# 3.8 – Linear & Exponential Growth

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…

• 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 $\frac{3}{1}$
• 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 $\frac{-3}{1}$
• When numbers (fractions) get smaller, it means you are multiplying by a fraction
• Ex. $4, 2, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}$