Vertex Form Calculator
Convert a quadratic between standard form and vertex form and find its vertex and axis of symmetry.
Vertex form
y = 2(x − 3)² − 11
- Standard form
- y = 2x² − 12x + 7
- Vertex (h, k)
- (3, −11)
- Axis of symmetry
- x = 3
- Opens
- Upward, so the vertex is the minimum
The vertex is (3, −11).
Calculation breakdown
| Step | Formula | Result |
|---|---|---|
| h, the x-coordinate of the vertex | −-12 ÷ (2 × 2) | 3 |
| k, the y-coordinate of the vertex | 7 − -12² ÷ (4 × 2) | -11 |
How it works
Choose the direction, then enter a, b and c for standard form, or a, h and k for vertex form.
Worked example
y = 2x² − 12x + 7:
- h = −(−12) ÷ (2 × 2) = 3
- k = 7 − (−12)² ÷ (4 × 2) = 7 − 18 = −11
- Vertex form: y = 2(x − 3)² − 11, with the vertex at (3, −11)
Sources
Frequently asked questions
What is vertex form?
Vertex form writes a quadratic as y = a(x − h)² + k. The point (h, k) is the vertex, the highest or lowest point of the parabola, and a tells you how wide it is and which way it opens.
How do you convert from standard form to vertex form?
Find h = −b ÷ (2a), then k = c − b² ÷ (4a). For y = 2x² − 12x + 7, h = 12 ÷ 4 = 3 and k = 7 − 144 ÷ 8 = −11, so y = 2(x − 3)² − 11.
How do you convert from vertex form to standard form?
Expand the square. With y = a(x − h)² + k, b = −2ah and c = ah² + k. For y = −(x + 1)² + 4, b = −2 and c = −1 + 4 = 3, so y = −x² − 2x + 3.
What is the axis of symmetry?
It is the vertical line x = h through the vertex. Each side of the parabola mirrors the other across it.
Does the parabola open up or down?
Up if a is positive, and down if a is negative. An upward parabola has its minimum at the vertex, and a downward one has its maximum there.