Algebra 2 · Unit 1 · Topic 1.2
Shifts, Reflections, and Scaling
The function g(x) = a f(x − h) + k shifts the graph of f right by h, scales its outputs by a, and shifts it up by k.
Definition and explanation
The function g(x) = a f(x − h) + k shifts the graph of f right by h, scales its outputs by a, and shifts it up by k. A negative a also reflects the graph across the x-axis before the vertical shift.
Worked example
Starting from f(x) = x², form g(x) = −3(x + 2)² + 5. The vertex moves to (−2, 5), and the graph opens downward with a vertical scale factor of 3. At x = −1, g(x) = 2.
Common mistake
The expression x + 2 inside a function shifts the graph left, not right.
Key takeaway
The function g(x) = a f(x − h) + k shifts the graph of f right by h, scales its outputs by a, and shifts it up by k.
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.