Find Equation Of Parabola Given Vertex And Focus Calculator

Welcome to our find equation of parabola given vertex and focus calculator. Enter the coordinates of the vertex and the focus to instantly determine the equation of the parabola and its properties.

Parabola Calculator

What is a Find Equation of Parabola Given Vertex and Focus Calculator?

A find equation of parabola given vertex and focus calculator is a specialized tool designed to determine the standard equation of a parabola when you know the coordinates of its vertex (h, k) and its focus (a, b). Parabolas are U-shaped curves, and their geometric properties are defined by these two points (and the derived directrix). This calculator simplifies the process of deriving the equation, whether the parabola opens vertically (up or down) or horizontally (left or right).

This tool is invaluable for students learning conic sections in algebra and pre-calculus, engineers, physicists, and anyone working with parabolic shapes, such as satellite dishes or reflector designs. The find equation of parabola given vertex and focus calculator helps in understanding the relationship between the vertex, focus, directrix, and the parabola's equation.

Common misconceptions are that any U-shaped curve is a parabola defined this way, but the specific relationship between the vertex, focus, and every point on the parabola (equidistant from focus and directrix) is key.

Find Equation of Parabola Given Vertex and Focus Calculator: Formula and Mathematical Explanation

The standard equations of a parabola depend on its orientation:

• Vertical Parabola (opens up or down): The x-coordinates of the vertex and focus are the same (h = a). The equation is `(x – h)^2 = 4p(y – k)`.
• Horizontal Parabola (opens left or right): The y-coordinates of the vertex and focus are the same (k = b). The equation is `(y – k)^2 = 4p(x – h)`.

In both cases, (h, k) are the coordinates of the vertex. The value 'p' is the directed distance from the vertex to the focus (and also from the vertex to the directrix, in the opposite direction).

Step-by-step derivation:

1. Identify the coordinates of the vertex (h, k) and the focus (a, b).
2. Compare h and a, and k and b.
3. If h = a, the parabola is vertical. Calculate p = b – k.
   • If p > 0, it opens upwards.
   • If p < 0, it opens downwards.
   • The equation is `(x – h)^2 = 4p(y – k)`.
   • Directrix: y = k – p
   • Axis of Symmetry: x = h
4. If k = b, the parabola is horizontal. Calculate p = a – h.
   • If p > 0, it opens to the right.
   • If p < 0, it opens to the left.
   • The equation is `(y – k)^2 = 4p(x – h)`.
   • Directrix: x = h – p
   • Axis of Symmetry: y = k
5. If h=a and k=b, the vertex and focus coincide, p=0, resulting in a degenerate case (not a standard parabola). Our find equation of parabola given vertex and focus calculator handles this.

Variables Table

Variables used in the find equation of parabola given vertex and focus calculator.

Variable	Meaning	Unit	Typical Range
h	x-coordinate of the vertex	Units of length	Any real number
k	y-coordinate of the vertex	Units of length	Any real number
a	x-coordinate of the focus	Units of length	Any real number
b	y-coordinate of the focus	Units of length	Any real number
p	Directed distance from vertex to focus	Units of length	Any non-zero real number (for non-degenerate parabola)

Practical Examples (Real-World Use Cases)

Using a find equation of parabola given vertex and focus calculator is useful in various scenarios.

Example 1: Vertical Parabola

Suppose a satellite dish is designed with a parabolic cross-section. The vertex is at (0, 0) and the focus (where the receiver is placed) is at (0, 2).

• Vertex (h, k) = (0, 0)
• Focus (a, b) = (0, 2)

Since h=a (0=0), it's a vertical parabola. p = b – k = 2 – 0 = 2. 4p = 8. Equation: (x – 0)^2 = 8(y – 0) => x^2 = 8y. Directrix: y = 0 – 2 = -2. Axis of Symmetry: x = 0. It opens upwards (p > 0).

Example 2: Horizontal Parabola

Consider a headlight reflector with a vertex at (1, 3) and the light source (focus) at (3, 3).

• Vertex (h, k) = (1, 3)
• Focus (a, b) = (3, 3)

Since k=b (3=3), it's a horizontal parabola. p = a – h = 3 – 1 = 2. 4p = 8. Equation: (y – 3)^2 = 8(x – 1). Directrix: x = 1 – 2 = -1. Axis of Symmetry: y = 3. It opens to the right (p > 0).

Our find equation of parabola given vertex and focus calculator quickly provides these results.

How to Use This Find Equation of Parabola Given Vertex and Focus Calculator

1. Enter Vertex Coordinates: Input the x-coordinate (h) and y-coordinate (k) of the parabola's vertex into the "Vertex (h)" and "Vertex (k)" fields.
2. Enter Focus Coordinates: Input the x-coordinate (a) and y-coordinate (b) of the parabola's focus into the "Focus (a)" and "Focus (b)" fields.
3. View Results: The calculator automatically updates and displays the equation of the parabola in standard form, the direction it opens, the value of 'p' and '4p', the equation of the directrix, and the axis of symmetry as you type.
4. Interpret the Chart: The visualization shows the relative positions of the vertex, focus, directrix, and the shape of the parabola.
5. Reset: Click the "Reset" button to clear the inputs and results to their default values.
6. Copy Results: Click "Copy Results" to copy the main equation and key parameters to your clipboard.

The find equation of parabola given vertex and focus calculator provides immediate feedback, allowing you to explore how changes in the vertex or focus affect the parabola's shape and equation.

Key Factors That Affect Parabola Equation Results

1. Vertex Position (h, k): The vertex directly sets the `(x-h)` and `(y-k)` terms in the equation, shifting the parabola's origin.
2. Focus Position (a, b): The focus, relative to the vertex, determines the orientation (vertical or horizontal) and the value of 'p'.
3. Relative Position of Vertex and Focus: If the x-coordinates match, it's vertical; if y-coordinates match, it's horizontal. The difference determines 'p'.
4. Value of 'p': This is the distance from the vertex to the focus (and vertex to directrix). A larger absolute value of 'p' means a wider parabola; a smaller absolute value means a narrower parabola.
5. Sign of 'p': Determines the direction of opening (up/down for vertical, right/left for horizontal).
6. Alignment: The calculator assumes the axis of symmetry is parallel to either the x-axis or y-axis. If the vertex and focus do not align this way, the parabola is rotated, and the standard equations used here don't directly apply. Our find equation of parabola given vertex and focus calculator works for non-rotated parabolas.

Frequently Asked Questions (FAQ)

What if the vertex and focus are the same point?

If the vertex and focus are the same, then p=0. This results in a degenerate parabola, either a line (for the vertical case (x-h)^2=0 => x=h) or two coincident lines (for the horizontal case (y-k)^2=0 => y=k). The find equation of parabola given vertex and focus calculator will indicate p=0.

How do I know if the parabola opens up, down, left, or right?

If the x-coordinates of the vertex and focus are the same, it's vertical. It opens up if p (b-k) is positive, down if negative. If the y-coordinates are the same, it's horizontal. It opens right if p (a-h) is positive, left if negative.

What is the directrix of a parabola?

The directrix is a line such that every point on the parabola is equidistant from the focus and the directrix. Its equation is y = k – p for vertical parabolas and x = h – p for horizontal parabolas.

What is the axis of symmetry?

It's a line that divides the parabola into two mirror images. It passes through the vertex and the focus. For vertical parabolas, it's x = h; for horizontal, it's y = k.

Can this calculator handle rotated parabolas?

No, this find equation of parabola given vertex and focus calculator is designed for parabolas with axes of symmetry parallel to the x or y axes. Rotated parabolas have more complex equations involving an 'xy' term.

What does 'p' represent?

'p' is the directed distance from the vertex to the focus. Its absolute value is the focal length.

How does the '4p' value relate to the equation?

The '4p' term is the coefficient of the linear term in the standard equation `(x-h)^2 = 4p(y-k)` or `(y-k)^2 = 4p(x-h)`. It determines the "width" or "latus rectum" of the parabola.

Can I input fractions or decimals?

Yes, you can input decimal numbers into the find equation of parabola given vertex and focus calculator fields.

Related Tools and Internal Resources

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• Vertex Form Calculator: Convert quadratic equations to vertex form.
• Focus and Directrix Calculator: Find parabola properties from focus and directrix.
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• Graphing Parabolas Tool: Visualize parabolas based on their equations.
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