Quadratic Function Graphing Calculator

Quadratic Function Graphing Calculator

Calculation Results:

Enter coefficients and click 'Calculate Properties'.

function calculateQuadratic() { var a = parseFloat(document.getElementById("coeffA").value); var b = parseFloat(document.getElementById("coeffB").value); var c = parseFloat(document.getElementById("coeffC").value); var functionTypeElem = document.getElementById("functionType"); var vertexCoordsElem = document.getElementById("vertexCoords"); var axisSymmetryElem = document.getElementById("axisSymmetry"); var yInterceptElem = document.getElementById("yIntercept"); var concavityElem = document.getElementById("concavity"); var discriminantValueElem = document.getElementById("discriminantValue"); var rootsElem = document.getElementById("roots"); // Clear previous results functionTypeElem.innerHTML = ""; vertexCoordsElem.innerHTML = ""; axisSymmetryElem.innerHTML = ""; yInterceptElem.innerHTML = ""; concavityElem.innerHTML = ""; discriminantValueElem.innerHTML = ""; rootsElem.innerHTML = ""; if (isNaN(a) || isNaN(b) || isNaN(c)) { functionTypeElem.innerHTML = "Please enter valid numbers for all coefficients."; return; } if (a === 0) { functionTypeElem.innerHTML = "Function Type: This is a linear function (y = " + b + "x + " + c + "), not a quadratic function."; yInterceptElem.innerHTML = "Y-intercept: y = " + c; return; } functionTypeElem.innerHTML = "Function Type: Quadratic Function (y = " + a + "x² + " + b + "x + " + c + ")"; // Vertex var vertexX = -b / (2 * a); var vertexY = a * Math.pow(vertexX, 2) + b * vertexX + c; vertexCoordsElem.innerHTML = "Vertex: (" + vertexX.toFixed(4) + ", " + vertexY.toFixed(4) + ")"; // Axis of Symmetry axisSymmetryElem.innerHTML = "Axis of Symmetry: x = " + vertexX.toFixed(4); // Y-intercept yInterceptElem.innerHTML = "Y-intercept: y = " + c; // Concavity if (a > 0) { concavityElem.innerHTML = "Concavity: Opens Up (minimum point)"; } else { concavityElem.innerHTML = "Concavity: Opens Down (maximum point)"; } // Discriminant var discriminant = Math.pow(b, 2) – 4 * a * c; discriminantValueElem.innerHTML = "Discriminant (Δ): " + discriminant.toFixed(4); // Roots if (discriminant > 0) { var root1 = (-b + Math.sqrt(discriminant)) / (2 * a); var root2 = (-b – Math.sqrt(discriminant)) / (2 * a); rootsElem.innerHTML = "Roots (x-intercepts): Two distinct real roots: x₁ = " + root1.toFixed(4) + ", x₂ = " + root2.toFixed(4); } else if (discriminant === 0) { var root = -b / (2 * a); rootsElem.innerHTML = "Roots (x-intercepts): One real root (repeated): x = " + root.toFixed(4); } else { var realPart = (-b / (2 * a)).toFixed(4); var imaginaryPart = (Math.sqrt(Math.abs(discriminant)) / (2 * a)).toFixed(4); rootsElem.innerHTML = "Roots (x-intercepts): Two complex conjugate roots: x₁ = " + realPart + " + " + imaginaryPart + "i, x₂ = " + realPart + " – " + imaginaryPart + "i"; } }

Understanding Quadratic Functions and Their Graphs

A quadratic function is a polynomial function of degree two. It is typically written in the standard form:

y = ax² + bx + c

where 'a', 'b', and 'c' are real numbers, and 'a' cannot be zero. The graph of a quadratic function is a U-shaped curve called a parabola.

Key Features of a Parabola:

  • Vertex: This is the highest or lowest point on the parabola. If the parabola opens upwards, the vertex is a minimum point; if it opens downwards, it's a maximum point. The x-coordinate of the vertex is given by the formula x = -b / (2a). Once you have the x-coordinate, you can find the y-coordinate by substituting x back into the original equation.
  • Axis of Symmetry: This is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = (vertex's x-coordinate).
  • Y-intercept: This is the point where the parabola crosses the y-axis. It occurs when x = 0, so the y-intercept is always (0, c).
  • Concavity (Direction of Opening):
    • If a > 0, the parabola opens upwards.
    • If a < 0, the parabola opens downwards.
  • Discriminant (Δ): Calculated as Δ = b² – 4ac. The discriminant tells us about the nature and number of the roots (x-intercepts) of the quadratic equation:
    • 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 (the parabola touches the x-axis at its vertex).
    • If Δ < 0, there are no real roots (the parabola does not cross or touch the x-axis; it has two complex conjugate roots).
  • Roots (X-intercepts): These are the points where the parabola crosses the x-axis (where y = 0). They can be found using the quadratic formula: x = [-b ± sqrt(b² – 4ac)] / (2a).

How to Use the Calculator:

Our Quadratic Function Graphing Calculator simplifies the process of finding these key properties. Simply input the coefficients 'a', 'b', and 'c' from your quadratic equation (y = ax² + bx + c) into the respective fields. Then, click the 'Calculate Properties' button. The calculator will instantly display:

  • The type of function (identifying if 'a' is zero, making it linear).
  • The coordinates of the vertex.
  • The equation of the axis of symmetry.
  • The y-intercept.
  • Whether the parabola opens upwards or downwards.
  • The value of the discriminant.
  • The real or complex roots (x-intercepts) of the function.

Examples:

Let's look at a few examples to illustrate:

  1. Equation: y = x² – 2x + 1
    • Input: a=1, b=-2, c=1
    • Output: Vertex (1, 0), Axis of Symmetry x=1, Y-intercept y=1, Concavity: Opens Up, Discriminant: 0, Roots: One real root x=1. (The parabola touches the x-axis at its vertex).
  2. Equation: y = -x² + 2x + 3
    • Input: a=-1, b=2, c=3
    • Output: Vertex (1, 4), Axis of Symmetry x=1, Y-intercept y=3, Concavity: Opens Down, Discriminant: 16, Roots: Two distinct real roots x₁=3, x₂=-1. (The parabola crosses the x-axis at two points).
  3. Equation: y = 2x² + 4x + 5
    • Input: a=2, b=4, c=5
    • Output: Vertex (-1, 3), Axis of Symmetry x=-1, Y-intercept y=5, Concavity: Opens Up, Discriminant: -24, Roots: Two complex conjugate roots. (The parabola does not cross the x-axis).

This calculator is a valuable tool for students, educators, and anyone working with quadratic functions, providing quick and accurate insights into their graphical properties.

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