1. What is Vertex Form?

The vertex form of a parabola equation is \((x-h)^2 = 4p(y-k)\), where (h,k) is the vertex of the parabola and p is the distance from the vertex to the focus. This form clearly shows the parabola's vertex and its orientation.

2. Conversion to Standard Form

The calculator converts vertex form to standard form (\(y = ax^2 + bx + c\)) using:

Where:

• \( h \) — x-coordinate of vertex
• \( k \) — y-coordinate of vertex
• \( p \) — distance from vertex to focus

Explanation: The conversion involves expanding the squared term and solving for y to get the standard quadratic form.

3. Importance of Parabola Forms

Details: Vertex form is useful for graphing and identifying key features, while standard form is better for finding roots and general analysis.

4. Using the Calculator

Tips: Enter the vertex coordinates (h,k) and the p value (must be non-zero). The calculator will provide the equivalent standard form equation.

5. Frequently Asked Questions (FAQ)

Q1: What if p is negative?
A: A negative p value indicates the parabola opens downward, while positive p means it opens upward.

Q2: Can this convert horizontal parabolas?
A: This calculator handles vertical parabolas. For horizontal parabolas \((y-k)^2 = 4p(x-h)\), a different conversion is needed.

Q3: Why would I need standard form?
A: Standard form is useful for finding y-intercepts (c), and for using the quadratic formula to find roots.

Q4: What's the relationship between p and a?
A: The coefficient a in standard form equals 1/(4p), showing how p affects the parabola's "width".

Q5: How precise are the results?
A: Results are displayed with 4 decimal places, but exact fractions might be more appropriate in some cases.