X-Axis Reflection Calculator

Example data table

Try these points in “Multiple points” mode to verify your setup.

Formula used

Reflection across the x-axis flips the vertical coordinate and keeps the horizontal coordinate unchanged.

• (x, y) → (x, −y)
• If y = f(x), then the reflection is y = −f(x).
• If an implicit relation is F(x, y) = 0, the reflection becomes F(x, −y) = 0.

How to use this calculator

1. Select a mode: point, list, segment, polygon, or equation.
2. Enter your coordinates or your function rule for y.
3. Choose decimal precision for tidy results.
4. Press “Reflect Across X-Axis” to compute the transformation.
5. Use the CSV or PDF buttons to save outputs.

X-axis reflection guide

1) What the transformation means

An x-axis reflection mirrors geometry across the horizontal axis. It changes “up” to “down” while keeping left–right placement fixed, so it is ideal for quick graph flips, coordinate checks, and verifying symmetry.

2) Core coordinate rule

For any point, the rule is (x, y) → (x, −y). The x-value stays identical, and the y-value changes sign. Example data: (2, 3)→(2, −3), (−4, −1)→(−4, 1), (0, 5)→(0, −5), and (7, 0)→(7, 0). Points on the x-axis are unchanged, so they act as “anchors” when you test results.

3) Working with point lists

When you paste many coordinates, apply the same rule row-by-row. If you are checking measured data, compare columns: the reflected y should be exactly the negative of the original y, even when x is a decimal like 1.25 or −0.6. Choosing a consistent precision helps you spot rounding issues and mismatched units.

4) Reflecting segments and polygons

For a line segment, reflect both endpoints to get the new segment. For polygons, reflect every vertex; lengths and angles remain the same, so the shape is congruent. The only “change” is orientation: a clockwise vertex order becomes counterclockwise after reflection, which matters in computational geometry and graphics.

5) Reflecting equations and graphs

For functions written as y = f(x), the reflected graph is y = −f(x). Keep parentheses: if f(x)=x^2−4x+1, then the reflection is y=−(x^2−4x+1). x-intercepts stay put because they occur where y=0. The y-intercept b flips to −b. For a line y=mx+b, slopes also flip sign: y=−mx−b. For a vertex form y=a(x−h)^2+k, the vertex moves from (h,k) to (h,−k) and “opens” the same way because a does not change.

6) Range, extrema, and inequalities

Range values invert: if the original outputs are between 2 and 9, the reflected outputs are between −9 and −2. Maxima become minima and vice versa. For piecewise rules, reflect each piece by negating the y-expression; for inequalities like y ≥ f(x), the reflected region becomes y ≤ −f(x).

7) Invariants and implicit relations

Distances are preserved because multiplying y by −1 does not change squared differences. Midpoints reflect the same way, and areas stay equal (unsigned area), although signed area changes sign. If you work with implicit relations F(x,y)=0, reflection corresponds to F(x,−y)=0, which also handles circles, ellipses, and other conics. For parametric curves (x(t), y(t)), the reflection is (x(t), −y(t)).

8) Matrix form and quick checks

In linear algebra, the reflection is a matrix transform: [x′; y′] = [[1,0],[0,−1]] [x; y]. It is its own inverse, meaning reflecting twice returns the original coordinates. Use quick checks: confirm x′=x for every row, confirm y′=−y, and test a known point like (3,2)→(3,−2). Then export your results as CSV or PDF for clean reporting and homework submission.

FAQs

1) What happens to the x-coordinate?

X stays the same. Only y changes sign, so (x, y) becomes (x, −y).

2) What if the point lies on the x-axis?

Nothing changes. If y = 0, then (x, 0) reflects to (x, 0), so the point remains on the axis.

3) How do I reflect a line y = mx + b?

Negate the entire expression: y = mx + b becomes y = −(mx + b) = −mx − b. Both the slope and y‑intercept change sign.

4) Do distances or areas change?

Lengths and unsigned areas stay the same, so the reflected shape is congruent. Orientation flips, meaning clockwise point order becomes counterclockwise.

5) How do I reflect an implicit equation like x² + y² = 25?

Replace y with −y: for F(x, y)=0 use F(x, −y)=0. For x^2 + y^2 = 25, the equation is unchanged because (−y)^2 = y^2.

6) Can I reflect a table of points at once?

Yes. Paste one coordinate pair per line, then the calculator reflects each row and builds a results table you can export.

7) Why should I keep parentheses in y = −(f(x))?

Without parentheses you might negate only the first term. Writing y = −(f(x)) ensures every term in f(x) is reflected correctly.