Quadratic Function Calculator: f(x) = ax² + bx + c
Function Evaluation:
'; outputHtml += 'f(' + x + ') = ' + fx.toFixed(4) + "; // Handle case where 'a' is zero (linear function) if (a === 0) { outputHtml += 'Vertex & Roots:
'; outputHtml += 'Since coefficient \'a\' is 0, this is a linear function (not quadratic).'; outputHtml += 'Vertex is not applicable for a linear function.'; if (b !== 0) { var linearRoot = -c / b; outputHtml += 'Root (for bx + c = 0): x = ' + linearRoot.toFixed(4) + "; } else if (c === 0) { outputHtml += 'The equation 0x + 0 = 0 is true for all x (infinite solutions).'; } else { outputHtml += 'The equation 0x + c = 0 (where c ≠ 0) has no solution.'; } resultDiv.innerHTML = outputHtml; return; } // Calculate Vertex var vertexX = -b / (2 * a); var vertexY = a * vertexX * vertexX + b * vertexX + c; outputHtml += 'Vertex:
'; outputHtml += 'Vertex (x, y): (' + vertexX.toFixed(4) + ', ' + vertexY.toFixed(4) + ')'; // Calculate Discriminant var discriminant = b * b – 4 * a * c; outputHtml += 'Discriminant (Δ):
'; outputHtml += 'Δ = ' + discriminant.toFixed(4) + "; // Calculate Roots outputHtml += 'Roots (x-intercepts):
'; if (discriminant > 0) { var root1 = (-b + Math.sqrt(discriminant)) / (2 * a); var root2 = (-b – Math.sqrt(discriminant)) / (2 * a); outputHtml += 'Two distinct real roots:'; outputHtml += 'x₁ = ' + root1.toFixed(4) + "; outputHtml += 'x₂ = ' + root2.toFixed(4) + "; } else if (discriminant === 0) { var root = -b / (2 * a); outputHtml += 'One real root (repeated):'; outputHtml += 'x = ' + root.toFixed(4) + "; } else { var realPart = (-b / (2 * a)).toFixed(4); var imaginaryPart = (Math.sqrt(Math.abs(discriminant)) / (2 * a)).toFixed(4); outputHtml += 'Two complex conjugate roots:'; outputHtml += 'x₁ = ' + realPart + ' + ' + imaginaryPart + 'i'; outputHtml += 'x₂ = ' + realPart + ' – ' + imaginaryPart + 'i'; } resultDiv.innerHTML = outputHtml; } .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; border-radius: 8px; padding: 20px; max-width: 600px; margin: 20px auto; font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; box-shadow: 0 4px 8px rgba(0,0,0,0.05); } .calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; font-size: 1.8em; } .calculator-content { display: flex; flex-direction: column; gap: 15px; } .input-group { display: flex; flex-direction: column; } .input-group label { margin-bottom: 5px; color: #555; font-weight: bold; } .input-group input[type="number"] { padding: 10px; border: 1px solid #ccc; border-radius: 4px; font-size: 1em; width: 100%; box-sizing: border-box; } .calculate-button { background-color: #007bff; color: white; padding: 12px 20px; border: none; border-radius: 5px; cursor: pointer; font-size: 1.1em; margin-top: 10px; transition: background-color 0.3s ease; } .calculate-button:hover { background-color: #0056b3; } .result-output { background-color: #e9ecef; border: 1px solid #dee2e6; border-radius: 5px; padding: 15px; margin-top: 20px; color: #333; } .result-output h3 { color: #0056b3; margin-top: 0; margin-bottom: 10px; font-size: 1.3em; } .result-output p { margin-bottom: 5px; line-height: 1.6; } .result-output .error { color: #dc3545; font-weight: bold; }Understanding the Quadratic Function: f(x) = ax² + bx + c
A quadratic function is a polynomial function of degree two. Its general form is f(x) = ax² + bx + c, where 'a', 'b', and 'c' are coefficients, and 'a' cannot be zero. The graph of a quadratic function is a parabola, which is a U-shaped curve that can open either upwards or downwards.
Key Components and Their Significance:
- Coefficient 'a': This coefficient determines the direction and width of the parabola.
- If
a > 0, the parabola opens upwards (it has a minimum point). - If
a < 0, the parabola opens downwards (it has a maximum point). - A larger absolute value of 'a' makes the parabola narrower, while a smaller absolute value makes it wider.
- If
- Coefficient 'b': Along with 'a', 'b' influences the position of the parabola's vertex.
- Constant 'c': This is the y-intercept of the parabola. When
x = 0,f(0) = c. - X-value for Evaluation: This is a specific point on the x-axis for which you want to find the corresponding y-value (the function's output).
What Our Calculator Determines:
Our Quadratic Function Calculator helps you analyze these functions by providing several key values:
1. Function Evaluation (f(x))
This calculates the value of the function f(x) for a specific x that you provide. It tells you the y-coordinate on the parabola corresponding to your chosen x-coordinate.
Formula: f(x) = ax² + bx + c
2. Vertex Coordinates (x, y)
The vertex is the turning point of the parabola. If the parabola opens upwards, the vertex is the minimum point; if it opens downwards, it's the maximum point. The vertex is crucial for understanding the range and symmetry of the function.
Formulas:
x_vertex = -b / (2a)y_vertex = f(x_vertex)(substitutex_vertexback into the original function)
3. Discriminant (Δ)
The discriminant is a part of the quadratic formula that tells us about the nature of the roots (x-intercepts) of the quadratic equation ax² + bx + c = 0.
Formula: Δ = b² - 4ac
- If
Δ > 0: There are two distinct real roots (the parabola crosses the x-axis at two different points). - If
Δ = 0: There is exactly one real root (a repeated root), meaning the parabola touches the x-axis at exactly one point (its vertex is on the x-axis). - If
Δ < 0: There are no real roots, but two complex conjugate roots (the parabola does not intersect the x-axis).
4. Roots (x-intercepts)
The roots are the values of x for which f(x) = 0. These are the points where the parabola intersects the x-axis. They are found using the quadratic formula.
Quadratic Formula: x = [-b ± sqrt(b² - 4ac)] / (2a)
The calculator will provide these roots, indicating if they are real or complex based on the discriminant.
Example Usage:
Let's consider the quadratic function f(x) = x² - 3x + 2.
- Coefficient 'a': 1
- Coefficient 'b': -3
- Constant 'c': 2
- X-value for evaluation: Let's use
x = 0
Using the calculator with these inputs, you would get:
- Function Evaluation f(0):
1*(0)² - 3*(0) + 2 = 2 - Vertex:
x_vertex = -(-3) / (2*1) = 3/2 = 1.5y_vertex = (1.5)² - 3*(1.5) + 2 = 2.25 - 4.5 + 2 = -0.25- Vertex: (1.5, -0.25)
- Discriminant:
Δ = (-3)² - 4*(1)*(2) = 9 - 8 = 1 - Roots: Since
Δ > 0, there are two real roots.x₁ = [ -(-3) + sqrt(1) ] / (2*1) = (3 + 1) / 2 = 4 / 2 = 2x₂ = [ -(-3) - sqrt(1) ] / (2*1) = (3 - 1) / 2 = 2 / 2 = 1- Roots: x = 1 and x = 2
This calculator is a powerful tool for students, engineers, and anyone working with quadratic equations, providing quick and accurate insights into their behavior.