1. What is the Parabola Standard Form Equation?

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

2. How Does the Calculator Work?

The calculator uses the standard form equation:

Where:

• \( p \) — Distance from vertex to focus (determines parabola's width)
• \( h \) — X-coordinate of the vertex
• \( k \) — Y-coordinate of the vertex

Explanation: The equation shows the relationship between any point (x,y) on the parabola and its geometric properties.

3. Importance of Standard Form

Details: The standard form makes it easy to identify the vertex, focus, axis of symmetry, and directrix of the parabola, which are essential for graphing and analysis.

4. Using the Calculator

Tips: Enter the focus distance (p) and vertex coordinates (h,k). The calculator will generate the standard form equation of the parabola.

5. Frequently Asked Questions (FAQ)

Q1: What does the 'p' value represent?
A: The p value is the distance from the vertex to the focus. It determines how "wide" or "narrow" the parabola appears.

Q2: How is this different from a vertical parabola?
A: A vertical parabola has the form \((x - h)^2 = 4p(y - k)\) and opens upward or downward instead of left/right.

Q3: What if p is negative?
A: A negative p value means the parabola opens to the left (for horizontal parabola) instead of to the right.

Q4: Can this be used for any parabola equation?
A: This converts to standard form for horizontal parabolas. Other forms may need to be rearranged first.

Q5: How do I find the focus and directrix from this form?
A: Focus is at (h+p, k). Directrix is the line x = h-p.