Function reflection calculator

Enter a formula such as f(x) = x^2 - 4x + 3, choose the x-axis or y-axis, and receive the reflected rule together with the coordinate mapping. You can also supply key points to see each original coordinate paired with its new location, making the symbolic change and the graphical movement easy to check together.

Reflecting a function across the x-axis

An x-axis reflection reverses every output of a function. If the original graph contains the point (x, y), its reflected graph contains (x, -y). The horizontal coordinate does not move, while the vertical coordinate changes sign. Symbolically, a rule y = f(x) becomes y = -f(x). The calculator displays this by placing a negative sign around the complete original expression, which matters when the rule has several terms. For example, reflecting f(x) = x^2 - 4x + 3 produces f(x) = -(x^2 - 4x + 3), not merely -x^2 - 4x + 3. The graph is flipped vertically: peaks become equally distant valleys, positive y-intercepts become negative y-intercepts, and every zero stays fixed because points on the x-axis already have y = 0. Supplying key points makes the movement explicit. A vertex at (2, -1) maps to (2, 1), while an intercept at (1, 0) remains (1, 0). This is useful for checking a sketch and for avoiding sign errors when expanding the transformed expression later.

Reflecting a function across the y-axis

A y-axis reflection reverses every input coordinate instead of every output. Each point (x, y) moves to (-x, y), so the graph travels horizontally while retaining its height. In function notation, y = f(x) becomes y = f(-x). The calculator performs that substitution everywhere the standalone variable x occurs and wraps the replacement in parentheses. Thus f(x) = x^3 + 2x becomes f(x) = (-x)^3 + 2(-x). Keeping the substitution visible is often more informative than immediately simplifying it because it shows precisely why each odd-powered term changes sign and each even-powered term does not. Points on the y-axis stay fixed because their x-coordinate is zero. Other features trade left and right positions: a local maximum at (-3, 5) appears at (3, 5), and an x-intercept at 4 moves to -4. This transformation is also a direct way to investigate symmetry. If substituting -x leaves a rule equivalent to the original, the function is even and its graph already has y-axis symmetry, so the reflected graph overlaps it.

Reading and checking the result

The result keeps three ideas separate: the transformed function rule, the general algebraic transformation, and the point-coordinate rule. Read the transformed rule as the formula you would graph. Use the transformation line to remember whether negation belongs outside the function, as in -f(x), or inside its input, as in f(-x). Then use the point rule to verify individual landmarks. Optional key points are returned as paired from-and-to coordinates, so a vertex, intercept, endpoint, or turning point can be checked without mental bookkeeping. The calculator treats the entered expression symbolically; it does not try to expand, factor, or numerically evaluate it. That preserves the structure you supplied and avoids changing an exact rule into an approximation. Standard algebraic text with x, numbers, operators, parentheses, and named functions is accepted, while unsupported characters and unbalanced parentheses are rejected. For reliable work, enter an explicit multiplication sign when ambiguity is possible, confirm that the listed points really belong to the original graph, and remember that reflecting a graph does not change distances or overall shape. It changes orientation and location only, according to the selected axis.