Parabola Equation Calculator

Parabola Equation Calculator

Enter the coordinates of the parabola's vertex and a point that lies on the parabola to determine its equation in both vertex and standard forms.

function calculateParabola() { var vertexX = parseFloat(document.getElementById("vertexX").value); var vertexY = parseFloat(document.getElementById("vertexY").value); var pointX = parseFloat(document.getElementById("pointX").value); var pointY = parseFloat(document.getElementById("pointY").value); var errorMessage = document.getElementById("errorMessage"); errorMessage.textContent = ""; // Clear previous errors document.getElementById("aValue").textContent = ""; document.getElementById("vertexForm").textContent = ""; document.getElementById("standardForm").textContent = ""; if (isNaN(vertexX) || isNaN(vertexY) || isNaN(pointX) || isNaN(pointY)) { errorMessage.textContent = "Please enter valid numbers for all fields."; return; } // Calculate 'a' var xMinusH = pointX – vertexX; var denominator = xMinusH * xMinusH; if (denominator === 0) { // This means pointX is equal to vertexX. // If pointY is also equal to vertexY, the point IS the vertex, which doesn't define 'a' uniquely. // If pointY is NOT equal to vertexY, it implies a vertical line, not a parabola in y=f(x) form. errorMessage.textContent = "The point's X-coordinate cannot be the same as the vertex's X-coordinate unless the point is the vertex itself. Please choose a point not on the axis of symmetry."; return; } var a = (pointY – vertexY) / denominator; // Display 'a' value document.getElementById("aValue").textContent = a.toFixed(4); // Construct Vertex Form: y = a(x – h)^2 + k var vertexFormEquation = "y = " + a.toFixed(4); if (vertexX === 0) { vertexFormEquation += "x^2"; } else { var hTerm = (vertexX < 0) ? "+ " + Math.abs(vertexX).toFixed(4) : "- " + vertexX.toFixed(4); vertexFormEquation += "(x " + hTerm + ")^2"; } if (vertexY !== 0) { var kTerm = (vertexY < 0) ? "- " + Math.abs(vertexY).toFixed(4) : "+ " + vertexY.toFixed(4); vertexFormEquation += " " + kTerm; } document.getElementById("vertexForm").textContent = vertexFormEquation; // Construct Standard Form: 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 b = -2 * a * vertexX; var c = a * vertexX * vertexX + vertexY; var standardFormEquation = "y = " + a.toFixed(4) + "x^2"; if (b !== 0) { var bTerm = (b < 0) ? "- " + Math.abs(b).toFixed(4) : "+ " + b.toFixed(4); standardFormEquation += " " + bTerm + "x"; } if (c !== 0) { var cTerm = (c < 0) ? "- " + Math.abs(c).toFixed(4) : "+ " + c.toFixed(4); standardFormEquation += " " + cTerm; } document.getElementById("standardForm").textContent = standardFormEquation; }

Understanding Parabola Equations

A parabola is a U-shaped curve that is symmetric. It is defined as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed straight line (the directrix).

Key Components of a Parabola:

• Vertex: The turning point of the parabola, where it changes direction. It is the point where the parabola is closest to its focus and directrix.
• Focus: A fixed point from which all points on the parabola are equidistant to the directrix.
• Directrix: A fixed straight line from which all points on the parabola are equidistant to the focus.
• Axis of Symmetry: A line passing through the vertex and the focus, perpendicular to the directrix. The parabola is symmetric about this line.
• 'a' Coefficient: In the equations, 'a' determines the width and direction of the parabola. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards. A larger absolute value of 'a' means a narrower parabola.

Forms of Parabola Equations:

There are two primary forms for the equation of a vertical parabola (one that opens up or down):

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

This form is particularly useful because it directly gives you the coordinates of the vertex, which are (h, k). The 'a' value, as mentioned, dictates the parabola's opening direction and width.

• (h, k): The coordinates of the vertex.
• a: Determines the direction and width of the parabola.

2. Standard Form: y = ax^2 + bx + c

This is a more general quadratic equation. While it doesn't directly show the vertex, it's a common form for quadratic functions. The vertex can be found using the formula x = -b / (2a), and then substituting this x-value back into the equation to find y.

• a: Same as in vertex form, determines direction and width.
• b: Influences the position of the vertex horizontally.
• c: The y-intercept of the parabola (where x=0).

How This Calculator Works:

This calculator uses the vertex form y = a(x - h)^2 + k. By providing the coordinates of the vertex (h, k) and one additional point (x, y) that lies on the parabola, the calculator can solve for the unique 'a' coefficient. Once 'a' is determined, it constructs the full vertex form equation. It then expands this vertex form into the standard form y = ax^2 + bx + c using the relationships b = -2ah and c = ah^2 + k.

Example Calculation:

Let's say you have a parabola with a vertex at (1, 2) and it passes through the point (3, 6).

1. Identify inputs:
   h = 1
k = 2
x = 3
y = 6
2. Use the vertex form: y = a(x - h)^2 + k
3. Substitute the values:
   6 = a(3 - 1)^2 + 2
6 = a(2)^2 + 2
6 = 4a + 2
4. Solve for 'a':
   6 - 2 = 4a
4 = 4a
a = 1
5. Vertex Form Equation:
   Substitute a=1, h=1, k=2 back into the vertex form:
   y = 1(x - 1)^2 + 2
y = (x - 1)^2 + 2
6. Standard Form Equation:
   Expand the vertex form:
   y = (x^2 - 2x + 1) + 2
y = x^2 - 2x + 3
   (Alternatively, using b = -2ah and c = ah^2 + k:
   b = -2 * 1 * 1 = -2
c = 1 * 1^2 + 2 = 1 + 2 = 3
   So, y = 1x^2 - 2x + 3)

Using the calculator with these values should yield a = 1, y = (x - 1)^2 + 2, and y = x^2 - 2x + 3.