Vertex Form Calculator
For y = 2x^2 + 8x + 3, the vertex form is 2(x + 2)^2 - 5, with a turning point at (-2, -5). This vertex form calculator converts any quadratic written in standard form, y = ax^2 + bx + c, into vertex form, y = a(x - h)^2 + k. Enter the three coefficients a, b, and c, and it returns h and k, the finished vertex form expression, the axis of symmetry, and whether the parabola opens upward or downward. This tool is a companion to the quadratic formula calculator, and the two answer different questions from the same a, b, and c. The quadratic formula finds the roots, meaning where the parabola crosses the x-axis, or where y equals zero. The vertex form calculator instead finds the parabola's turning point, the single highest or lowest point on the curve, along with the vertical line that splits the parabola into two mirror-image halves. A quadratic can have zero, one, or two real roots, but it always has exactly one vertex, so this calculator always returns an answer as long as a is not zero. Vertex form is the version of a quadratic that graphs fastest by hand, since the vertex (h, k) can be plotted directly without solving anything further, and the sign of a tells you immediately whether that point is a minimum or a maximum.
Quick answer
Standard form is y = ax^2 + bx + c. Vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex.
What this tells you
- •Standard form is y = ax^2 + bx + c. Vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex.
- •The horizontal shift h equals negative b divided by 2a. A positive b produces a negative h, which is a common sign trap.
- •The vertical shift k equals c minus b squared divided by 4a, which is the same number you get by plugging h back into the original equation.
- •The leading coefficient a does not change between standard form and vertex form. It multiplies the squared term in both versions.
- •If a is positive, the parabola opens upward and the vertex is a minimum. If a is negative, it opens downward and the vertex is a maximum.
- •The axis of symmetry is the vertical line x = h, passing straight through the vertex.
- •When b is zero, there is no horizontal shift, so h is 0 and the vertex sits directly on the y-axis.
How to Use
- 1Enter the coefficient a, the number multiplying x squared. It cannot be zero, or the equation stops being quadratic.
- 2Enter the coefficient b, the number multiplying x. Enter 0 if there is no linear term.
- 3Enter the constant term c.
- 4Click Calculate to get the vertex form expression, the vertex point (h, k), the axis of symmetry, and the direction the parabola opens.
- 5Compare h and k against your own scratch work if you are checking homework. Expanding the returned vertex form should multiply back out to your original a, b, and c.
How It Works
Formula
y = a(x - h)^2 + k, where h = -b / (2a) and k = c - b^2 / (4a)Vertex form rewrites a quadratic so its turning point (h, k) is visible directly in the expression. The horizontal coordinate h comes from h = -b / (2a), which locates the axis of symmetry between the parabola's two branches. The vertical coordinate k is the y-value at that axis, computed as k = c - b^2 / (4a). An equivalent way to find k is to substitute h back into the original y = ax^2 + bx + c and simplify, which returns the same number and is a useful way to double check the result by hand. The coefficient a is copied unchanged from standard form into vertex form, since it controls how steeply the parabola opens in both versions.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Upward-opening parabola: y = 2x^2 + 8x + 3
h = -8 / (2 x 2) = -2, and k = 3 - 64 / 8 = 3 - 8 = -5. Since h is negative, the term reads (x + 2) rather than (x - 2). Expanding 2(x + 2)^2 - 5 gives 2x^2 + 8x + 8 - 5 = 2x^2 + 8x + 3, which matches the original equation. Because a is positive, the vertex (-2, -5) is the minimum point of the parabola.
Downward-opening parabola: y = -x^2 + 4x + 1
h = -4 / (2 x -1) = 2, and k = 1 - 16 / -4 = 1 + 4 = 5. Since h is positive, the term reads (x - 2). Expanding -(x - 2)^2 + 5 gives -(x^2 - 4x + 4) + 5 = -x^2 + 4x - 4 + 5 = -x^2 + 4x + 1, which matches. Because a is negative, the vertex (2, 5) is the maximum point.
No horizontal shift: y = x^2 - 9
With b = 0, h = -0 / 2 = 0, so there is no horizontal shift and the (x - h) term simplifies away entirely, leaving x^2 - 9. k equals c directly in this case, since the b^2 / (4a) term also drops to zero when b is 0. The vertex sits at (0, -9), directly on the y-axis, which is a common edge case students run into when a quadratic has no linear term.
Fractional coefficient: y = 0.5x^2 - 3x + 4
h = -(-3) / (2 x 0.5) = 3 / 1 = 3, and k = 4 - 9 / 2 = 4 - 4.5 = -0.5. Since h is positive, the term reads (x - 3), and since k is negative, the trailing term reads minus 0.5. This example shows the calculator working correctly with a fractional leading coefficient and a negative b, both of which are common in textbook problems.
Standard Form to Vertex Form Conversions
A quick-reference set of common quadratics converted to vertex form, useful for checking your own algebra or spotting a pattern before you plug in your own numbers.
| Standard form (a, b, c) | h | k | Vertex form | Vertex | Opens |
|---|---|---|---|---|---|
| 1, 0, 0 | 0 | 0 | x^2 | (0, 0) | Upward |
| 1, -2, 1 | 1 | 0 | (x - 1)^2 | (1, 0) | Upward |
| 2, 8, 3 | -2 | -5 | 2(x + 2)^2 - 5 | (-2, -5) | Upward |
| -1, 4, 1 | 2 | 5 | -(x - 2)^2 + 5 | (2, 5) | Downward |
| 1, 0, -9 | 0 | -9 | x^2 - 9 | (0, -9) | Upward |
| 0.5, -3, 4 | 3 | -0.5 | 0.5(x - 3)^2 - 0.5 | (3, -0.5) | Upward |
| -2, 4, -6 | 1 | -4 | -2(x - 1)^2 - 4 | (1, -4) | Downward |
The first row, y = x^2, is the base parabola with no shift in either direction, which is worth memorizing as the reference shape every other row transforms.
Completing the square, and how this differs from the quadratic formula calculator
The h and k values in vertex form come from a technique called completing the square. Starting from y = ax^2 + bx + c, you factor a out of the first two terms, then add and subtract the exact amount needed to turn the expression inside the parentheses into a perfect square trinomial. That amount works out to (b / 2a) squared. Once the parentheses form a perfect square, the expression collapses into a(x - h)^2 plus whatever constant is left outside, which is k. The formulas h = -b / (2a) and k = c - b^2 / (4a) are simply the shortcut result of carrying out that process algebraically once, so you never have to repeat the full expansion by hand for every new set of coefficients.
This tool is deliberately built as a companion to the quadratic formula calculator, not a replacement for it. Both tools start from the same three numbers, a, b, and c, but they answer different questions. The quadratic formula calculator solves ax^2 + bx + c = 0 for x, which tells you where the parabola crosses the x-axis, meaning where y equals zero. Depending on the discriminant, that can be two real roots, one repeated root, or two complex roots. The vertex form calculator instead never touches y = 0. It finds the single point where the parabola turns around, which always exists and is always a real point as long as a is not zero, regardless of whether the parabola ever crosses the x-axis at all.
In practice, use the quadratic formula calculator when a problem asks for zeros, x-intercepts, or solutions to an equation. Use the vertex form calculator when a problem asks for a minimum or maximum value, the axis of symmetry, or the coordinates of the turning point, such as the peak height of a thrown object or the minimum cost point on a revenue curve. Many textbook problems ask for both in sequence, first the vertex to sketch the general shape, then the roots to pin down where the curve meets the axis.
Common mistakes
- Getting the sign of h backward. Vertex form subtracts h, so h = -b / (2a) means a positive b always produces a negative h, and the term in parentheses ends up written as (x + something) rather than (x - something).
- Dropping the leading coefficient a when rewriting the equation. A common error is writing (x - h)^2 + k with no a in front at all, but a carries over unchanged from standard form and multiplies the entire squared term.
- Confusing k, the vertex's y-coordinate, with c, the standard form y-intercept. They are only equal to each other when b is 0, since k also depends on both b and a.
- Forgetting to square b before dividing by 4a when computing k by hand, which is easy to miss since the formula has b squared in the numerator, not b alone.
- Reporting the vertex as (k, h) instead of (h, k). The horizontal coordinate always comes first, so a mixed-up order flips the point across the diagonal and gives a wrong location entirely.
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