Desmo Graphing Calculator

Quadratic Equation & Parabola Calculator

Find the roots and vertex of any quadratic equation, perfect for graphing with tools like Desmos.

function calculateQuadratic() { 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"); var output = ""; if (isNaN(a) || isNaN(b) || isNaN(c)) { output = "Please enter valid numbers for all coefficients."; resultDiv.innerHTML = output; return; } if (a === 0) { output = "Coefficient 'a' cannot be zero for a quadratic equation. This would be a linear equation."; resultDiv.innerHTML = output; return; } var discriminant = b * b – 4 * a * c; var vertexX = -b / (2 * a); var vertexY = a * vertexX * vertexX + b * vertexX + c; output += "Equation: " + a + "x² + " + b + "x + " + c + " = 0″; output += "Discriminant (Δ): " + discriminant.toFixed(4) + ""; if (discriminant > 0) { var root1 = (-b + Math.sqrt(discriminant)) / (2 * a); var root2 = (-b – Math.sqrt(discriminant)) / (2 * a); output += "Roots (x-intercepts):"; output += "x₁ = " + root1.toFixed(4) + ""; output += "x₂ = " + root2.toFixed(4) + ""; output += "(Two distinct real roots)"; } else if (discriminant === 0) { var root = -b / (2 * a); output += "Root (x-intercept):"; output += "x = " + root.toFixed(4) + ""; output += "(One real root, repeated)"; } else { var realPart = (-b / (2 * a)).toFixed(4); var imaginaryPart = (Math.sqrt(Math.abs(discriminant)) / (2 * a)).toFixed(4); output += "Roots (x-intercepts):"; output += "x₁ = " + realPart + " + " + imaginaryPart + "i"; output += "x₂ = " + realPart + " – " + imaginaryPart + "i"; output += "(Two complex conjugate roots)"; } output += "Vertex:"; output += "(x, y) = (" + vertexX.toFixed(4) + ", " + vertexY.toFixed(4) + ")"; output += "(The turning point of the parabola)"; resultDiv.innerHTML = output; }

Understanding Quadratic Equations and Their Graphs

Quadratic equations are fundamental in mathematics, appearing in various fields from physics to engineering. They are polynomial equations of the second degree, meaning the highest power of the variable (usually 'x') is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients, and 'a' cannot be zero.

The Parabola: Graphing Quadratic Equations

When you graph a quadratic equation on a coordinate plane, the result is a distinctive U-shaped curve called a parabola. The direction of the parabola (opening upwards or downwards) depends on the sign of the coefficient 'a':

• If a > 0, the parabola opens upwards, and its vertex is the minimum point.
• If a < 0, the parabola opens downwards, and its vertex is the maximum point.

Key Features of a Parabola

To fully understand and graph a parabola, two key features are crucial:

1. Roots (x-intercepts): These are the points where the parabola intersects the x-axis. At these points, the value of 'y' (or f(x)) is zero. A quadratic equation can have:
   • Two distinct real roots: The parabola crosses the x-axis at two different points. This occurs when the discriminant (Δ = b² - 4ac) is greater than zero.
   • One real root (repeated): The parabola touches the x-axis at exactly one point (its vertex lies on the x-axis). This happens when the discriminant is equal to zero.
   • Two complex conjugate roots: The parabola does not intersect the x-axis at all. This occurs when the discriminant is less than zero.
   The roots are found using the quadratic formula: x = [-b ± sqrt(b² - 4ac)] / 2a.
2. Vertex: This is the turning point of the parabola. It's either the lowest point (minimum) if the parabola opens upwards or the highest point (maximum) if it opens downwards. The coordinates of the vertex are given by:
   • x-coordinate: x = -b / 2a
   • y-coordinate: y = f(-b / 2a) (substitute the x-coordinate back into the original equation)

How This Calculator Helps with Graphing (e.g., using Desmos)

While powerful graphing tools like Desmos can instantly visualize any quadratic equation, understanding the underlying calculations for roots and the vertex is essential for deeper comprehension. This calculator provides these critical values:

• Quickly find roots: Determine where your parabola will cross (or touch) the x-axis.
• Identify the vertex: Pinpoint the exact minimum or maximum point of your graph.
• Verify your graphs: Use the calculated roots and vertex to check the accuracy of a graph you've drawn manually or to confirm the features displayed by a graphing calculator like Desmos.

Example Usage:

Let's consider the equation x² - 3x + 2 = 0.

• Coefficient 'a': 1
• Coefficient 'b': -3
• Coefficient 'c': 2

Using the calculator above with these values, you would find:

• Discriminant: (-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 = 2
  • x₂ = [3 - sqrt(1)] / 2(1) = (3 - 1) / 2 = 1
• Vertex:
  • x = -(-3) / 2(1) = 3 / 2 = 1.5
  • y = (1.5)² - 3(1.5) + 2 = 2.25 - 4.5 + 2 = -0.25

So, the parabola y = x² - 3x + 2 crosses the x-axis at x=1 and x=2, and its lowest point (vertex) is at (1.5, -0.25). You can then input y = x^2 - 3x + 2 into Desmos to visually confirm these calculated features.