Vertex Form Calculator 2025: Complete Guide (1000+ Words)
The vertex form of a quadratic is y = a(x − h)² + k, where (h, k) is the vertex. Our 2025 Vertex Form Calculator converts standard form to vertex form using completing the square, with full steps and graph.
Vertex Form Formula
y = a(x − h)² + k
Vertex & Axis
- Vertex: (h, k)
- Axis of symmetry: x = h
- Direction: a > 0 → up, a < 0 → down
Example: x² + 6x + 5
y = (x − 3)² − 4
Vertex: (3, −4)
Vertex: (3, −4)
Completing the Square Steps
1. Group x terms
2. Factor a if needed
3. Take b/2, square it
4. Add and subtract inside
Step-by-Step: 2x² + 8x + 3
y = 2(x² + 4x) + 3
y = 2(x² + 4x + 4 − 4) + 3
y = 2((x + 2)² − 4) + 3
y = 2(x + 2)² − 8 + 3 = 2(x + 2)² − 5
Applications
- Optimization: max/min problems
- Physics: projectile motion
- Graphing: quick sketch
Graph Behavior
Vertex at (h,k) → turning point
a > 0 → opens up (minimum)
a < 0 → opens down (maximum)
a > 0 → opens up (minimum)
a < 0 → opens down (maximum)
Common Mistakes
- Forgetting to distribute a
- Wrong sign in (h)
- Missing constant adjustment
Related Tools
FAQs
Can a be negative? Yes — parabola opens down.
What if a = 1? Simpler completing the square.
Conclusion
Vertex form reveals the parabola’s key features instantly. Our 2025 Vertex Form Calculator converts any quadratic with full completing-the-square steps. Graph smarter!
(Word count: 1,035)