Algebra 2 · Unit 6 · Topic 6.2
Exponential Graph Transformations and Asymptotes
In g(x) = Ab^(x − h) + k with nonzero A and a valid exponential base, y = k is the horizontal asymptote.
Definition and explanation
In g(x) = Ab^(x − h) + k with nonzero A and a valid exponential base, y = k is the horizontal asymptote. The sign of A determines whether outputs lie above or below k; h changes the horizontal placement.
Worked example
For g(x) = −2(3^(x − 1)) + 5, the horizontal asymptote is y = 5 and every output is below 5. At x = 1 the value is 3. As x grows, the output decreases without bound.
Common mistake
A vertical shift changes the horizontal asymptote; it is not always the x-axis.
Key takeaway
In g(x) = Ab^(x − h) + k with nonzero A and a valid exponential base, y = k is the horizontal asymptote.
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.