Reflecting Functions in MYP Extended Maths

What Is a Reflection?

A reflection flips a graph across an axis, producing a mirror image. Unlike translations, reflections change the position of points in a way that involves negation — you negate either the input or the output. There are two distinct reflections you need to master at Extended level.

Reflection in the x-axis: −f(x)

Applying a negative sign outside the function gives y = −f(x). This reflects the graph across the x-axis. Every point (x, y) becomes (x, −y). A maximum becomes a minimum and vice versa. The x-intercepts stay fixed because −f(x) = 0 wherever f(x) = 0.

Example: if f(x) = x², then −f(x) = −x² is a downward-opening parabola with vertex still at the origin.

Reflection in the y-axis: f(−x)

Replacing x with −x inside the function gives y = f(−x). This reflects the graph across the y-axis. Every point (x, y) becomes (−x, y). The y-intercept stays fixed because f(−0) = f(0).

Example: if f(x) = 2x + 1, then f(−x) = −2x + 1. The gradient reverses sign, and the line is reflected across the y-axis.

Symmetry and Reflections

Some functions are unchanged by a reflection — these have symmetry. An even function satisfies f(−x) = f(x) and is symmetric about the y-axis. An odd function satisfies f(−x) = −f(x) and has rotational symmetry about the origin. Recognising these properties is a mark of strong mathematical reasoning at Extended level.

Common Mistakes

• Confusing which axis the reflection is across — −f(x) is across the x-axis (y-values negate), f(−x) is across the y-axis (x-values negate).
• Moving x-intercepts when applying −f(x) — they do not move.
• Not checking whether the reflected graph still passes the vertical line test.

Linking to Other Transformations

Reflections are often combined with translations in multi-step problems (covered in Transformation Functions 4). Practise each reflection type in isolation before combining.