Exponential Functions and Their Graphs

Algebra Foundation

Algebra 1

Study expressions, equations, inequalities, functions, linear models, systems, polynomials, quadratics, and exponential growth with original explanations and worked examples.

Learn with original explanations and worked examples. OpenStax Elementary Algebra 2e was consulted for topic coverage.

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Algebra 1 · Unit 10 · Topic 10.3

Exponential Functions and Their Graphs

An exponential function has the form f(x) = Abˣ with A ≠ 0, b > 0, and b ≠ 1.

Definition and explanation

An exponential function has the form f(x) = Abˣ with A ≠ 0, b > 0, and b ≠ 1. For A > 0 it grows when b > 1 and decays when 0 < b < 1; its unshifted graph approaches y = 0 without reaching it.

Worked example

For f(x) = 5(2ˣ), values at x = −1, 0, 1, and 2 are 2.5, 5, 10, and 20. Each increase of 1 in x doubles the output, and the y-intercept is (0, 5).

Common mistake

An exponential changes by a constant ratio over equal input steps, not by a constant difference.

Key takeaway

An exponential function has the form f(x) = Abˣ with A ≠ 0, b > 0, and b ≠ 1.

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.