Vertex Form Calculator

Every parabola that opens up or down can be written in vertex form, y = a(x − h)² + k. The point (h, k) is the vertex, the single turning point. The line x = h is the axis of symmetry, and the number a controls how wide the parabola is and which way it opens.

Compare that with standard form, y = ax² + bx + c. Standard form tells you the y-intercept right away, because it is c, but the vertex is hidden. Vertex form flips this. The vertex is visible and the y-intercept takes a little arithmetic. OpenStax Algebra and Trigonometry describes vertex form as the version in which the vertex appears directly, and uses y = -3(x + 2)² + 4 as an example with h = -2, k = 4 and a downward opening.

The table below shows nine standard form quadratics converted to vertex form, so you can see the pattern before doing it yourself.

The method is completing the square. Take y = 2x² − 8x + 3. First factor the leading coefficient out of the x terms: y = 2(x² − 4x) + 3. Half of -4 is -2, and -2 squared is 4. Add and subtract 4 inside the bracket, remembering that the 4 you subtract inside is multiplied by 2 outside, so it costs 8: y = 2(x² − 4x + 4) − 8 + 3. The bracket is now a perfect square, (x − 2)², and the constants combine to -5. The result is y = 2(x − 2)² − 5.

The steps view of the calculator prints exactly these moves for whatever coefficients you enter, including fractions. If a is 1 it skips the factoring step, and it ends with a check that f(h) equals k.

The table lays out the same example one move at a time.

You do not need to complete the square to find the vertex. Two formulas do it. The x coordinate is h = -b / (2a). Put that x back into the equation to get the y coordinate, k = f(h), or use k = c - b² / (4a).

For y = 2x² − 6x + 7, h = 6 / 4 = 3/2. Then k = 2(3/2)² − 6(3/2) + 7 = 5/2. This matches the OpenStax Algebra and Trigonometry worked example for the same quadratic, which gives h = 3/2 and k = 5/2.

To find the vertex in a calculator like this one, choose standard form, enter a, b and c, and the vertex is the first two results. On a graphing calculator you would graph the function and use its minimum or maximum tool, which finds a decimal approximation. The formulas here are exact. The h formula also gives the axis of symmetry, x = h, because the parabola is a mirror image on either side of it.

To go back, expand the square. a(x − h)² + k becomes ax² − 2ah x + (a h² + k), so b = -2ah and c = a h² + k. For y = -2(x − 3)² + 5 that gives b = 12 and c = -13, so y = -2x² + 12x − 13.

Sometimes you know the vertex and one more point instead of a. Put the vertex into y = a(x − h)² + k, then substitute the point and solve for a. With vertex (2, -1) and point (4, 7) you get a = 2. The point cannot sit directly above or below the vertex, and it cannot be level with it, since the first is not a function and the second would force a = 0.

If you have three points on the parabola and no vertex, solve for a, b and c first. For (0, 3), (1, -3) and (2, -5), the constant is c = 3, then a + b = -6 and 4a + 2b = -8, so a = 2 and b = -8. Enter those into the standard form option to get the vertex form.

Once you have h, k and a, the rest of the parabola follows. The axis of symmetry is x = h. If a is positive the parabola opens upward and k is its minimum. If a is negative it opens downward and k is its maximum.

The y-intercept is c. The x-intercepts are where y = 0, which in vertex form means x = h ± √(-k / a). That square root exists only when -k / a is zero or positive, which matches the sign of the discriminant b² − 4ac. A positive discriminant gives two x-intercepts, zero gives one, and a negative value gives none that are real.

Symmetry is a good check. For y = 2(x − 2)² − 5, the points at x = 1 and x = 3 are the same distance from the axis, and both give y = -3. The tables on this page show the effect of a, h and k, the symmetry check, and the discriminant cases.

Pick what you have. Standard form needs a, b and c. Vertex form needs a, h and k. The third option needs a vertex and one more point. Every field accepts whole numbers, decimals and fractions such as 1/2 or -3/4.

The results show the vertex (h, k), the standard form coefficients a, b and c, the discriminant, and the x-intercepts when they exist. Below them, the solution view writes out the steps in plain text, and the button lets you switch to a result-only view. The solution always includes the vertex form, the axis of symmetry, whether the vertex is a maximum or minimum, the y-intercept and the roots.

This tool handles quadratics only, and it cannot take an arbitrary function. It is a calculator for algebra practice and checking homework: reading your own working against the printed steps is the quickest way to find where a sign went wrong.

The most common error is the sign of h. Vertex form is written (x − h), so (x + 3) means h = -3. The next is skipping the factor a. If a is not 1, factor it out of the x terms before halving, or the square you add is wrong by a factor of a. A third is forgetting that anything added inside a bracket multiplied by a must be balanced outside by a times that amount.

Check by expanding. Multiply your vertex form back out and compare with the original. For y = 2(x − 2)² − 5 you get 2(x² − 4x + 4) − 5 = 2x² − 8x + 3, which matches. You can also test one point: set x = 0 and the answer should equal c.

The table lists these mistakes and the fix for each.

Vertex form is one of several ways to reshape an expression so its structure is visible. When a quadratic factor shows up inside a larger problem, you may need to divide polynomials, split fractions into simpler pieces, or evaluate a polynomial at a point.

For dividing by a linear factor, use the synthetic division calculator. For longer divisors, the polynomial long division calculator shows each round. To break a rational expression into simpler fractions, the partial fraction decomposition calculator handles it with steps. If you only need a straight line through two points, the linear interpolation calculator finds the value in between.

The formulas in this guide agree with the OpenStax and Lamar University algebra notes listed in the references. Keep your own derivation next to the calculator's steps when you study, because exams ask you to show the method.