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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

Alg1A 3.8 - Linear Exp Graphs Alg1A 3.8 - Linear Exp Graphs2

 


  • 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}

Alg1A 3.8 - Exp Growth Decay

 


Alg1A 3.8 - Linear Exponential

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