A reflection flips a graph across a line, like a mirror. The equation y = −f(x) reflects the graph in the x-axis, and y = f(−x) reflects it in the y-axis. The shape stays the same, but every point changes position.
This lesson follows translating a simple graph in graphs and transformations. Together the two give you the core moves.
How do you tell the two reflections apart?
Ask which coordinate the minus sign touches.
Minus on the whole function: y = −f(x). The output is negated, so each y-value changes sign. The graph turns upside down across the x-axis.
Minus on the input: y = f(−x). The input is negated, so each x-value changes sign. The graph turns sideways across the y-axis.
A good habit is to pick one point on the original graph and work out where it lands. One test point settles the direction.
Worked example
Let f(x) = x² + 2x. Find the equations and the vertices of y = −f(x) and y = f(−x).
Step 1, original: f(x) = x² + 2x = (x + 1)² − 1. The vertex is (−1, −1). The roots are x = 0 and x = −2, because x(x + 2) = 0.
Step 2, y = −f(x): y = −x² − 2x. Every y changes sign, so the vertex is (−1, 1). The roots stay at 0 and −2. The curve now opens downward.
Step 3, y = f(−x): replace x with −x: y = (−x)² + 2(−x) = x² − 2x. Every x changes sign, so the vertex is (1, −1). The roots become x = 0 and x = 2, because x(x − 2) = 0.
Step 4, check with a point: on the original, f(1) = 1 + 2 = 3. On y = −f(x), at x = 1 the value is −3. On y = f(−x), the value 3 appears at x = −1, since (−1)² − 2(−1) = 3.
The mistake to watch for
A common slip is to think that f(−x) flips the graph upside down.
Mistaken answer: “y = f(−x) is a reflection in the x-axis, because there is a minus sign.”
The student noticed a minus and chose the axis by feel.
Test a point to correct it. If (1, 3) is on y = f(x), then for y = f(−x) the output 3 is now reached at x = −1, giving (−1, 3). The height stays 3 and only x changed sign, which is a reflection in the y-axis.
Check yourself
Try these without a calculator, then open each answer.
1. The point (3, 5) is on y = f(x). Where is the corresponding point on y = −f(x) and on y = f(−x)?
Show answer
On y = −f(x), the y-value changes sign: (3, −5). On y = f(−x), the x-value changes sign: (−3, 5).
2. The graph of y = x² − 4 is reflected in the x-axis. Give the new equation and the y-intercept.
Show answer
New equation: y = −(x² − 4) = y = 4 − x². The y-intercept is 4, so (0, 4). The original intercept was −4.
3. The line y = 3x − 6 is reflected in the y-axis. Find the equation and check it.
Show answer
Replace x with −x: y = 3(−x) − 6 = y = −3x − 6. Check: the original x-intercept (2, 0) becomes (−2, 0), and −3 × (−2) − 6 = 0. The y-intercept stays at −6.
Where this leads next
Now try both moves together in the graphs and transformations practice set, and move on to using intersections to approximate a solution. The quadratic structure explorer shows a quadratic and its mirror image side by side, and the graph model explorer gives extra practice with other shapes.
Some students understand reflections but lose track once a translation is added as well. That is the kind of step-by-step reasoning our teachers look at in online one-to-one Mathematics tuition.