function calculateFunction() {
var a = parseFloat(document.getElementById('coeffA').value);
var b = parseFloat(document.getElementById('coeffB').value);
var c = parseFloat(document.getElementById('coeffC').value);
var xVal = parseFloat(document.getElementById('valX').value);
if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(xVal)) {
alert("Please enter valid numerical values for all coefficients.");
return;
}
if (a === 0) {
alert("Coefficient 'a' cannot be 0 for a quadratic function. This would be a linear function.");
return;
}
// Equation string
var eq = "f(x) = " + a + "x² " + (b >= 0 ? "+ " : "- ") + Math.abs(b) + "x " + (c >= 0 ? "+ " : "- ") + Math.abs(c);
document.getElementById('resEquation').innerText = eq;
// Discriminant
var disc = (b * b) – (4 * a * c);
document.getElementById('resDisc').innerText = disc.toFixed(4);
// Roots
var rootsSpan = document.getElementById('resRoots');
if (disc > 0) {
var x1 = (-b + Math.sqrt(disc)) / (2 * a);
var x2 = (-b – Math.sqrt(disc)) / (2 * a);
rootsSpan.innerText = "x₁ = " + x1.toFixed(4) + ", x₂ = " + x2.toFixed(4);
} else if (disc === 0) {
var x = -b / (2 * a);
rootsSpan.innerText = "x = " + x.toFixed(4) + " (Double Root)";
} else {
rootsSpan.innerText = "No real roots (Complex roots)";
}
// Vertex
var h = -b / (2 * a);
var k = (a * h * h) + (b * h) + c;
document.getElementById('resVertex').innerText = "(" + h.toFixed(4) + ", " + k.toFixed(4) + ")";
// Evaluation
var fX = (a * xVal * xVal) + (b * xVal) + c;
document.getElementById('resEval').innerText = "f(" + xVal + ") = " + fX.toFixed(4);
// Direction
document.getElementById('resDir').innerText = a > 0 ? "Opens Upward (Minimum)" : "Opens Downward (Maximum)";
document.getElementById('function-results').style.display = 'block';
}
Understanding Quadratic Functions
A quadratic function is a polynomial function of degree two. The general form is represented as f(x) = ax² + bx + c, where a, b, and c are constants and a is not equal to zero. These functions create a U-shaped curve known as a parabola.
Key Components of the Analysis
The Discriminant (Δ): Calculated as b² – 4ac. This value determines the nature of the roots. If Δ > 0, there are two distinct real roots. If Δ = 0, there is one real root. If Δ < 0, the function does not cross the x-axis (complex roots).
The Vertex: This is the highest or lowest point on the parabola. The x-coordinate (h) is found using -b / 2a, and the y-coordinate (k) is found by plugging h back into the function.
Direction of Opening: If the leading coefficient a is positive, the parabola opens upward like a cup. If a is negative, it opens downward.
Practical Example
Suppose you have the function f(x) = 1x² + 2x – 8. Using the calculator above:
Identify Coefficients: a = 1, b = 2, c = -8.
Find Discriminant: 2² – 4(1)(-8) = 4 + 32 = 36. Since 36 is positive, we expect two real roots.
Calculate Roots: Using the quadratic formula, we find x = 2 and x = -4.
Locate Vertex: h = -2 / (2*1) = -1. k = f(-1) = 1(-1)² + 2(-1) – 8 = -9. The vertex is at (-1, -9).
Common Applications
Quadratic functions are widely used in physics to model projectile motion, in economics to find profit maximization points, and in engineering to design parabolic reflectors. This calculator simplifies the algebraic process, providing instant insights into the geometric properties of any standard quadratic equation.