Unit 3 Linear and Exponential Functions

1. Unit 3 Linear and Exponential Functions Lesson Linear and Exponential Number Patterns

2. Sequences A set of numbers written in a specific order or pattern Examples 2, 4, 6, 8 -- finite sequence (only 4 terms) 0, 5, 10, 15, … -- infinite sequence (infinite # of terms)

3. Terminology/Notation Term – a number in a sequence -- represents the first term in a sequence (1 is the counter) -- the nth term in a sequence or the formula -- the term before the nth term

4. Sequence Formulas Ex. For the sequence formula, , solve the first 5 terms.

5. Consider the series 4, 8, 12, 16, 20, 24, … The third term in the series, a3, is 12. What is the first term? What is the sixth term? What is the seventh term?

6. In the previous example, we came up with the seventh term by looking at the pattern and applying it to the next term in the sequence. But what if we are asked to find the 150th term? Today, we are going to look at 2 ways to write a rule (an equation) for finding the nth term in a series: closed formula and recursive formula.

7. Closed or Explicit Formula With a closed formula, we do not need to know what the previous terms are in order to calculate the next term. Let’s practice: Find the 150th term: an = n2 – 4

8. Linear Sequences You’ve already been using “closed formula” sequences, for example linear functions: y= 2x - 2 graphs the linear function, an = 2n - 2 graphs linear sequence dots.

9. Ex. an = 2n – 2 Chart Graph

10. Linear Sequences= Arithmetic Sequences For linear sequences, you add the same amount each time Arithmetic Sequence Explicit Formula or

11. 1, 3, 5, 7, 9, … The closed formula involves a linear function with slope of _________ and a y-intercept (0 term) of ___________. Answer:

12. Let’s look at another example:4, 8, 12, 16, 20, … The slope is _______________. The y-intercept is _____________. Answer:

13. Example Write the explicit formula for: -22, -2, 18, 38 …

14. Example Solve the explicit formula for an arithmetic sequence with and d = -10.

15. Example Solve the explicit formula if

16. 1, 3, 9, 27, 81, … What is the sequence pattern? Chart Graph

17. 1, 3, 9, 27, 81, … Exponential Closed Sequence We can say the exponential function’s ratio (or base) is ______. The function’s 1st term or is _________. Answer:

18. Exponential Sequences = Geometric Sequences If r is between 0 and 1 --- decays If r > 1 --- grows

19. Recursive Formula What does recursive mean? The dictionary defines recursive as pertaining to or using a rule or procedure that can be applied repeatedly. So, to simplify things, when you see recursive, I want you to think repeat.

20. Example of a recursive formula a1 = 4 an = an-1 + 4 Notice that the formula has 2 parts: • It defines the first term. In the above example, a1 = 4. • It defines the remaining terms. In our example, an=an-1 + 4. What does an-1 mean???

21. Find the first six terms of the recursive sequence a1 = 3 an = an-1 + 5 a1 = 3 a2 = 3 + 5 = 8 a3 = 8 + 5 = 13 a4 = 13 + 5 = 18 a5 = 18 + 5 = 24 a6 = 24 + 5 = 29 Repeat Repeat Repeat Repeat

22. If you wanted to find the 50th term, you would have to repeat your calculations 50 times. Can you see why we associate recursive with repeat?

23. Just to summarize A closed formula has only one equation. You can easily find any term in the sequence. ***** A recursive formula has 2 parts: the first term, and a rule, or equation, for finding the remaining terms based on knowing the previous term. In order to find the 50th term, you will first have to find the previous 49 terms.

24. 1, 3, 5, 7, 9, … In the series above, we notice that each term in the series is just 2 more than the term before it. We can say that a1 = 1 a2 = a1 + 2 a3 = a2 + 2 a4 = a3 + 2 Instead of writing a2, a3, a4, etc, we can shorten it to: an = an-1 + 2

25. 1, 3, 5, 7, 9, … So our recursive formula is: a1 = 1 an = an-1 + 2 Notice our formula has both parts: a1 and an

26. You try… Write the closed linear function and the recursive formula for the following sequences: • 0, 5, 10, 15, 20, … • 3, 10, 17, 24, … • 1, 11, 21, 31, … 4) 2, 4, 6, 8, …

27. Let’s write the closed formula for the four sequences we just defined. Pay attention to the pattern… Problem 1: 0, 5, 10, 15, 20, … Recursive: a1 = 0 an = an-1 + 5 Closed: an = 5n - 5 Problem 2: 3, 10, 17, 24, … Recursive: a1 = 3 an = an-1 + 7 Closed: an = 7n - 4

28. Can you see the pattern??? Problem 3: 1, 11, 21, 31, … Recursive: a1 = 1 an = an-1 + 10 Closed: an = 10(n-1) + 1 Problem 4: 2, 4, 6, 8, … Recursive: a1 = 2 an = an-1 + 2 Closed: an = 2(n-1) + 2

29. 1, 3, 9, 27, 81, … In the series above, we notice that each term in the series is 3 times the term before it. We can say that a1 = 1 a2 = 3a1 a3 = 3a2 a4 = 3a3 Instead of writing a2, a3, a4, etc, we can shorten it to: an = 3an-1

30. 1, 3, 9, 27, 81, … So our recursive formula is: a1 = 1 an = 3an-1 Notice our formula has both parts: a1 and an

31. You try… Write the closed and recursive formula for the following sequences: • 1, 5, 25, 125, … • 3, 12, 48, 192, … • 4, 24, 144, 864, … 4) 2, 20, 200, 2000, …

32. Let’s write the closed formula for the four sequences we just defined. Pay attention to the pattern… Problem 1: 1, 5, 25, 125, … Recursive: a1 = 1 an = 5an-1 Closed: an = 5(n-1) Check: a4 = 53 = 125 Problem 2: 3, 12, 48, 192, … Recursive: a1 = 3 an = 4an-1 Closed: an = 3*4n-1 Check: a4 = 3(4)3 = 192

33. Let’s write the closed formula for the four sequences we just defined. Pay attention to the pattern… Problem 3: 4, 24, 144, 864, … Recursive: a1 = 4 an = 6an-1 Closed: an = 4*6(n-1) Check: a3 = 4(6)2 = 144 Problem 4: 2, 20, 200, 2000, … Recursive: a1 = 2 an = 10an-1 Closed: an = 2*10n-1 Check: a4 = 2(10)3 = 2000