 30∙28 2 Doubles: A = 30∙(256) 56 days (8 weeks) Time (x): A = 7,680

15. GRAPHING: To provide the graph of the equation we can go back to basics and create a table. Ex: What is the graph of y = 3∙2x?

16. GRAPHING: = -2 3∙2(-2) = 3∙2(-1) -1 = 3∙1 3 3∙2(0) 0 = 3∙2 6 3∙2(1) 1 = 3∙4 12 3∙2(2) 2

17. GRAPHING: -2 -1 3 0 6 1 12 2 This graph grows fast = Exponential Growth

18. YOU TRY IT: Ex: What is the graph of y = 3∙x?

19. GRAPHING: 3∙(-2) 12 -2 =3∙(2)2 3∙(-1) -1 =3∙(2)1 6 3∙(0) 0 3 = 3∙1 3∙(1) =3∙ 1 3∙(2) 2 =3∙

20. GRAPHING: -2 12 -1 6 3 0 1 2 This graph goes down = Exponential Decay

21. VIDEOS: Exponential Functions Growth https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/exponential-growth-functions Graphing https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/graphing-exponential-functions

22. VIDEOS: Exponential Functions Decay https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/word-problem-solving--exponential-growth-and-decay

23. CLASSWORK:Page 450-451: Problems: As many as needed to master the concept.