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Combining Transformations of Functions

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).

Reading the transformations from the equation

In g(x) = a × f(x − h) + k: a negative sign reflects and/or stretches, h shifts horizontally (opposite of its sign), and k shifts vertically. This builds on single transformations like reflection across the x-axis, reflection across the y-axis, and horizontal translations, applied all at once.

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Algebra