Evaluate f(x) = ax² + bx + c and analyze key properties
Result f(x):0
Discriminant (Δ):0
Vertex (h, k):0
Roots (x-intercepts):None
Understanding Mathematical Functions
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Our calculator specifically focuses on Quadratic Functions, which are polynomial functions of the second degree.
Standard Form: f(x) = ax² + bx + c
Key Components of a Quadratic Function
The Coefficients: 'a' determines the width and direction of the parabola. If a > 0, it opens upwards; if a < 0, it opens downwards.
The Discriminant: Calculated as b² – 4ac. This value tells us the nature of the roots. If positive, there are two real roots; if zero, one real root; if negative, the roots are complex.
The Vertex: This is the peak or the lowest point of the parabola, represented by the coordinates (h, k).
Example Calculation
Consider the function f(x) = 1x² – 4x + 4. To find the value when x = 3:
Square the x value: 3² = 9
Multiply by coefficient 'a': 1 * 9 = 9
Multiply 'b' by x: -4 * 3 = -12
Add the constant 'c': 9 – 12 + 4 = 1
Thus, f(3) = 1. The vertex of this specific function is at (2, 0), and since the discriminant is 0, it has exactly one real root at x = 2.
Applications of Quadratic Functions
Quadratic functions are widely used in physics to model projectile motion, in economics to find maximum profit or minimum cost, and in engineering to design parabolic reflectors and bridges. By understanding the relationship between the coefficients and the resulting graph, professionals can predict outcomes and optimize systems efficiently.
function calculateFunction() {
var a = parseFloat(document.getElementById('coeff_a').value);
var b = parseFloat(document.getElementById('coeff_b').value);
var c = parseFloat(document.getElementById('const_c').value);
var x = parseFloat(document.getElementById('val_x').value);
if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(x)) {
alert("Please enter valid numeric values for all fields.");
return;
}
if (a === 0) {
alert("For a quadratic function, 'a' cannot be zero. If a=0, this is a linear function.");
}
// Calculate f(x)
var fx = (a * Math.pow(x, 2)) + (b * x) + c;
// Calculate Discriminant
var disc = (b * b) – (4 * a * c);
// Calculate Vertex
var h = -b / (2 * a);
var k = (a * Math.pow(h, 2)) + (b * h) + c;
// Calculate Roots
var rootsText = "";
if (disc > 0) {
var r1 = (-b + Math.sqrt(disc)) / (2 * a);
var r2 = (-b – Math.sqrt(disc)) / (2 * a);
rootsText = "x₁ = " + r1.toFixed(2) + ", x₂ = " + r2.toFixed(2);
} else if (disc === 0) {
var r = -b / (2 * a);
rootsText = "x = " + r.toFixed(2);
} else {
rootsText = "Complex Roots (No Real Intercepts)";
}
// Display results
document.getElementById('res_fx').innerText = fx.toLocaleString(undefined, {minimumFractionDigits: 0, maximumFractionDigits: 4});
document.getElementById('res_disc').innerText = disc.toFixed(2);
document.getElementById('res_vertex').innerText = "(" + h.toFixed(2) + ", " + k.toFixed(2) + ")";
document.getElementById('res_roots').innerText = rootsText;
document.getElementById('resultsArea').style.display = 'block';
}