The Parabola Calculator finds the vertex, axis of symmetry, focus and directrix from three coefficients. It supports a vertical form based on y = ax² + bx + c and a horizontal counterpart based on x = ay² + by + c.
These features describe the geometry of the parabola. The value of a must be nonzero; if a were zero, the expression would be linear and would not have a parabola’s focus or directrix.
For y = ax² + bx + c, h = −b ÷ (2a) and k = c − b² ÷ (4a). The vertex is (h, k), and p = 1 ÷ (4a).
For a vertical parabola, focus = (h, k + p), directrix is y = k − p, and the axis is x = h.
For x = ay² + by + c, the calculator uses vertex (k, h), focus (k + p, h), directrix x = k − p, and axis y = h.
| Term | Meaning on this page |
|---|---|
| Orientation | Vertical for y as a quadratic in x; horizontal for x as a quadratic in y. |
| a | Nonzero quadratic coefficient. Its sign controls opening direction, and its magnitude affects width. |
| b | Linear coefficient, including its sign. |
| c | Constant term. |
| h and k | Intermediate coordinates obtained by completing the square. |
| p | Signed vertex-to-focus distance, equal to 1/(4a) in these coordinate forms. |
For the vertical parabola y = x² − 4x + 3, a = 1, b = −4 and c = 3. Then h = −(−4)/(2 × 1) = 2.
Next, k = 3 − (−4)²/(4 × 1) = 3 − 4 = −1, and p = 1/(4 × 1) = 0.25.
The vertex is (2, −1), the axis is x = 2, the focus is (2, −0.75), and the directrix is y = −1.25. Substituting x = 2 into the original equation confirms the vertex height of −1.
The vertex is the turning point. A positive a makes a vertical parabola open upward and a horizontal one open right; a negative a reverses those directions. The axis passes through the vertex and focus and divides the curve into mirror halves.
The focus and directrix provide an equivalent geometric definition: every point on the parabola is equally distant from the focus and directrix. The signed p value positions both features on opposite sides of the vertex. A larger absolute a gives a smaller absolute p and a narrower graph in the selected form.
This page does not list x-intercepts or y-intercepts as separate results. For a vertical equation, c is the y-intercept, and real x-intercepts depend on the discriminant b² − 4ac. Keep more than six decimal places in further work when rounded output could affect a sensitive construction.
You can check the vertex without relying on the displayed geometry. Substitute the reported coordinate into the original equation, and test equal offsets on both sides of the symmetry axis. For a vertical parabola, x values h − r and h + r should produce the same y value, apart from rounding.
Changing c shifts the graph along its output direction without changing a or b, while changing b also moves the symmetry axis. Comparing one coefficient at a time can make the formulas easier to understand. Always retain the chosen orientation when describing that movement, since horizontal and vertical forms exchange the coordinate roles.
It treats the coefficients as y = ax² + bx + c. The squared variable is x, so the symmetry axis is a vertical line written x = h.
It treats the coefficients as x = ay² + by + c. The squared variable is y, so the symmetry axis is a horizontal line written y = h.
When a equals zero, there is no squared term. The expression describes a line rather than a parabola, and the focus formula would divide by zero.
Positive a opens a vertical graph upward or a horizontal graph rightward. Negative a opens it downward or leftward.
It is the signed distance from the vertex to the focus along the symmetry axis. The directrix lies the same distance from the vertex on the opposite side.
The page rounds displayed coordinates to six decimal places. Compare unrounded algebra, signs and orientation before deciding that two close results disagree.
No. It reports geometric features. For a vertical quadratic, roots can be found separately with factoring, completing the square or the quadratic formula when appropriate.