Graph Transformations: Reflections

Statement

Graph transformations describe how the graph of a function moves or changes when it is reflected, translated, or stretched. Reflections are one of the most common transformations and involve flipping a graph across an axis.

\[ y = -f(x) \quad \text{reflects the graph in the x-axis} \]

\[ y = f(-x) \quad \text{reflects the graph in the y-axis} \]

Why it’s true

• If you replace \(f(x)\) with \(-f(x)\), every output value (y-value) is multiplied by \(-1\). This inverts the graph vertically, so all positive values become negative and vice versa. This is reflection in the x-axis.
• If you replace \(x\) with \(-x\), every input value is reversed. This means points that were to the right of the y-axis move to the left, and vice versa. This is reflection in the y-axis.

Recipe (how to use it)

1. Start with the original graph \(y = f(x)\).
2. For reflection in the x-axis, replace \(f(x)\) with \(-f(x)\). Flip every point across the x-axis.
3. For reflection in the y-axis, replace every \(x\) with \(-x\). Flip every point across the y-axis.
4. Sketch the new graph by plotting some key points and applying the transformation.

Spotting it

Whenever you see a minus sign outside the function (e.g. \(-f(x)\)), it reflects in the x-axis. Whenever you see a minus sign inside the function (e.g. \(f(-x)\)), it reflects in the y-axis.

Common pairings

• Translations (shifts left/right, up/down).
• Stretches (scaling the graph vertically or horizontally).
• Combinations of multiple transformations, where reflection is one step.

Mini examples

1. Given: \(y = f(x)\) passes through (2,3). Reflect in x-axis: New point is (2,-3).
2. Given: \(y = f(x)\) passes through (-4,1). Reflect in y-axis: New point is (4,1).

Pitfalls

• Mixing up axes: A minus outside flips vertically (x-axis), while a minus inside flips horizontally (y-axis).
• Forgetting both signs: \(y = -f(-x)\) combines both reflections, which is the same as a 180° rotation around the origin.
• Not testing key points: Always check by substituting a few values of x.

Exam strategy

• Write out the transformation clearly before sketching.
• Mark a few easy points and reflect them first to get the new shape.
• Remember: minus outside = flip vertically, minus inside = flip horizontally.

Summary

Reflections are simple but powerful transformations. The rule is: minus outside the function (\(-f(x)\)) reflects in the x-axis, and minus inside (\(f(-x)\)) reflects in the y-axis. Mastering this makes sketching transformed graphs much quicker and more reliable.