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Vertex Form Calculator

Convert a quadratic from standard form to vertex form, see each completing-the-square step, and read the vertex for graphing.

Result

1(x - 2)² - 1

1x² - 4x + 3 in vertex form is 1(x - 2)² - 1, with vertex at (2, -1)

Quick Answer

A vertex form calculator rewrites ax² + bx + c as a(x - h)² + k. The vertex is (h, k), where h = -b/(2a) and k = c - b²/(4a). For x² - 4x + 3, completing the square gives (x - 2)² - 1, so the vertex is (2, -1) on the graph.

How Standard Form Converts To Vertex Form Step By Step

Vertex form shows the turning point of a parabola at once. In a(x - h)² + k, the vertex is (h, k). Standard form, ax² + bx + c, does not show that point as clearly. The tool changes the form by completing the square. Start by taking a outside the x terms when a is not 1. Then add and subtract the number that makes the x terms a perfect square. The added and subtracted parts keep the value the same. Group the perfect-square trinomial. Write the trinomial as a squared binomial. The last number becomes k. This method gives the same h and k as the direct formulas. The new form also shows the axis of symmetry, x = h, without extra algebra.

How Vertex Form Is Calculated By Completing The Square

Converting a quadratic expression to vertex form utilizes completing the square or direct evaluation of the vertex coordinates.

  • h = -b ÷ (2a).
  • k = c − b² ÷ (4a).
  • Vertex Form: y = a(x − h)² + k.

a: controls the width and opening direction.

b: helps set the horizontal shift.

c: is the standard-form constant.

h: is the x-coordinate of the vertex.

k: is the y-coordinate of the vertex.

Getting Started With Vertex Form And Quadratic Inputs

Inputs

  • a: enter the coefficient of x².
  • b: enter the coefficient of x.
  • c: enter the constant term.

Steps

  1. Write the quadratic as ax² + bx + c.
  2. Enter a, b, and c with their signs.
  3. Calculate h and k.
  4. Read the vertex form.
  5. Expand the result to check the first standard form.

Vertex Form Example With Every Algebra Step Shown Clearly

Quadratic standard form is x² − 4x + 3, so a = 1, b = −4, and c = 3.

  1. Group the x terms: x² − 4x.
  2. Take half of the x coefficient: −4 ÷ 2 = −2.
  3. Square that value: (−2)² = 4.
  4. Add and subtract 4 inside standard form: x² − 4x + 4 − 4 + 3.
  5. Write the perfect square trinomial as a squared binomial: (x − 2)² − 4 + 3.
  6. Combine the remaining constants: −4 + 3 = −1.

The vertex form is (x − 2)² − 1, with vertex (2, −1). All values are exact, so no rounding is needed. Formula check: h = −(−4)/(2×1) = 2, k = 3 − 16/4 = −1.

Where Vertex Form Is Most Useful For Graphing A Parabola

Use vertex form for the turning point, axis of symmetry, or a quick graph. Vertex form also helps check shifts from y = x².

For nearby graphing, quadratic, and algebraic factoring tools, see the math calculators hub and look up concepts in the glossary.

Assumptions

  • The input is a quadratic in one variable.
  • The coefficient a is not zero.
  • Coefficients use the same standard-form order.
  • Full values are kept until display.

Limitations

  • If a = 0, the expression is not quadratic.
  • Decimal inputs may lead to rounded h or k values.
  • The tool does not draw the full graph.
  • Complex coefficients are outside the stated scope.

In Practice

Watch the sign inside (x - h), because readers can easily read h backward. A vertex with h = 2 uses (x - 2), while h = -2 uses (x + 2). Read h from the opposite sign inside the bracket.

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Related Guides

Vertex Form Calculator Questions With Clear Answers

Why Does Vertex Form Show The Vertex Directly?

Vertex form shows the vertex directly because the squared term becomes zero at x = h. At that input, y = k, so the turning point is (h, k). The form also shows the axis of symmetry.

Is Completing The Square The Same As Using The Formula?

Completing the square and the direct formulas produce the same h and k values. Completing the square shows why the form works, while the formulas give a shorter path when only the result is needed.

What Does The Coefficient A Change In Vertex Form?

The coefficient a controls the parabola's opening direction and vertical stretch in vertex form. A positive value opens upward, while a negative value opens downward. The size of a also changes how narrow or wide the graph looks.

How Can I Check A Vertex Form Conversion?

Check a vertex form conversion by expanding the square and combining like terms. The result must match ax² + bx + c term by term. This check catches sign errors, constant errors, and a wrong value of h.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.