Equation of a Parabola Calculator

Parabola Equation Calculator (Vertex Form)

(Determines width and direction)
(Horizontal shift of vertex)
(Vertical shift of vertex)

Results:

function calculateParabola() { var a = parseFloat(document.getElementById("coeffA").value); var h = parseFloat(document.getElementById("vertexH").value); var k = parseFloat(document.getElementById("vertexK").value); var resultDiv = document.getElementById("parabolaResult"); var standardFormElem = document.getElementById("standardForm"); var vertexCoordsElem = document.getElementById("vertexCoords"); var axisSymmetryElem = document.getElementById("axisSymmetry"); var directionElem = document.getElementById("direction"); var focusCoordsElem = document.getElementById("focusCoords"); var directrixEqElem = document.getElementById("directrixEq"); // Clear previous results standardFormElem.innerHTML = ""; vertexCoordsElem.innerHTML = ""; axisSymmetryElem.innerHTML = ""; directionElem.innerHTML = ""; focusCoordsElem.innerHTML = ""; directrixEqElem.innerHTML = ""; if (isNaN(a) || isNaN(h) || isNaN(k)) { standardFormElem.innerHTML = "Please enter valid numbers for all fields."; return; } if (a === 0) { standardFormElem.innerHTML = "Coefficient 'a' cannot be zero for a parabola. It would be a straight line."; return; } // Calculate standard form coefficients: y = Ax^2 + Bx + C // From y = a(x – h)^2 + k // y = a(x^2 – 2hx + h^2) + k // y = ax^2 – 2ahx + ah^2 + k var A = a; var B = -2 * a * h; var C = a * h * h + k; // Format standard form equation var standardFormString = "Standard Form: y = "; standardFormString += (A === 1 ? "" : (A === -1 ? "-" : A.toFixed(2))) + "x2"; if (B !== 0) { standardFormString += (B > 0 ? " + " : " – ") + Math.abs(B).toFixed(2) + "x"; } if (C !== 0) { standardFormString += (C > 0 ? " + " : " – ") + Math.abs(C).toFixed(2); } if (B === 0 && C === 0) { // Special case for y = ax^2 standardFormString = "Standard Form: y = " + (A === 1 ? "" : (A === -1 ? "-" : A.toFixed(2))) + "x2"; } // Other properties var vertex = "(" + h.toFixed(2) + ", " + k.toFixed(2) + ")"; var axisOfSymmetry = "x = " + h.toFixed(2); var direction = (a > 0) ? "Upward" : "Downward"; // Focus and Directrix for vertical parabola y = a(x-h)^2 + k // Focus: (h, k + 1/(4a)) // Directrix: y = k – 1/(4a) var p = 1 / (4 * a); var focusY = k + p; var directrixY = k – p; var focus = "(" + h.toFixed(2) + ", " + focusY.toFixed(2) + ")"; var directrix = "y = " + directrixY.toFixed(2); // Display results standardFormElem.innerHTML = standardFormString; vertexCoordsElem.innerHTML = "Vertex: " + vertex; axisSymmetryElem.innerHTML = "Axis of Symmetry: " + axisOfSymmetry; directionElem.innerHTML = "Direction of Opening: " + direction; focusCoordsElem.innerHTML = "Focus: " + focus; directrixEqElem.innerHTML = "Directrix: " + directrix; }

Understanding the Equation of a Parabola

A parabola is a U-shaped curve that is symmetrical. It 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). Parabolas are fundamental in mathematics and physics, appearing in the trajectories of projectiles, the design of satellite dishes, and the paths of comets.

Forms of a Parabola's Equation

There are several ways to express the equation of a parabola, but two of the most common for parabolas opening vertically (up or down) are the Standard Form and the Vertex Form.

1. Vertex Form: y = a(x - h)2 + k

This form is incredibly useful because it directly gives you the coordinates of the parabola's vertex and provides insight into its shape and direction:

  • (h, k): Represents the coordinates of the vertex of the parabola. The vertex is the turning point of the parabola.
  • a: The coefficient 'a' determines the width and direction of the parabola.
    • If a > 0, the parabola opens upward.
    • If a < 0, the parabola opens downward.
    • The absolute value of a dictates the width: a larger |a| means a narrower parabola, while a smaller |a| means a wider parabola.

2. Standard Form: y = Ax2 + Bx + C

This is another common form, often encountered when solving quadratic equations. While it doesn't directly show the vertex, it can be derived from it. The coefficients A, B, C are related to a, h, k from the vertex form.

Key Properties of a Parabola

  • Vertex: The highest or lowest point on the parabola, given by (h, k) in vertex form.
  • Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two mirror images. Its equation is x = h.
  • Focus: A fixed point from which all points on the parabola are equidistant to the directrix. For a vertical parabola, its coordinates are (h, k + 1/(4a)).
  • Directrix: A fixed line from which all points on the parabola are equidistant to the focus. For a vertical parabola, its equation is y = k - 1/(4a).
  • Direction of Opening: Determined by the sign of 'a'. Upward if a > 0, downward if a < 0.

How to Use the Calculator

Our Parabola Equation Calculator simplifies the process of understanding these properties. Simply input the values for a, h, and k from your parabola's vertex form equation:

  1. Coefficient 'a': Enter the value that determines the parabola's width and direction.
  2. Vertex x-coordinate 'h': Input the x-coordinate of the parabola's vertex.
  3. Vertex y-coordinate 'k': Input the y-coordinate of the parabola's vertex.

Click "Calculate Parabola Properties," and the calculator will instantly provide you with the standard form equation, the vertex coordinates, the axis of symmetry, the direction of opening, the focus, and the directrix.

Example Calculation

Let's consider a parabola with the equation y = 2(x - 1)2 + 5.

  • a = 2
  • h = 1
  • k = 5

Using the calculator with these inputs, you would get:

  • Standard Form: y = 2x2 - 4x + 7 (derived from 2(x2 - 2x + 1) + 5 = 2x2 - 4x + 2 + 5)
  • Vertex: (1, 5)
  • Axis of Symmetry: x = 1
  • Direction of Opening: Upward (since a = 2 > 0)
  • Focus: (1, 5 + 1/(4*2)) = (1, 5 + 0.125) = (1, 5.125)
  • Directrix: y = 5 – 1/(4*2) = 5 – 0.125 = 4.875

This calculator is a powerful tool for students, educators, and anyone working with quadratic functions and parabolas, making complex calculations straightforward and understandable.

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