Calculate key features of a parabola given its quadratic equation in the form ax² + bx + c.
Results:
Vertex:
Axis of Symmetry:
Y-intercept:
Opens Up/Down:
Understanding Parabolas
A parabola is a U-shaped curve, one of the fundamental conic sections. In mathematics, it's most commonly represented by a quadratic equation of the form:
y = ax² + bx + c
where a, b, and c are coefficients. The value of 'a' dictates the parabola's orientation and width. The vertex represents the minimum or maximum point of the parabola. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror images.
Key Properties and Formulas:
Vertex (h, k): The x-coordinate of the vertex (h) is found using the formula h = -b / (2a). The y-coordinate (k) is found by substituting this 'h' value back into the original equation: k = a(h)² + b(h) + c.
Axis of Symmetry: This is a vertical line passing through the vertex, so its equation is x = h, or x = -b / (2a).
Y-intercept: This is the point where the parabola crosses the y-axis. It occurs when x = 0. Substituting x=0 into the equation gives y = a(0)² + b(0) + c, so the y-intercept is simply the value of c, or the point (0, c).
Direction:
If a > 0, the parabola opens upwards (U-shape).
If a < 0, the parabola opens downwards (∩-shape).
Width: A larger absolute value of 'a' results in a narrower parabola, while a smaller absolute value of 'a' results in a wider parabola.
Use Cases for Parabola Calculations:
Parabolas appear in various real-world scenarios:
Physics: The trajectory of projectiles (like a ball thrown in the air) follows a parabolic path under gravity, ignoring air resistance. The vertex represents the highest point of the trajectory.
Engineering: Parabolic reflectors are used in satellite dishes, telescopes, and headlights because they can focus parallel rays of light or radio waves to a single point (the focus), or broadcast them in parallel beams.
Architecture: The shape of suspension bridge cables and arches often approximates a parabola.
Economics: Some economic models involve parabolic functions to represent cost or revenue curves.
This calculator helps visualize and understand these mathematical properties quickly.
function calculateParabola() {
var a = parseFloat(document.getElementById("coefficientA").value);
var b = parseFloat(document.getElementById("coefficientB").value);
var c = parseFloat(document.getElementById("coefficientC").value);
var resultDiv = document.getElementById("result-value");
var vertexSpan = document.getElementById("vertex");
var axisOfSymmetrySpan = document.getElementById("axisOfSymmetry");
var yInterceptSpan = document.getElementById("yIntercept");
var opensDirectionSpan = document.getElementById("opensDirection");
// Clear previous results
resultDiv.innerHTML = "";
vertexSpan.textContent = "";
axisOfSymmetrySpan.textContent = "";
yInterceptSpan.textContent = "";
opensDirectionSpan.textContent = "";
// Input validation
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = "Error: Please enter valid numbers for all coefficients.";
return;
}
if (a === 0) {
resultDiv.innerHTML = "Error: Coefficient 'a' cannot be zero for a parabola.";
return;
}
// Calculate Vertex x-coordinate (h)
var h = -b / (2 * a);
// Calculate Vertex y-coordinate (k)
var k = a * (h * h) + b * h + c;
// Determine direction
var direction = (a > 0) ? "Upwards (U-shape)" : "Downwards (∩-shape)";
// Format results
var formattedVertex = `(${h.toFixed(4)}, ${k.toFixed(4)})`;
var formattedAxisOfSymmetry = `x = ${h.toFixed(4)}`;
var formattedYIntercept = `(0, ${c})`;
// Display results
resultDiv.innerHTML = "