Algebra 2 · Unit 6 · Topic 6.1
Exponential Functions and Constant Ratios
For f(x) = Abˣ with A > 0, b > 0, and b ≠ 1, equal input increments produce equal output ratios.
Definition and explanation
For f(x) = Abˣ with A > 0, b > 0, and b ≠ 1, equal input increments produce equal output ratios. Growth has b > 1 and decay has 0 < b < 1; the unshifted real domain is all real numbers and range is positive numbers.
Worked example
A model gives 12, 18, and 27 at x = 0, 1, and 2. The ratio is 1.5, so f(x) = 12(1.5)ˣ fits these observations. It predicts f(3) = 40.5 if the same ratio continues.
Common mistake
A few matching data points do not prove an exponential law remains valid indefinitely.
Key takeaway
For f(x) = Abˣ with A > 0, b > 0, and b ≠ 1, equal input increments produce equal output ratios.
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 Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.