Vertex Form Calculator

What vertex form shows that standard form hides

Every quadratic can be written two ways. The expanded form, y = ax^2 + bx + c, is convenient for adding terms and reading the y-intercept. The vertex form, y = a(x - h)^2 + k, is convenient for graphing, because the turning point of the parabola sits right in the letters: the vertex is at (h, k). OpenStax College Algebra 2e (Rice University, 2021) calls this the standard form of a quadratic function and notes that many texts name it vertex form for that reason.

Read the transformations directly. The squared term built from (x - h) slides the base parabola y = x^2 sideways, k lifts or drops it, and a stretches or flips it. A sketch needs three numbers: h, k and the sign of a. For y = -(x - 3)^2 + 8 you can say at a glance that the parabola opens downward and peaks at (3, 8).

The axis of symmetry is the vertical line x = h. Every pair of points placed at h - 1 and h + 1 shares a y value, a quick test after any conversion.

The conversion formulas: h, k, and completing the square

Starting from y = ax^2 + bx + c, two short formulas relocate the vertex. The x coordinate is h = -b / 2a, and the y coordinate is k = f(h), the function value at that input. Substituting the expression for h gives the closed form k = c - b^2 / 4a, which the calculator uses to skip a second evaluation.

Completing the square reaches the same place by algebra. Factor a out of the x terms, add and subtract (b / 2a)^2 inside the parentheses, then fold the square together:

y = a(x^2 + (b/a)x) + c = a(x + b/2a)^2 + c - b^2/4a.

LibreTexts Intermediate Algebra (OpenStax, accessed 2026) walks this conversion in its parabolas chapter and tabulates the same vertex rule, x = -b/2a, for the general form. The formulas are faster, and completing the square shows where they come from.

Example: converting y = 2x^2 - 8x + 3

Take a = 2, b = -8, c = 3. The vertex x coordinate is h = -(-8) / (2 x 2) = 8/4 = 2. The vertex y coordinate is k = f(2) = 2(4) - 16 + 3 = -5, and the closed form agrees: k = 3 - 64/8 = 3 - 8 = -5. The vertex form is therefore y = 2(x - 2)^2 - 5.

Check it by expanding: 2(x - 2)^2 - 5 = 2(x^2 - 4x + 4) - 5 = 2x^2 - 8x + 8 - 5, which is the original expression. The parabola opens upward because a is positive, and the minimum value is -5, reached at x = 2.

The roots follow from the quadratic formula on the original form. The discriminant is (-8)^2 - 4(2)(3) = 40, so x = (8 plus or minus 6.3246) / 4, giving 0.4189 and 3.5811 to four decimals. The vertex sits halfway between the roots at x = 2, a quick consistency check on any conversion.

Each of the three common forms carries the same curve, and each answers a different question fastest:

What each coefficient controls

Vertex form separates the three jobs that the expanded form tangles together. The table reads each piece off y = 2(x - 2)^2 - 5.

Reading (x + 3) as a shift to the right is the classic sign error; the shift moves left. Test with a point when in doubt: (x + 3)^2 - 10 at x = -3 gives -10, so the vertex sits at (-3, -10), on the left side of the y axis.

Example: building the equation from a vertex and a point

The reverse direction matters in modeling, where you often know the turning point and one other point. Start from y = a(x - h)^2 + k, substitute the known vertex, and use the point to pin down a:

a = (y - k) / (x - h)^2

Suppose a cable follows a parabola with vertex (1, 2) and passes through the point (3, 10). Then a = (10 - 2) / (3 - 1)^2 = 8/4 = 2, and the equation is y = 2(x - 1)^2 + 2. Verify at the given point: 2(3 - 1)^2 + 2 = 8 + 2 = 10, and at x = 0: 2(1) + 2 = 4.

Expanding turns the result into standard form for further algebra: 2(x - 1)^2 + 2 = 2x^2 - 4x + 4. The calculator's second panel runs this vertex-and-point solve for any inputs you enter.

The same solve, run on three more vertex-and-point pairs:

The same solve appears in pricing models: if revenue follows R(p) = -20p^2 + 400p for a price p in dollars, then h = -400/(2 x -20) = 10 and k = -20(100) + 4,000 = 2,000, so the model peaks at a price of 10 dollars with revenue of 2,000.

Traps that skew a vertex conversion

• Sign of h. The formula is h = -b/2a. With b = -8 the two negatives cancel, giving h = +2. Dropping the minus on b is the most frequent slip.
• k is not c. The constant c is the y-intercept, the value at x = 0. The vertex height k = f(h) differs from c whenever h is not 0: the example has c = 3 and k = -5.
• Dropping terms when expanding. After squaring, expand 2(x - 2)^2 as 2x^2 - 8x + 8 and then carry the -5. Leaving the -5 out returns 2x^2 - 8x + 8 instead of the original 2x^2 - 8x + 3.
• Dividing by 2 instead of 2a. When a = 2, h = -b/4. Halving b alone moves the axis to the wrong line.
• Expecting real roots from every parabola. Roots exist only when a and k have opposite signs, so that -k/a is positive. y = 2(x - 1)^2 + 2 never touches the x axis, so the calculator reports no real roots.

Related algebra pages for the same equation

The site's other algebra pages carry the neighboring jobs. The quadratic formula calculator solves ax^2 + bx + c = 0 directly, which is the roots half of this page's output. The standard form calculator handles the expanded side of the conversion. For lines rather than parabolas, the slope intercept form calculator, the point slope form calculator and the equation of a line calculator mirror the same convert-between-forms workflow.