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Vertex Form of Quadratic Calculator

Online vertex form calculator: enter a, h, k in y=a(x−h)²+k to get the vertex, axis of symmetry, opening direction and maximum/minimum value, with an optional function value at any x. Completing the square is the key to quadratic graphs.

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y=a(x−h)²+k; vertex (h,k), axis x=h
(h,k)x=h
Controls opening direction and width
Axis of symmetry is x=h
k is also the extremum
Leave blank for vertex only

📖 Tutorial | Vertex Form

1. Definition
The expression y=a(x−h)²+k (a≠0) is the vertex form of a quadratic, showing the vertex (h,k) directly. It is obtained from y=ax²+bx+c by completing the square, with h=−b/(2a).
2. Formula & Notation
aQuadratic coefficient; direction and width
hVertex x-coordinate; axis x=h
kVertex y-coordinate; the extremum
(h,k)The vertex
3. Properties & Laws
  • a>0 opens up with minimum k;
  • a<0 opens down with maximum k;
  • Axis of symmetry is x=h;
  • Larger |a| gives a narrower parabola;
  • h shifts horizontally and k vertically.
4. Steps
  1. Enter a, h, k (add x for a function value);
  2. Click "Calculate" for vertex, axis, opening and extremum;
  3. Or click "Load Example" first.
5. Worked Examples
Example: Find the vertex, axis and extremum of y=(x−2)²+3.
Solution: a=1,h=2,k=3, so the vertex is (2,3), axis x=2; a>0 opens up, and the minimum is 3.
6. Common Pitfalls
Inside the parentheses is x−h, so the vertex x is h, not −h;
a cannot be 0;
The extremum is k, not h.

❓ FAQ | Vertex Form

What is vertex form?
y=a(x−h)²+k (a≠0), which encodes the vertex directly.
What is the vertex?
(h,k); the axis of symmetry is x=h.
Convert from standard form?
Complete the square, or use h=−b/2a and k=(4ac−b²)/4a.
Sign of a?
a>0 opens up with a minimum; a<0 opens down with a maximum, both equal k.
Given vertex and a point?
Set y=a(x−h)²+k and substitute the point to find a.
Axis of symmetry?
x=h in vertex form, x=−b/2a in standard form.
Interchangeable forms?
Yes: expand vertex form or complete the square.