Vertex Form Calculator
Convert between standard and vertex forms
Understanding Vertex Form
The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form makes it easy to identify key features of the parabola at a glance.
Standard Form vs Vertex Form
Standard Form: y = ax² + bx + c
- • Easy to find y-intercept (0, c)
- • Direct application of quadratic formula
- • Common for algebraic manipulation
Vertex Form: y = a(x - h)² + k
- • Vertex (h, k) visible immediately
- • Easy to graph and transform
- • Shows axis of symmetry: x = h
Components of Vertex Form
a: Controls the width and direction of the parabola. If a > 0, opens upward (minimum at vertex). If a < 0, opens downward (maximum at vertex). Larger |a| means narrower parabola.
h: x-coordinate of the vertex. The parabola is symmetric about the line x = h.
k: y-coordinate of the vertex. This is the minimum value (if a > 0) or maximum value (if a < 0) of the function.
Conversion Formulas
Standard to Vertex:
- h = -b / (2a)
- k = c - b² / (4a)
Vertex to Standard:
- b = -2ah
- c = ah² + k
Frequently Asked Questions
When should I use vertex form?
Use vertex form when graphing parabolas, finding maximum/minimum values, or when the vertex information is most important. It's ideal for optimization problems and transformations.
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's the point where the parabola changes direction.
Why is there a minus sign in (x - h)?
The form is a(x - h)² + k, so if you want the vertex at x = 3, you write (x - 3)². The minus sign is part of the standard form. For a vertex at x = -2, you'd write (x - (-2))² = (x + 2)².
How do I find the axis of symmetry?
In vertex form, the axis of symmetry is simply x = h. In standard form, it's x = -b/(2a). This vertical line passes through the vertex and divides the parabola into mirror images.
Can I find roots from vertex form?
Yes, but it requires solving a(x - h)² + k = 0, which gives (x - h)² = -k/a. Take the square root of both sides and solve for x. If -k/a is negative, there are no real roots.
What if k is zero?
If k = 0, the vertex lies on the x-axis, meaning the parabola touches the x-axis at exactly one point. This indicates a repeated root at x = h.
How does changing 'a' affect the graph?
Larger |a| makes the parabola narrower (steeper). Smaller |a| (between -1 and 1) makes it wider (flatter). Negative 'a' flips the parabola upside down. The vertex position doesn't change.
Is vertex form unique?
Yes, for a given quadratic, there's exactly one vertex form (just as there's one standard form). The vertex is a unique point, and the form completely describes the parabola's position and shape.