Convert a quadratic function between standard form y = ax² + bx + c and vertex form y = a(x − h)² + k. The calculator finds the vertex, the axis of symmetry, the y-intercept, and the zeros (x-intercepts), and draws the parabola. Choose which form you have and enter its coefficients.
The vertex form calculator converts a quadratic function between its two most useful shapes — standard form, y = ax² + bx + c, and vertex form, y = a(x − h)² + k — and works in both directions. Give it the coefficients of one form and it instantly returns the other, along with the parabola's vertex, axis of symmetry, y-intercept, and zeros, plus a graph so you can see the curve. Whether you are completing the square by hand and want to check your answer, graphing a parabola, or solving a quadratic, this tool does the algebra for you and shows the key features at a glance.
Quadratic functions are everywhere in mathematics, physics, and engineering, and the vertex form is often the most revealing way to write them because it exposes the parabola's turning point directly. Below you will find clear explanations of both forms, the conversion formulas, worked examples you can reproduce, and answers to the questions students ask most.
A quadratic function is any function whose highest power of x is two. It can be written in several equivalent ways, and two of them are especially important.
y = ax² + bx + k → y = ax² + bx + c
In standard form, y = ax² + bx + c, the three coefficients a, b, and c fully define the parabola. Here a controls how wide or narrow the curve is and whether it opens upward (a > 0) or downward (a < 0), while c is the y-intercept — the value of y when x is zero. Standard form is convenient for reading off the y-intercept and for applying the quadratic formula.
y = a(x − h)² + k
In vertex form, y = a(x − h)² + k, the same parabola is written so that its turning point — the vertex — is visible directly. The vertex sits at the point (h, k), and the vertical line x = h is the axis of symmetry. The coefficient a is identical to the a in standard form, so the parabola's width and direction are unchanged. Vertex form is ideal for graphing and for finding the maximum or minimum value of the function.
To convert y = ax² + bx + c into vertex form, you find the vertex coordinates with these formulas:
h = −b / (2a) k = c − b² / (4a)
Once you have h and k, the vertex form is simply y = a(x − h)² + k, keeping the same value of a. The formula for h is the well-known "axis of symmetry" formula, and k is just the function evaluated at x = h — the height of the vertex.
To go the other way, from y = a(x − h)² + k back to standard form, expand the square. This gives:
b = −2ah c = ah² + k
So the standard form is y = ax² + (−2ah)x + (ah² + k). Both conversions are exact and reversible, which is why the calculator can move freely between the two representations.
Take the quadratic y = x² − 4x + 3, so a = 1, b = −4, and c = 3 (these are the calculator's default values). Find the vertex:
So the vertex is (2, −1) and the vertex form is y = (x − 2)² − 1. Because a = 1 is positive, the parabola opens upward and the vertex is a minimum. Its zeros are x = 1 and x = 3 (where the curve crosses the x-axis), and its y-intercept is (0, 3). Press Calculate to see all of this, then try your own coefficients.
Now take a = 0.25, h = −17, k = −54 (the default in the second tab). Convert to standard form:
So the vertex form y = 0.25(x + 17)² − 54 becomes the standard form y = 0.25x² + 8.5x + 18.25. The vertex is (−17, −54), the y-intercept is (0, 18.25), and the zeros are approximately x = −31.6969 and x = −2.3031. The calculator reproduces every one of these figures.
Whichever direction you convert, the calculator reports the full picture of the parabola:
The great advantage of vertex form is that it hands you the parabola's most important point for free. The vertex is the turning point — the lowest point of an upward-opening parabola or the highest point of a downward-opening one — which makes it the key to a huge range of problems. In optimization, the vertex gives the maximum or minimum value of a quadratic, answering questions like "what price maximizes profit?" or "what launch angle gives the greatest range?" In graphing, knowing the vertex and the direction lets you sketch the curve immediately. And because the axis of symmetry passes through the vertex, vertex form also reveals the parabola's mirror symmetry at a glance. Standard form hides all of this inside its coefficients; vertex form puts it on display.
Converting from standard form to vertex form by hand is done through a process called completing the square. The idea is to rewrite ax² + bx + c so that part of it becomes a perfect square trinomial, which can be factored as a(x − h)². The steps are: factor a out of the first two terms, add and subtract the square of half the resulting x-coefficient inside, and simplify. The formulas h = −b/(2a) and k = c − b²/(4a) are simply the end result of completing the square in general. This calculator effectively completes the square for you and, just as importantly, lets you check your own work when you do it by hand — a core skill in any algebra course.
The zeros (also called roots or x-intercepts) of a quadratic are the x-values where y = 0 — where the parabola crosses the x-axis. The calculator finds them using the quadratic formula:
x = (−b ± √(b² − 4ac)) / (2a)
The quantity under the square root, b² − 4ac, is called the discriminant, and it tells you how many real zeros there are. If it is positive, the parabola crosses the x-axis at two points and there are two real zeros. If it is exactly zero, the parabola just touches the x-axis at its vertex, giving one repeated zero. If it is negative, the parabola never reaches the x-axis and there are no real zeros (the roots are complex). The calculator handles all three cases and tells you which one applies.
The coefficient a is the same in both forms and shapes the entire parabola. Its sign decides the direction: a positive a opens the parabola upward like a valley, so the vertex is a minimum, while a negative a opens it downward like a hill, making the vertex a maximum. Its magnitude controls the width: a large |a| makes a narrow, steep parabola, while a small |a| (close to zero) makes a wide, gentle one. When a = 1, the parabola has the "standard" width of y = x². One thing a can never be is zero — if it were, the x² term would vanish and the function would be linear, not quadratic, so the calculator requires a non-zero value of a.
Standard form and vertex form are the two shapes this calculator works with, but a quadratic can also be written in a third way: factored form, y = a(x − x₁)(x − x₂), where x₁ and x₂ are the zeros. Each form highlights a different feature. Standard form shows the y-intercept (c) directly; vertex form shows the vertex (h, k) directly; and factored form shows the zeros directly. They all describe the same parabola, and you can move between them: expanding factored form gives standard form, and completing the square on standard form gives vertex form. Factored form only exists neatly when the quadratic has real zeros, whereas vertex form always exists. That universality, plus the way it exposes the turning point, is a big reason vertex form is so widely taught and used. This calculator focuses on the standard-to-vertex conversion, but it also reports the zeros, giving you the information you would need to write the factored form as well.
Once you have a quadratic in vertex form, sketching it by hand takes only a few seconds. Start by plotting the vertex at (h, k). Note the direction from the sign of a — upward if a is positive, downward if negative — and draw the axis of symmetry as a dashed vertical line through the vertex. Then plot the y-intercept at (0, c) and use symmetry to mirror it to the other side of the axis, giving a second point for free. If the parabola has real zeros, mark them on the x-axis; they are also symmetric about the axis. Finally, connect the points with a smooth U-shaped curve. The width of that curve follows |a|: steeper for large values, gentler for small ones. The graph this calculator draws does exactly this, plotting the vertex, y-intercept, and zeros so you can check your own sketch at a glance.
Vertex form of a quadratic is y = a(x − h)² + k, where (h, k) is the vertex of the parabola and a determines its width and direction. It makes the turning point immediately visible.
Use h = −b/(2a) and k = c − b²/(4a) to find the vertex, then write y = a(x − h)² + k with the same a. This is equivalent to completing the square.
Expand the square. Using b = −2ah and c = ah² + k gives y = ax² + bx + c. Alternatively, multiply out a(x − h)² + k directly.
The vertex is the parabola's turning point — its lowest point if it opens upward, or its highest point if it opens downward. In vertex form it is the point (h, k).
If a were zero, the x² term would disappear and the equation would be linear, not quadratic. A parabola requires a non-zero a, so the calculator does not accept a = 0.
The discriminant b² − 4ac shows how many real zeros the quadratic has: two if it is positive, one if it is zero, and none (complex roots) if it is negative.
This Vertex Form Calculator is provided for general educational and informational purposes. It uses standard algebraic formulas for quadratic functions. While results are computed accurately, you should verify important work independently, especially for exams or professional use.