ACT® · Math: Functions — Composition, Transformation, and Inverse Functions
Reflections Across the Axes
Putting a minus sign on the outside of a function flips its graph upside down (across the x-axis); putting it on the inside flips the graph sideways (across the y-axis). Those are two different moves and they usually give different pictures.
A reflection across the x-axis takes y = f(x) to y = -f(x). Every output gets negated, so a point that used to sit at height 5 now sits at height -5, same x-coordinate, mirrored vertically over the x-axis. A reflection across the y-axis takes y = f(x) to y = f(-x). This time the input gets negated before the function ever sees it, so the graph flips left-right: whatever happened at x = 3 now happens at x = -3.
These two moves attack different variables, which is why they almost never look the same. Take y = √x, which lives entirely in the first quadrant since x cannot be negative. The graph of y = -√x keeps the same domain, x ≥ 0, but drops every point straight down into the fourth quadrant. The graph of y = √(-x) keeps the same shape but needs -x ≥ 0, so it only exists for x ≤ 0 and sits in the second quadrant. Same parent function, two completely different regions of the plane, because one flip acts on outputs and the other acts on inputs.
Composition and transformation are really the same operation seen from two angles: reflecting is just composing f with the negation function, either before or after f runs. Do both reflections in a row, negate the output and negate the input, and the combined effect sends every point (p, q) to (-p, -q): a 180 degree rotation about the origin. For an odd function this double reflection gives back the exact same graph, which is the shortcut for recognizing odd symmetry, while even functions already match their own y-axis reflection.
Key terms
- Reflection across the x-axis
- The transformation y = -f(x), which negates every output and flips the graph vertically.
- Reflection across the y-axis
- The transformation y = f(-x), which negates every input and flips the graph horizontally.
- Odd function
- A function where f(-x) = -f(x), so it looks the same after a 180 degree rotation about the origin.
- Even function
- A function where f(-x) = f(x), so its graph is already symmetric across the y-axis.
- Rotation about the origin
- A move sending each point (p, q) to (-p, -q), equivalent to reflecting across both axes in a row.
Watch it explained
- How it worksHow does the negative of f(x) mirror every output across the x-axis point by point?
- Worked exampleDraw y = -√x beside y = √(-x) and show how the two pictures differ.
- Common mistakeWhy doesn't f(-x) give the same picture as the negative of f(x) for most functions?
- Going furtherWhat happens when a graph is reflected across both axes one after the other?
Check yourself
End of topic.