Function transformations such as reflections and shifts can be combined into a single equation. Learn how each part of g(x) = a times f(x minus h) plus k affects the graph, with a worked example showing a parabola reflected and shifted to a new vertex.
Combining more than one transformation
Function transformations — shifts, reflections, and stretches — can be applied together. When you combine them, the order matters, and it helps to track each one's effect on the vertex or a few key points.
Worked example
Start with the base function f(x) = x², whose vertex sits at the origin (0, 0). Now apply three transformations together: reflect over the x-axis, shift right 1 unit, and shift up 4 units. The result is g(x) = −(x − 1)² + 4.
f(x) = x² (dashed) and its combined transformation g(x) = −(x−1)²+4 (solid).
The reflection flips the parabola to open downward, the "−1" inside the parentheses shifts it right 1, and the "+4" shifts it up 4 — so the vertex moves from (0, 0) to (1, 4).