Reflections

A reflection in the x-axis is represented by the transformation $y = -f(x)$, where the negative sign is applied to the entire output of the function.

This transformation affects the vertical position of points; every point $(x, y)$ on the original graph is mapped to $(x, -y)$ on the reflected graph.

Geometrically, the graph is flipped vertically. Points above the x-axis move below it, and points below the x-axis move above it.

Any point where the graph intersects the x-axis (the roots or x-intercepts) remains unchanged because the y-coordinate is zero, and $-0 = 0$.

A reflection in the y-axis is represented by the transformation $y = f(-x)$, where the negative sign is applied directly to the input variable.

This transformation affects the horizontal position of points; every point $(x, y)$ on the original graph is mapped to $(-x, y)$ on the reflected graph.

Geometrically, the graph is flipped horizontally. The right side of the graph moves to the left, and the left side moves to the right.

Any point where the graph intersects the y-axis (the y-intercept) is an invariant point because the x-coordinate is zero, and $-0 = 0$.

Asymptotes are treated as part of the graph and must be reflected according to the same rules as the function itself.

For a reflection in the x-axis ($y = -f(x)$), horizontal asymptotes change sign (e.g., $y = k$ becomes $y = -k$), while vertical asymptotes remain unaffected.

For a reflection in the y-axis ($y = f(-x)$), vertical asymptotes change sign (e.g., $x = h$ becomes $x = -h$), while horizontal asymptotes remain unaffected.

If an asymptote lies exactly on the axis of reflection (e.g., the x-axis itself during an x-axis reflection), its position does not change.