Vertex Form Calculator: Standard Form to Vertex Form and Back

This vertex form calculator converts a quadratic between standard form, y = ax² + bx + c, and vertex form, y = a(x - h)² + k. It gives the vertex (h, k), the axis of symmetry, whether the parabola opens up or down, its minimum or maximum value, the discriminant, the roots and the y-intercept.

The vertex form of a quadratic equation puts the vertex on display: in y = a(x - h)² + k the turning point of the parabola is (h, k). The number a is the same in both forms. It sets how wide the parabola is and which way it opens: up when a is positive, down when it is negative. Standard form, in turn, shows the y-intercept, c, directly. Each form answers different questions, which is why courses ask you to move between them.

All coefficients are kept as exact fractions, so a vertex at (5/2, -1/4) is reported that way rather than as rounded decimals. Roots that are irrational are shown as decimals to 15 significant figures.

1. Choose whether you have the standard form or the vertex form.
2. Enter a, then b and c for standard form, or h and k for vertex form. Note that in y = 3(x + 1)² - 12, h is -1, because x + 1 is x - (-1).
3. Read the vertex at the top and both forms of the equation below, with the axis, direction, extreme value, discriminant, roots and y-intercept.

1. Make the coefficients exact. Each typed number becomes a fraction. If a is 0, the equation is a straight line, and the calculator says so.

1. Standard form to vertex form, by completing the square. The vertex is at h = -b/(2a), and k is the value of y there, which simplifies to c - b²/(4a).

1. Vertex form to standard form, by expanding. Multiplying out a(x - h)² + k gives b = -2ah and c = ah² + k.

1. Discriminant and roots. The discriminant b² - 4ac decides the roots (DLMF 1.11.11). Positive: two real roots from the quadratic formula. Zero: one repeated root at x = h. Negative: no real roots, and the two complex roots h ± i√(4ac - b²)/(2|a|) are given. When the discriminant is the square of a fraction, the roots are given exactly; otherwise as decimals.

1. The rest. The axis of symmetry is the vertical line x = h. The y-intercept is (0, c). The vertex is a minimum when a is positive and a maximum when a is negative, and that value is k.

Using the defaults, y = 2x² - 12x + 19:

1. h = -(-12)/(2 × 2) = 12/4 = 3.
2. k = 19 - (-12)²/(4 × 2) = 19 - 144/8 = 19 - 18 = 1.
3. Vertex form: y = 2(x - 3)² + 1. Vertex (3, 1), axis x = 3.
4. a = 2 is positive, so the parabola opens upward and the minimum value is y = 1 at x = 3.
5. Discriminant: (-12)² - 4 × 2 × 19 = 144 - 152 = -8, so there are no real roots. The complex roots are 3 ± 0.707106781186548i.
6. y-intercept: (0, 19).

Expand the square and add k. The y = 3(x + 1)² - 12 preset has a = 3, h = -1 and k = -12. Then b = -2 × 3 × (-1) = 6 and c = 3 × 1 + (-12) = -9, so the standard form is y = 3x² + 6x - 9.

Its discriminant is 144, a perfect square, so the roots are exact: x = -3 and x = 1.

h = -b/(2a) and k = c - b²/(4a). Some books write k as f(h), the value of the function at h, which is the same number. For x² - 5x + 6, h = 5/2 and k = 6 - 25/4 = -1/4, so the vertex form is y = (x - 5/2)² - 1/4.

The parabola crosses the x-axis at x = 2 and x = 3, which are equally spaced either side of 5/2, as the axis of symmetry requires.

The reference table lists the vertex form of a dozen quadratics. Look at x² + 2x + 1, whose vertex form is simply (x + 1)²: when k is 0, the vertex sits on the x-axis and there is one repeated root. And 3x² - 12 needs no completing at all, because with no x term the vertex is already on the y-axis.

• Factored form. y = a(x - r₁)(x - r₂) is not written out, but the roots card gives r₁ and r₂ whenever they are real.
• Exact surds. Irrational roots are shown as decimals. In exact form, -x² + 4x + 1 has roots 2 ± √5; the calculator gives -0.23606797749979 and 4.23606797749979.
• Graphs. The parabola is not drawn.
• Contact lens vertex distance. In optometry, vertex distance is the gap between a spectacle lens and the eye, and a vertex calculator adjusts a prescription for it. That is unrelated to this page; ask an optician.
• Cubic and higher equations. Only quadratics are handled.

• Sign of h. In y = a(x - h)² + k, the vertex x-coordinate is h, so (x + 3)² means h = -3.
• Forgetting a when completing the square. Factor a out of the x terms first; the vertex form keeps the same a as the standard form.
• Dropping k. The constant outside the square is the y-coordinate of the vertex.
• Reading a negative discriminant as an error. It means the parabola does not reach the x-axis. The roots are complex, not missing.
• Mixing up maximum and minimum. Positive a opens upward and has a minimum; negative a opens downward and has a maximum.

• NIST Digital Library of Mathematical Functions, section 1.11(iii), Polynomials of Degrees Two, Three, and Four: roots of the quadratic equation and the discriminant (1.11.11). dlmf.nist.gov/1.11
• OpenStax, College Algebra 2e, section 5.1, Quadratic Functions: standard and vertex form, and the vertex at h = -b/(2a). openstax.org/books/college-algebra-2e/pages/5-1-quadratic-functions