Vertex, Focus, and Directrix Calculator
Find the vertex, focus, and directrix for a parabola given its equation in standard form using our Vertex, Focus, and Directrix Calculator.
Parabola Calculator
x = a(y – k)² + h
What is a Vertex, Focus, and Directrix Calculator?
A Vertex, Focus, and Directrix Calculator is a tool used to determine the key characteristics of a parabola: its vertex (the point where the parabola turns), its focus (a point inside the parabola used in its geometric definition), and its directrix (a line outside the parabola used in its geometric definition). A parabola is defined as the set of all points that are equidistant from the focus and the directrix. Our Vertex, Focus, and Directrix Calculator simplifies finding these elements from the standard form of a parabola's equation.
This calculator is useful for students learning about conic sections, engineers, physicists, and anyone working with parabolic shapes, such as satellite dishes or reflector telescopes. A common misconception is that the 'a' value directly gives the focus, but it's used to calculate the distance to the focus from the vertex.
Vertex, Focus, and Directrix Formula and Mathematical Explanation
The standard equations of a parabola with vertex (h, k) are:
- y = a(x – h)² + k: This parabola opens vertically. If 'a' is positive, it opens upwards; if 'a' is negative, it opens downwards.
- x = a(y – k)² + h: This parabola opens horizontally. If 'a' is positive, it opens to the right; if 'a' is negative, it opens to the left.
For the form y = a(x – h)² + k:
- Vertex: (h, k)
- The distance from the vertex to the focus and from the vertex to the directrix is |p|, where p = 1/(4a).
- Focus: (h, k + p) = (h, k + 1/(4a))
- Directrix: y = k – p = k – 1/(4a)
For the form x = a(y – k)² + h:
- Vertex: (h, k)
- The distance p = 1/(4a) is measured along the x-axis.
- Focus: (h + p, k) = (h + 1/(4a), k)
- Directrix: x = h – p = h – 1/(4a)
The Vertex, Focus, and Directrix Calculator uses these formulas based on the selected equation form.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Coefficient determining the width and direction of the parabola | None | Any non-zero real number |
| h | x-coordinate of the vertex (for y=…) or part of vertex (for x=…) | None | Any real number |
| k | y-coordinate of the vertex (for y=…) or part of vertex (for x=…) | None | Any real number |
| p | Focal distance (1/(4a)) | None | Any non-zero real number |
Practical Examples
Let's see how our Vertex, Focus, and Directrix Calculator works with some examples.
Example 1: y = 2(x – 3)² + 1
- Form: y = a(x – h)² + k
- a = 2, h = 3, k = 1
- Vertex: (3, 1)
- p = 1 / (4 * 2) = 1/8 = 0.125
- Focus: (3, 1 + 0.125) = (3, 1.125)
- Directrix: y = 1 – 0.125 = 0.875
Using the Vertex, Focus, and Directrix Calculator with a=2, h=3, k=1 and form 'y=…' gives these results.
Example 2: x = -0.5(y + 1)² – 2 (which is x = -0.5(y – (-1))² + (-2))
- Form: x = a(y – k)² + h
- a = -0.5, k = -1, h = -2
- Vertex: (-2, -1)
- p = 1 / (4 * -0.5) = 1 / -2 = -0.5
- Focus: (-2 + (-0.5), -1) = (-2.5, -1)
- Directrix: x = -2 – (-0.5) = -2 + 0.5 = -1.5
The Vertex, Focus, and Directrix Calculator helps verify these quickly.
How to Use This Vertex, Focus, and Directrix Calculator
- Select the Form: Choose the form of your parabola's equation: `y = a(x – h)² + k` or `x = a(y – k)² + h`.
- Enter 'a': Input the coefficient 'a'. It cannot be zero.
- Enter 'h': Input the value of 'h' from the equation.
- Enter 'k': Input the value of 'k' from the equation.
- View Results: The calculator automatically updates the Vertex, Focus, Directrix, and the value of p (1/4a). The results are also shown in a table and visualized in a basic chart.
- Interpret: The vertex is the turning point. The focus is a point, and the directrix is a line that defines the parabola. 'p' is the distance from the vertex to the focus and vertex to the directrix.
Key Factors That Affect Vertex, Focus, and Directrix Results
- Value of 'a': Determines how wide or narrow the parabola is and its direction of opening. A smaller |a| means a wider parabola, larger |a| means narrower. The sign of 'a' determines the opening direction and thus the relative positions of the focus and directrix.
- Sign of 'a': If positive for y=…, opens up; negative, opens down. If positive for x=…, opens right; negative, opens left. This directly affects the focus and directrix calculation relative to the vertex.
- Values of 'h' and 'k': These directly give the coordinates of the vertex (h, k). Any change in h or k shifts the entire parabola, including its vertex, focus, and directrix, without changing its shape or orientation.
- Form of the Equation: Whether the equation is in terms of `(x-h)²` or `(y-k)²` determines if the parabola opens vertically or horizontally, completely changing the formulas for focus and directrix.
- Non-zero 'a': 'a' cannot be zero because if it were, the equation would become linear (y=k or x=h), not quadratic, and would not represent a parabola. The Vertex, Focus, and Directrix Calculator will show an error if a=0.
- Accuracy of h and k: Especially when the equation is not in perfect standard form (e.g., y = 2(x+3)²+1, where h=-3), correctly identifying h and k is crucial.
Frequently Asked Questions (FAQ)
What is a parabola?
A parabola is a U-shaped curve that is defined as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
What happens if 'a' is zero in the Vertex, Focus, and Directrix Calculator?
If 'a' is zero, the equation is no longer quadratic and does not represent a parabola. The formulas for focus (1/4a) and directrix would involve division by zero. Our Vertex, Focus, and Directrix Calculator will indicate an error if 'a' is 0.
How do I find 'a', 'h', and 'k' from a general form equation like y = 2x² + 8x + 9?
You need to complete the square to convert it to the standard form y = a(x – h)² + k. For y = 2x² + 8x + 9, it becomes y = 2(x² + 4x) + 9 = 2(x² + 4x + 4 – 4) + 9 = 2((x + 2)² – 4) + 9 = 2(x + 2)² – 8 + 9 = 2(x + 2)² + 1. So, a=2, h=-2, k=1. You can then use the Vertex, Focus, and Directrix Calculator.
What does the 'p' value signify?
The value p = 1/(4a) represents the directed distance from the vertex to the focus and from the vertex to the directrix. Its absolute value |p| is the focal length.
Can the focus be the same as the vertex?
No, the focus is distinct from the vertex unless 'a' goes to infinity, which isn't a standard parabola. The distance 'p' is non-zero if 'a' is finite and non-zero.
Why is the directrix a line and the focus a point?
That's based on the geometric definition of a parabola: the set of points equidistant from a single point (focus) and a single line (directrix).
Does this Vertex, Focus, and Directrix Calculator work for rotated parabolas?
No, this calculator is designed for parabolas whose axes of symmetry are parallel to the x-axis or y-axis, represented by the standard forms y = a(x – h)² + k and x = a(y – k)² + h. Rotated parabolas have an 'xy' term in their equation.
What are real-world applications of parabolas?
Parabolic shapes are used in satellite dishes, car headlights, telescopes, and the paths of projectiles under gravity (ignoring air resistance).