Vertex Form Calculator

y = ax² + bx + c → h = -b / (2a) → k = c - b² / (4a) → y = a(x - h)² + k

a

The stretch, and the direction. It is the same number as in the standard form — the vertex form does not change it — and it now sits in front of the bracket. Positive opens upward, negative opens downward, and its size decides how narrow the parabola is: |a| greater than 1 is narrower than the plain x², less than 1 is wider.

h

The x coordinate of the vertex, which is what the bracket is built around. Its sign flips when it goes into the bracket: a vertex at x = -3 is written (x + 3). It is also the axis of symmetry, and the highest or lowest point of the curve sits on the vertical line x = h.

k

The y coordinate of the vertex: the value the curve takes there, and so the smallest value it reaches when it opens upward, or the largest when it opens downward. It is what is left over after the square accounts for the rest, and k = 0 is the special case where the vertex lands exactly on the x-axis.

x, y

The two variables of the finished equation. Any pair that satisfies it sits on the curve, so the vertex form is a complete description of the parabola — the same description as the standard form, arranged so that the interesting point is the one you can read.

Use it when you have a quadratic in standard form and what you want to know is where the curve turns around — the maximum of a profit, the minimum of a cost, the top of a thrown ball — or when you need to sketch it, because the vertex plus the direction of opening is most of the shape. It is also the form to reach for when the vertex is the thing you were given and the standard form is what you want, read backwards. Every quadratic has one, and the three coefficients are all it takes to find it.

1. y = 2x² - 4x + 1 — a leading coefficient other than 1

   1. The vertex x: h = -b / (2a) = 4 / 4 = 1
   2. The vertex y: k = c - b² / (4a) = 1 - 16 / 8 = 1 - 2 = -1
   3. The bracket is built around h = 1, so it reads (x - 1)
   4. The leading coefficient 2 goes in front of the bracket: y = 2(x - 1)² - 1

   The check is to expand it back: 2(x - 1)² = 2x² - 4x + 2, and adding the -1 returns 2x² - 4x + 1, which is where we started. That is what makes this a rewrite rather than a different curve — the same parabola, described by its turning point instead of by its coefficients. The 2 in front is doing real work and does not disappear; a leading coefficient of 1 is the only one that leaves the bracket standing alone.

2. y = -x² + 4 — opening downward, with no x term at all

   1. The vertex x: h = -0 / (2 × -1) = 0
   2. The vertex y: k = 4 - 0 = 4
   3. With h = 0 the bracket would say (x - 0), so it collapses to a bare x²
   4. A leading coefficient of -1 is written as a single minus sign: y = -x² + 4

   This is the case where the vertex form and the standard form are the same string, and that is the point: when b is zero the vertex already sits on the y-axis, so there was nothing to move and nothing to complete. Read the answer as a description rather than a change — the curve opens downward and turns around at (0, 4), which is four units above the origin.

3. y = 0.5x² + 3x + 0.5 — a fractional leading coefficient

   1. The vertex x: h = -3 / (2 × 0.5) = -3
   2. The vertex y: k = 0.5 - 9 / 2 = 0.5 - 4.5 = -4
   3. The bracket is built around h = -3, so it reads (x + 3) — the sign flips
   4. Half is printed as a fraction and bracketed, because 1/2(x + 3)² would read as 1 divided by 2(x + 3)²: y = (1/2)(x + 3)² - 4

   Expanding confirms it: (1/2)(x + 3)² = 0.5x² + 3x + 4.5, and 4.5 - 4 brings the constant back to 0.5. The fraction appears because the coefficients were fractions, not because anything was approximated — 1/2 is exact where 0.5 is the same number, and the printed form is the one a person writes by hand.

The leading coefficient may not be zero. With a = 0 there is no square to complete and no vertex to report, because the graph is a straight line: the page refuses the input rather than inventing an answer. The coefficients are limited to a million in either direction, and a very small a combined with a very large b is refused too, since the vertex would be pushed past the point where the number means anything. The equation is printed, not solved: the page does not tell you the roots, where the curve crosses the x-axis, or the y value at some other x — the linked quadratic formula calculator does the roots, and expanding the vertex form back out is what the FOIL calculator does. Numbers are printed as fractions when the denominator is small enough to read and as six-decimal values otherwise, so a vertex at one third comes out as 0.333333. Nothing here carries units, because the coefficients of a quadratic are numbers rather than measurements.

What is vertex form?

It is y = a(x - h)² + k, a way of writing a quadratic that puts the vertex in plain sight: the turning point of the parabola is the point (h, k), and both coordinates can be read off the equation without doing any work. The standard form ax² + bx + c describes the same curve — the two are rearrangements of each other, and a is the same number in both.

How is this different from the completing the square calculator?

The arithmetic is the same and the vertex comes out identical, because the two pages take the very same three coefficients. What differs is what they hand back: that page prints the rewritten expression, a(x - h)² + k, and explains the method of getting there; this page prints the equation of the parabola, y = a(x - h)² + k, and is about reading the graph — where it turns, which way it opens, how wide it is.

How do I tell whether the parabola opens up or down?

By the sign of a, the number in front of the bracket. Positive opens upward, so the vertex is the lowest point on the curve and k is the minimum value; negative opens downward, so the vertex is the highest point and k is the maximum. The page does not print a verdict, because the sign of a is not one of its outputs — but it is right there in the equation, in front of the bracket.

Where is the axis of symmetry?

At x = h, the vertical line through the vertex — the same h that appears in the bracket. The parabola is a mirror image across that line, which is why the vertex is the turning point at all: everything to the left of it is the reflection of something to its right. It also means the two roots, when the curve has them, sit the same distance either side of it, so their average is h.

Why does the equation sometimes show fractions?

Because the exact value was a fraction. A decimal that ends within a few digits and has a small denominator is printed as a fraction — 0.5 as 1/2, 0.25 as 1/4 — since that is how it would be written by hand, and because 1/2 is exact rather than approximately 0.5. Anything longer is printed as a decimal rounded to six places, so a vertex at one third appears as 0.333333.

Why is the last term missing from the answer sometimes?

Because the vertex y is zero, and adding nothing to the equation would only add a term that is always zero. That happens exactly when the curve touches the x-axis instead of crossing it — the vertex sits on the axis, and the quadratic is a perfect square. So y = (x + 1)² is a complete vertex form, not a truncated one.

• Parabola vertex — the point this page reports, and why the curve turns around there — Wolfram MathWorld (United States)
• Vertex — the same notion for curves in general, of which the parabola's turning point is one case — Wolfram MathWorld (United States)
• Quadratic polynomial — ax² + bx + c and the several ways it can be written, vertex form among them — Wolfram MathWorld (United States)