Answers:
- Have a common difference
- Have a common factor
- Have a linear first difference
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Explanation:
Linear functions, such as y = 2x+5, have a common difference. In that example, the common difference would be 2 since we add 2 to each term to get the next one. Consider when x = 1 and that leads to y = 7. Then when x = 2, we have y = 9. Then when x = 3, we have y = 11, and so on. The sequence of y values {7,9,11,...} shows we add 2 each time.
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Exponential functions have us multiply the previous term by some common factor to get the next term. A sequence like {2,6,18,...} has us multiply by 3 each time and the exponential function modeling this sequence is y = 2(3)^(x-1) where x is a natural number. So this is an example showing that exponential functions have a common factor, or we can call this a common ratio.
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Quadratic functions are a bit strange.
Let's consider the quadratic of y = x^2. The sequence of y values it generates is {1,4,9,16,25...} for positive integers x = 1, 2, 3, 4, 5....
Let's subtract the adjacent terms (larger - smaller)
- 4-1 = 3
- 9-4 = 5
- 16-9 = 7
- 25-16 = 9
Note the results of 3,5,7,9 are in a linear sequence (we're adding 2 to each increment value, so the next increment would be +11). We can consider this a linear first difference.
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For any of these functions, I'm restricting x to be an integer and x = 1 is the smallest input possible.
