Normal Form to Vertex Form Calculator – Quadratic Conversion
Convert standard form quadratics to vertex form.
Table of Contents
How to Use
- Enter coefficient a (the x² coefficient, cannot be zero)
- Enter coefficient b (the x coefficient)
- Enter coefficient c (the constant term)
- Click calculate to convert to vertex form
- View the vertex, axis of symmetry, and other properties
What is Vertex Form?
Vertex form is a way of writing a quadratic function that makes it easy to identify the vertex (the highest or lowest point) of the parabola. The vertex form is written as: f(x) = a(x - h)² + k, where (h, k) is the vertex.
The standard form ax² + bx + c can be converted to vertex form using the completing the square method or the formulas: h = -b/(2a) and k = c - b²/(4a).
How to Convert
To convert from standard form to vertex form:
- Calculate h = -b/(2a) to find the x-coordinate of the vertex
- Calculate k = c - b²/(4a) to find the y-coordinate of the vertex
- Write the vertex form as a(x - h)² + k
- The coefficient 'a' remains the same in both forms
Properties from Vertex Form
The vertex form reveals important properties of the parabola:
- Vertex: The point (h, k) is the minimum or maximum of the function
- Axis of symmetry: The vertical line x = h divides the parabola symmetrically
- Direction: If a > 0, the parabola opens upward; if a < 0, it opens downward
- Width: Larger |a| means a narrower parabola; smaller |a| means wider
Applications
Vertex form is useful in many contexts:
- Finding maximum or minimum values of quadratic functions
- Graphing parabolas quickly by identifying the vertex
- Solving optimization problems in physics and economics
- Analyzing projectile motion (maximum height)
- Designing parabolic structures in engineering
Frequently Asked Questions
- Why can't coefficient 'a' be zero?
- If a = 0, the equation becomes linear (bx + c), not quadratic. A quadratic equation must have an x² term, which requires a ≠ 0.
- What does the vertex represent?
- The vertex (h, k) is the turning point of the parabola. If a > 0, it's the minimum point; if a < 0, it's the maximum point. It represents the extreme value of the quadratic function.
- How do I find the x-intercepts from vertex form?
- Set the equation equal to zero and solve: a(x - h)² + k = 0. This gives x = h ± √(-k/a). Real solutions exist only when -k/a ≥ 0.
- What is completing the square?
- Completing the square is an algebraic technique used to convert standard form to vertex form. It involves adding and subtracting a term to create a perfect square trinomial, which can then be factored.