**Objectives**

- Determine if a table of data represents a linear or exponential function.
- Identify the common difference of a linear function.
- Identify the common ratio for an exponential function.
- Distinguish between exponential growth and decay.
- Interpret graphs of linear and exponential functions in a context.

**Key Terms**

- Sequence – An ordered pattern of numbers.
- Ex. Each term in the following sequence is 4 less than the previous term: 11, 7, 3, –1, –5, –9, –13, …

- Arithmetic Sequence – A sequence of numbers in which the difference between any two consecutive terms is the same (constant).
- That constant number is the common difference.

- Common Difference – The constant value between any two consecutive numbers in an arithmetic sequence.
- Linear Function – A term that describes a function in which the y-values form an arithmetic sequence.
- There is a common difference between each y-value.

- Geometric Sequence – A sequence of numbers in which the ratio of any two successive (consecutive) terms is equal.
- Common Ratio – The constant ratio between any two consecutive numbers in a geometric sequence.
- The change from one number to the next.

- Exponential Function – A term that describes a function in which the y-values form a geometric sequence.
- Exponential Decay – A situation in which a quantity decreases by a common ratio at regular intervals.
- Exponential Growth – A situation in which a quantity increases by a common ratio at regular intervals.

**Notes**

- Patterns & Sequences
- A pattern is something that repeats in a predictable manner.
- A sequence is a list of numbers in a specific order.
- A sequence is the result of a pattern.

- Arithmetic Sequences
- To find the common difference, subtract two numbers next to each other (consecutive numbers). Do this a few times to make sure the results are always the same value.
- Ex. 4, 7, 10, 13, 16, 19
- 7 – 4 = 3
- 10 – 7 = 3
- 13 – 10 = 3
- 16 – 13 = 3
- Result: 3 is the common difference!

- Add the common difference (same amount) to the last result to get the next result
- 19 + 3 = 22
- 22 + 3 = 25
- 25 + 3 = 28, and so on…

- To find the common difference, subtract two numbers next to each other (consecutive numbers). Do this a few times to make sure the results are always the same value.

- Geometric Sequences
- To find the common ratio, divide two numbers next to each other (consecutive numbers). Do this a few times to make sure the results are always the same value.
- Ex. 3, 9, 27, 81
- 9 ÷ 3 = 3
- 27 ÷ 9 = 3
- 81 ÷ 27 = 3

- Multiply the common ratio (usually a fraction) by the last result to get the next result
- 81 • 3 = 243
- 243 • 3 = 729, and so on…
- Note: 3 is the same as

- Negative Signs: Sometimes the sign will flip.
- Multiply by a negative to get a positive, then by a negative to get a negative, and so on…
- Ex. 2, -6, 18, – 54, 162
- The common ratio is

- When numbers (fractions) get smaller, it means you are multiplying by a fraction
- Ex.

- To find the common ratio, divide two numbers next to each other (consecutive numbers). Do this a few times to make sure the results are always the same value.