Algebra 1 · Unit 10 · Topic 10.5
Comparing Linear, Quadratic, and Exponential Patterns
For equally spaced inputs, constant first differences suggest a linear pattern, constant nonzero second differences suggest a quadratic pattern, and constant ratios of nonzero outputs suggest an exponential pattern.
Definition and explanation
For equally spaced inputs, constant first differences suggest a linear pattern, constant nonzero second differences suggest a quadratic pattern, and constant ratios of nonzero outputs suggest an exponential pattern. A finite pattern alone does not establish a unique law beyond the observations.
Worked example
At x = 0, 1, 2, 3, outputs 2, 5, 8, 11 have first differences 3 and fit y = 3x + 2. Outputs 2, 4, 8, 16 have ratio 2 and fit y = 2(2ˣ). At x = 4 these models predict 14 and 32 respectively, if their assumptions remain valid.
Common mistake
Equal input spacing is necessary for these difference and ratio checks; predictions beyond the data require justification.
Key takeaway
For equally spaced inputs, constant first differences suggest a linear pattern, constant nonzero second differences suggest a quadratic pattern, and constant ratios of nonzero outputs suggest an exponential pattern.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Elementary Algebra 2e and Algebra and Trigonometry 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.