Vertex Form Calculator

Type the coefficients of a quadratic and get it back as y = a(x − h)² + k, with the vertex, axis of symmetry and intercepts worked out. It runs both directions, standard form to vertex form and vertex form to standard form, with the steps for either one click away. The graph redraws on every keystroke, so you can see what each coefficient does to the parabola instead of taking it on faith.

y = ax² + bx + c

a x² coefficient

b x coefficient

c constant

Vertex form

y = (x − 2)² − 1

y = x² − 4x + 3

Vertex: (2, −1)
Axis of symmetry: x = 2
minimum: y = −1 at x = 2
Opens: up (a > 0)
y-intercept: (0, 3)
x-intercepts: (1, 0) (3, 0)

Graph

a1b-4c3

The gold point is the vertex (2, −1) and the dashed ochre line is the axis of symmetry x = 2. Drag a to watch the curve narrow, widen or flip over against the gray dashed curve, which is the same vertex with a = 1. The view holds still while you drag; Fit recenters it.

The vertex form formula

y = a(x − h)² + k

Going standard to vertex form takes two substitutions on y = ax² + bx + c. The a is the same in both forms.

h = −b ÷ 2a k = c − b² ÷ 4a

Going from vertex form to standard form is expansion instead: b = −2ah and c = ah² + k.

Standard form to vertex form, step by step

Converting a quadratic to vertex form by hand means completing the square, and the two formulas above are the shortcut for it. Here is the long version for y = 2x² + 8x + 5, a case where a is not 1, which is where most of the sign mistakes happen.

Step 1. Start with standard form

y = 2x² + 8x + 5
Step 2. Factor 2 out of the x² and x terms

y = 2(x² + 4x) + 5
Step 3. Halve the x coefficient and square it

(4 ÷ 2)² = 2² = 4
Step 4. Add and subtract that inside, then factor the perfect square

y = 2(x + 2)² − 8 + 5
Step 5. Collect the constants: k = −3

y = 2(x + 2)² − 3

Check it with the shortcut: h = −8 ÷ (2 × 2) = −2, and k = 5 − 8² ÷ (4 × 2) = −3. Same vertex, no rearranging. Enter your own a, b and c above and the calculator writes out these same lines.

Vertex form to standard form, step by step

The reverse conversion is plain expansion. Square the bracket, multiply through by a, then collect the constants. Worked out for y = 2(x − 3)² + 5:

Step 1. Start with vertex form

y = 2(x − 3)² + 5
Step 2. Square the bracket

(x − 3)² = x² − 6x + 9
Step 3. Multiply every term by 2

y = 2x² − 12x + 18 + 5
Step 4. Collect the constants: c = 23

y = 2x² − 12x + 23

Or skip to the coefficients: b = −2ah = −2 × 2 × 3 = −12, and c = ah² + k = 2 × 9 + 5 = 23. Switch the calculator to Vertex form → standard form to run it on your own numbers.

More quadratics converted to vertex form

Four common shapes, converted both ways. Every row is generated by the calculator's own functions at build time.

Standard form	Vertex form	Vertex	x-intercepts
y = x² − 4x + 3	y = (x − 2)² − 1	(2, −1)	1, 3
y = −x² + 6x − 5	y = −(x − 3)² + 4	(3, 4)	1, 5
y = 2x² + 8x + 5	y = 2(x + 2)² − 3	(−2, −3)	−3.224744871, −0.7752551286
y = 0.5x² + 3x + 1	y = 0.5(x + 3)² − 3.5	(−3, −3.5)	−5.645751311, −0.3542486889

What each letter does to the graph

Vertex form is popular because all three numbers are visible on the curve. Change one in the calculator and watch which part of the picture moves.

Change	What happens to the parabola
a > 0	Opens upward. The vertex is the lowest point, so k is the minimum.
a < 0	Opens downward. The vertex is the highest point, so k is the maximum.
\|a\| larger	Narrower and steeper. Every y value is stretched away from the vertex.
\|a\| between 0 and 1	Wider and flatter, but still the same U shape.
h	Slides the whole curve left or right. Note the minus sign in (x − h): h = 3 shifts right, h = −3 shifts left.
k	Slides the curve up or down, which is what decides whether it crosses the x-axis at all.

Reading the graph

The gold point is the vertex (h, k), and the dashed gold line through it is the axis of symmetry x = h.
The teal points are the x-intercepts, the solutions of the equation. No points means the discriminant is negative and the roots are complex.
The view stays centered on the vertex and keeps a square scale, so a steeper a really does draw a narrower curve. Use the zoom buttons if a tall curve runs off the top.

When vertex form is the useful one

Standard form is convenient for the quadratic formula and for reading the y-intercept, which is just c. Vertex form is the one to reach for when the question is about the turning point: the maximum height of a thrown ball, the production level that minimizes cost, the profit-maximizing price, or graphing by hand from a shifted parent function. Any question phrased as "what is the largest" or "what is the smallest" is asking for k.

How to use this calculator

Choose a direction: Standard form → vertex form or Vertex form → standard form.
Enter the three values, a, b and c or a, h and k. The readout and the graph update as you type.
Read the converted equation, plus the vertex, axis of symmetry, minimum or maximum, and both intercepts.
Tap Show the steps for the completing-the-square or expansion work, or switch direction to convert the answer straight back.

Frequently Asked Questions

What is vertex form?

Vertex form writes a quadratic as y = a(x − h)² + k, where (h, k) is the vertex of the parabola. It's the same curve as standard form, y = ax² + bx + c, arranged so you can read the turning point straight off the equation instead of calculating it.

How do you convert standard form to vertex form?

Use h = −b ÷ 2a and k = c − b² ÷ 4a, then write y = a(x − h)² + k with the same a. For y = x² − 4x + 3 that gives h = 4 ÷ 2 = 2 and k = 3 − 16 ÷ 4 = −1, so the vertex form is y = (x − 2)² − 1. Completing the square is the longhand version of those two formulas, and the calculator shows every line of it.

How do you convert vertex form to standard form?

Expand the square and collect terms: a(x − h)² + k becomes ax² − 2ahx + ah² + k, so b = −2ah and c = ah² + k. For y = 2(x − 3)² + 5, b = −2 × 2 × 3 = −12 and c = 2 × 9 + 5 = 23, giving y = 2x² − 12x + 23. Switch the calculator to Vertex form → standard form and it expands it for you, line by line.

Do you have to complete the square to convert a quadratic to vertex form?

No. Completing the square is the method taught in class, but the two shortcut formulas h = −b ÷ 2a and k = c − b² ÷ 4a give the same vertex in one step, because they are what completing the square works out to in general. Use the shortcut to check your answer, and the steps to show your work.

What do h and k tell you about the graph?

h is the x-coordinate of the vertex and also the axis of symmetry, the vertical line x = h that the parabola mirrors across. k is the y-coordinate of the vertex, which is the minimum value of y when a is positive and the maximum when a is negative.

What does a do to the parabola?

The sign of a decides which way it opens: positive opens upward, negative opens downward. The size of a decides how tight it is. Values above 1 stretch it vertically into a narrow curve, values between 0 and 1 flatten it into a wide one. The graph on this page redraws as you change a, so you can see the effect.

Can you find the x-intercepts from vertex form?

Yes. Set y = 0 and solve to get x = h ± √(−k ÷ a). There are two intercepts when −k ÷ a is positive, one when the vertex sits on the x-axis, and none when the vertex is above the axis on an upward parabola or below it on a downward one.

Why won't the calculator accept a = 0?

With a = 0 the x² term disappears and the equation becomes y = bx + c, a straight line. A line has no vertex and no axis of symmetry, so there is nothing to convert. Enter any nonzero a and the conversion works.

Is my data sent anywhere?

No. The calculator and the graph both run in your browser. Nothing you type is stored or sent to a server.