Vertex Form Calculator

A vertex form calculator converts a quadratic equation from standard form, y = ax² + bx + c, to vertex form, y = a(x − h)² + k. It can also convert vertex form back to standard form. This tool shows every step and gives the vertex, axis of symmetry, roots, and y-intercept.

Decimals (0.5) and fractions (1/2) both work.

What Is a Vertex Form Calculator?

A vertex form calculator is a tool that rewrites a quadratic equation so you can see its vertex right away. The vertex is the lowest or highest point of the curve. This curve is called a parabola.

Vertex form looks like this:

y = a(x − h)² + k

  • (h, k) is the vertex.
  • a tells you which way the parabola opens and how wide it is.

How to Use the Vertex Form Calculator

  1. Pick a direction. Choose “Standard form to vertex form” or “Vertex form to standard form.”
  2. Enter the values. For standard form, enter a, b, and c. For vertex form, enter a, h, and k.
  3. Use 0 for a missing term. For y = x² − 4, enter b = 0.
  4. Press Convert.

You can type whole numbers, decimals like 0.5, or fractions like 1/2. The tool keeps answers as exact fractions.

Vertex Form Formulas

Standard Form to Vertex Form

The value of a stays the same. You find h and k with these formulas:

h = −b ÷ (2a)
k = c − b² ÷ (4a)

You can also find k by putting h back into the equation for x.

Vertex Form to Standard Form

Expand the square and combine like terms. The formulas are:

b = −2ah
c = ah² + k

Worked Examples

Example 1: y = x² + 6x + 5

  1. Here a = 1, b = 6, and c = 5.
  2. h = −6 ÷ (2 × 1) = −3
  3. k = 5 − 6² ÷ (4 × 1) = 5 − 9 = −4
  4. Vertex form: y = (x + 3)² − 4

The vertex is (−3, −4). Since a is positive, the parabola opens up. So −4 is the lowest value.

Example 2: y = −3x² + 12x − 7

  1. Here a = −3, b = 12, and c = −7.
  2. h = −12 ÷ (2 × −3) = −12 ÷ −6 = 2
  3. k = −7 − 12² ÷ (4 × −3) = −7 − 144 ÷ −12 = −7 + 12 = 5
  4. Vertex form: y = −3(x − 2)² + 5

The vertex is (2, 5). Since a is negative, the parabola opens down. So 5 is the highest value.

Example 3: Fractions

Take y = x² + 3x + 1.

  1. h = −3 ÷ 2 = −3/2
  2. k = 1 − 9 ÷ 4 = −5/4
  3. Vertex form: y = (x + 3/2)² − 5/4

The tool keeps these as exact fractions. It also shows the decimals: −1.5 and −1.25.

Example 4: Back to Standard Form

Take y = 3(x + 1)² + 4. Here a = 3, h = −1, and k = 4.

  1. b = −2 × 3 × (−1) = 6
  2. c = 3 × (−1)² + 4 = 3 + 4 = 7
  3. Standard form: y = 3x² + 6x + 7

What Else the Tool Shows

  • Axis of symmetry: the line x = h. The parabola is a mirror image on each side.
  • Direction: it opens up if a is positive and down if a is negative.
  • Minimum or maximum value: this is k. It is a minimum if the parabola opens up and a maximum if it opens down.
  • Y-intercept: the point (0, c), where the curve crosses the y-axis.
  • Roots: the x-values where the curve crosses the x-axis.

Roots and the Discriminant

The discriminant is b² − 4ac. It tells you how many real roots there are:

  • Positive: two real roots
  • Zero: one root, at the vertex
  • Negative: no real roots, so the curve never touches the x-axis

For y = x² + 6x + 5, the discriminant is 36 − 20 = 16. That is positive, so there are two roots: x = −5 and x = −1.

Why Use Vertex Form?

  • Graph faster. Plot the vertex first, then a few points on each side.
  • Solve max and min problems. Word problems about the highest point of a ball or the lowest cost use the vertex.
  • Check your homework. Compare each step with your own work.
  • Avoid mistakes with fractions. The tool keeps exact values, so you do not lose accuracy by rounding.

Vertex Form Calculator FAQs

Is This Tool Free?

Yes. It is free, and you do not need to sign up. It runs in your browser.

Why Can’t a Be Zero?

If a = 0, there is no x² term. The equation is then a straight line, not a parabola, so it has no vertex.

Does It Use Completing the Square?

It uses the formulas h = −b/(2a) and k = c − b²/(4a). These give the same answer as completing the square, with fewer steps.

Can It Handle Decimals and Fractions?

Yes. Type 0.5 or 1/2. The tool turns decimals into exact fractions and shows a decimal version of the answer too. For whole-number math, try our integer calculator.

More Free Calculators

Final Thoughts

A vertex form calculator turns any quadratic into a form where you can read the vertex at a glance. It shows each step, keeps fractions exact, and gives the roots and axis of symmetry too. Use it to graph parabolas, solve word problems, and check your homework.