Find Equation of Parabola Given Vertex and Directrix Calculator
Enter the vertex (h, k) and the directrix equation (x=d or y=d) to find the equation of the parabola and other properties.
p:
Focus:
Axis of Symmetry:
Equation Form:
| Property | Value |
|---|---|
| Vertex (h, k) | |
| Directrix | |
| p | |
| Focus | |
| Axis of Symmetry | |
| Standard Equation |
Graph of the parabola, vertex, focus, and directrix.
What is a Find Equation of Parabola Given Vertex and Directrix Calculator?
A "find equation of parabola given vertex and directrix calculator" is a tool used to determine the standard equation of a parabola when you know the coordinates of its vertex (h, k) and the equation of its directrix (either y=d or x=d). Parabolas are U-shaped curves, and each point on a parabola is equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex is the point on the parabola that is closest to the directrix and lies exactly midway between the focus and the directrix.
This calculator is useful for students studying conic sections in algebra and geometry, engineers, physicists, and anyone working with parabolic shapes, such as in the design of satellite dishes, reflectors, or optical lenses. By inputting the vertex and directrix, the calculator finds the focal length 'p', the focus coordinates, the axis of symmetry, and the parabola's equation.
Common misconceptions include thinking any U-shaped curve is a parabola or that the directrix passes through the parabola. The directrix never intersects the parabola, and the parabola's shape is precisely defined by the distance 'p' between the vertex and the directrix/focus.
Find Equation of Parabola Given Vertex and Directrix Calculator Formula and Mathematical Explanation
The key to finding the equation of a parabola from its vertex and directrix is the distance 'p', which is the directed distance from the vertex to the focus (and also from the vertex to the directrix, but with opposite sign relative to focus distance along the axis).
Let the vertex be V(h, k).
- If the directrix is a horizontal line y = d: The parabola opens either upwards or downwards. The axis of symmetry is x = h. The distance 'p' from the vertex (h, k) to the directrix y = d is k – d. So, p = k – d. The focus F is at (h, k + p). The standard equation is: (x – h)2 = 4p(y – k) If p > 0 (k > d), the parabola opens upwards. If p < 0 (k < d), it opens downwards.
- If the directrix is a vertical line x = d: The parabola opens either to the right or to the left. The axis of symmetry is y = k. The distance 'p' from the vertex (h, k) to the directrix x = d is h – d. So, p = h – d. The focus F is at (h + p, k). The standard equation is: (y – k)2 = 4p(x – h) If p > 0 (h > d), the parabola opens to the right. If p < 0 (h < d), it opens to the left.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| h | x-coordinate of the vertex | Length units | -∞ to +∞ |
| k | y-coordinate of the vertex | Length units | -∞ to +∞ |
| d | Constant in the directrix equation (y=d or x=d) | Length units | -∞ to +∞ |
| p | Directed distance from vertex to focus/directrix | Length units | -∞ to +∞ (not zero) |
Practical Examples (Real-World Use Cases)
The principles used by the find equation of parabola given vertex and directrix calculator are seen in various applications.
Example 1: Satellite Dish Design
A satellite dish is designed with a parabolic cross-section. The vertex is at (0, 0), and the receiver (focus) needs to be 2 meters from the vertex along the axis of symmetry. If the dish opens upwards, the focus is (0, 2), so p=2. The vertex is (0,0), so directrix is y = 0-2 = -2.
- Vertex (h, k) = (0, 0)
- Directrix y = -2
- p = k – d = 0 – (-2) = 2
- Equation: (x – 0)2 = 4 * 2 * (y – 0) => x2 = 8y
Example 2: Headlight Reflector
A car headlight reflector has a parabolic shape. The light bulb is placed at the focus. Suppose the vertex of the parabola is at (-1, 3) and the directrix is x = -4.
- Vertex (h, k) = (-1, 3)
- Directrix x = -4
- p = h – d = -1 – (-4) = 3
- Equation: (y – 3)2 = 4 * 3 * (x – (-1)) => (y – 3)2 = 12(x + 1)
Using the find equation of parabola given vertex and directrix calculator with these values would confirm these equations.
How to Use This Find Equation of Parabola Given Vertex and Directrix Calculator
- Enter Vertex Coordinates: Input the h and k values of the vertex into the respective fields.
- Select Directrix Type: Choose whether the directrix is a horizontal line (y=) or a vertical line (x=) from the dropdown.
- Enter Directrix Value: Input the constant 'd' from the directrix equation.
- Calculate: The calculator automatically updates as you type or change the selection. You can also click "Calculate".
- Read Results: The primary result shows the equation of the parabola. Intermediate values like 'p', focus, and axis of symmetry are also displayed, along with a table and a graph.
- Reset: Click "Reset" to return to default values.
- Copy: Click "Copy Results" to copy the main equation and key values to your clipboard.
Understanding the results from the find equation of parabola given vertex and directrix calculator helps visualize the parabola's orientation and width.
Key Factors That Affect Find Equation of Parabola Given Vertex and Directrix Calculator Results
- Vertex Position (h, k): This directly sets the h and k values in the standard equation, shifting the parabola's origin.
- Directrix Equation (y=d or x=d): This determines whether the parabola opens vertically or horizontally and, along with the vertex, defines 'p'.
- Value of 'd': The constant in the directrix equation, combined with 'k' or 'h', determines the magnitude and sign of 'p'.
- Magnitude of 'p': The absolute value of 'p' (|p|) determines the "width" of the parabola. Smaller |p| means a narrower parabola, larger |p| means a wider one. The latus rectum length is |4p|.
- Sign of 'p': If the directrix is y=d, p > 0 means it opens up, p < 0 opens down. If x=d, p > 0 opens right, p < 0 opens left.
- Orientation (Vertical or Horizontal Axis): Determined by whether the directrix is y=d (vertical axis, opens up/down) or x=d (horizontal axis, opens left/right).
The find equation of parabola given vertex and directrix calculator relies on these inputs to derive the correct equation.
Frequently Asked Questions (FAQ)
- 1. What if the directrix passes through the vertex?
- If the directrix passed through the vertex, the distance 'p' would be zero, leading to a degenerate parabola (a line). The definition of a parabola requires the focus and directrix to be distinct and not contain the vertex in that manner.
- 2. Can I enter the focus instead of the directrix?
- This specific find equation of parabola given vertex and directrix calculator requires the directrix. If you have the vertex and focus, you can first calculate 'p' and the directrix and then use the calculator, or look for a "parabola from vertex and focus calculator".
- 3. What does p=0 mean?
- If p=0, the vertex is on the directrix, which isn't a standard parabola. The equation would become (x-h)^2 = 0 or (y-k)^2 = 0, representing the lines x=h or y=k.
- 4. How is the axis of symmetry determined?
- The axis of symmetry is perpendicular to the directrix and passes through the vertex and the focus. If the directrix is y=d (horizontal), the axis is x=h (vertical). If the directrix is x=d (vertical), the axis is y=k (horizontal).
- 5. Can the vertex and directrix be the same?
- No, the vertex is a point (h,k) and the directrix is a line (y=d or x=d). The vertex is never on the directrix in a non-degenerate parabola.
- 6. Does the find equation of parabola given vertex and directrix calculator handle very large numbers?
- Yes, it can handle large numbers, but extremely large values might affect the visual representation on the graph due to scaling.
- 7. How do I know if the parabola opens up/down or left/right?
- If the directrix is y=d, it opens up (p>0) or down (p<0). If the directrix is x=d, it opens right (p>0) or left (p<0). The find equation of parabola given vertex and directrix calculator determines this from your inputs.
- 8. What is the latus rectum?
- The latus rectum is a line segment passing through the focus, perpendicular to the axis of symmetry, with endpoints on the parabola. Its length is |4p|.
Related Tools and Internal Resources
- Parabola from Focus and Directrix Calculator: If you know the focus and directrix instead of the vertex.
- Vertex Form Calculator: Work with the vertex form of quadratic equations.
- Standard Form of Parabola Calculator: Convert between different forms of parabola equations.
- Conic Sections Calculator: Explore other conic sections like ellipses and hyperbolas.
- Graphing Parabolas Guide: Learn more about graphing parabolas and their properties.
- Algebra Calculators: A collection of calculators for various algebra problems.