Math
Vertex Form Calculator
Pick a direction: enter a, b, c to get vertex form a(x - h)^2 + k, or enter a, h, k to expand back to standard form.
Either way the vertex form calculator gives the vertex, axis of symmetry, whether the parabola opens up or down, the maximum or minimum value, the y-intercept, and the roots in exact form.
Cannot be 0. Negative a opens the parabola downward.
Try an example
Result
Converted form
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- Original form
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- Vertex (h, k)
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- Axis of symmetry
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- Direction
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- Maximum or minimum
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- y-intercept
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More details (4 more)
- Roots (exact)
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- Roots (decimal)
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- Factored form
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- Width compared with y = x^2
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
Standard form and vertex form
The same parabola can be written two ways. Standard form y = ax^2 + bx + c makes the y-A point where a graph crosses an axis. obvious (it is c). The turning point of a parabola, or a corner point of a shape. form y = a(x - h)^2 + k makes the vertex obvious (it is (h, k)). Converting between them is a routine skill in Algebra 1 and 2.
Reading the vertex form
| Part | What it tells you |
|---|---|
| a | Opens up if a > 0, down if a < 0; |a| > 1 is narrower than y = x^2, |a| < 1 is wider |
| h | x-coordinate of the vertex and the axis of symmetry x = h; the sign inside the parentheses is flipped |
| k | y-coordinate of the vertex; the minimum value when a > 0, the maximum when a < 0 |
How to use it
- Choose Standard → vertex form and enter a, b, c, or choose Vertex → standard form and enter a, h, k.
- Read the converted equation at the top and the vertex, The vertical line that splits a parabola into two mirror-image halves., and direction underneath.
- Open the steps to see h = -b/(2a), k = c - b^2/(4a), or the expansion of a(x - h)^2 + k.
How to read the answer
The vertex (h, k) is the turning point of the parabola. If a is positive the vertex is the lowest point and k is the minimum; if a is negative the vertex is the highest point and k is the maximum. The roots tell you where the graph crosses the x-axis, if it does.
Common mistakes and edge cases
- Flipping the sign of h: (x - 2)^2 has h = 2, (x + 2)^2 has h = -2.
- Forgetting to multiply the expanded square by a before adding k.
- Using k = c instead of k = c - b^2/(4a). The constant of standard form is the y-intercept, not the vertex height.
- Entering a = 0. The graph would be a line with no vertex.
Worked examples
Standard to vertex
x^2 - 4x + 1 → (x - 2)^2 - 3
Converted form
y = (x - 2)^2 - 3
Vertex to standard
(x - 2)^2 - 3 → x^2 - 4x + 1
Converted form
y = x^2 - 4x + 1
Opens downward with fractions
-2x^2 + 3x + 1 → -2(x - 3/4)^2 + 17/8
Converted form
y = -2(x - 3/4)^2 + 17/8
Negative h
3(x + 1)^2 - 4 → 3x^2 + 6x - 1
Converted form
y = 3x^2 + 6x - 1
No real roots
x^2 + 2x + 5 → (x + 1)^2 + 4
Converted form
y = (x + 1)^2 + 4
Decimal a
0.5(x - 4)^2 + 2 → (1/2)x^2 - 4x + 10
Converted form
y = (1/2)x^2 - 4x + 10
a = 0 is not a parabola
Linear input is rejected
Converted form
Error
Frequently asked questions
What is vertex form?+
Vertex form of a quadratic is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola and a controls the direction and width.
How do I find h and k from standard form?+
h = -b/(2a). Then substitute h back into the equation to get k, or use k = c - b^2/(4a). Both give the same value.
How do I convert vertex form back to standard form?+
Expand (x - h)^2 = x^2 - 2hx + h^2, multiply every term by a, then add k. The result is ax^2 - 2ahx + (ah^2 + k).
Does vertex form tell me the roots?+
Set a(x - h)^2 + k = 0 and solve: (x - h)^2 = -k/a, so x = h ± √(-k/a). If -k/a is negative there are no real roots. This calculator lists the roots exactly.
What does a do in vertex form?+
The sign of a sets the direction (positive opens up, negative opens down). Its size sets the width: |a| > 1 makes the parabola narrower than y = x^2, |a| < 1 makes it wider.
Is the axis of symmetry the same as h?+
Yes. The axis of symmetry is the vertical line x = h through the vertex.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026