1. What is the Vertex Form of a Parabola?

The vertex form of a parabola is \((x - h)^2 = 4p(y - k)\), where (h,k) is the vertex and p is the distance from the vertex to the focus. This form is particularly useful when the vertex and one other point on the parabola are known.

2. How Does the Calculator Work?

The calculator uses the vertex form equation:

Where:

• \( (h, k) \) — Coordinates of the vertex
• \( (x, y) \) — Coordinates of a point on the parabola
• \( p \) — Distance from vertex to focus (and to directrix)

Explanation: Given the vertex and another point, we can solve for p and then write the complete equation of the parabola.

3. Understanding the Parameters

Details: The parameter p determines the "width" of the parabola and the location of its focus. A larger |p| makes the parabola wider, while a smaller |p| makes it narrower.

4. Using the Calculator

Tips: Enter the coordinates of the vertex and any other point on the parabola. The point must not have the same y-coordinate as the vertex.

5. Frequently Asked Questions (FAQ)

Q1: What if my parabola opens sideways?
A: For horizontal parabolas, use \((y - k)^2 = 4p(x - h)\). This calculator is for vertical parabolas only.

Q2: What does a negative p value mean?
A: A negative p means the parabola opens downward. Positive p means it opens upward.

Q3: Can I use this for any point on the parabola?
A: Yes, as long as it's not the vertex itself and lies exactly on the parabola.

Q4: How do I find the focus and directrix?
A: Focus is at (h, k + p). Directrix is the line y = k - p.

Q5: What if I get an error message?
A: This likely means the point you entered has the same y-coordinate as the vertex, making the denominator zero.