Graphing Quadratic Functions Calculator

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Quadratic Function Graphing Calculator

Graph Properties:

Vertex (x, y): N/A, N/A

Axis of Symmetry: N/A

Y-intercept: N/A

Discriminant (Δ): N/A

Nature of Roots: N/A

X-intercept 1: N/A

X-intercept 2: N/A

function calculateQuadratic() { var a = parseFloat(document.getElementById('inputA').value); var b = parseFloat(document.getElementById('inputB').value); var c = parseFloat(document.getElementById('inputC').value); var errorMessage = document.getElementById('errorMessage'); var vertexXElem = document.getElementById('vertexX'); var vertexYElem = document.getElementById('vertexY'); var axisOfSymmetryElem = document.getElementById('axisOfSymmetry'); var yInterceptElem = document.getElementById('yIntercept'); var discriminantValueElem = document.getElementById('discriminantValue'); var natureOfRootsElem = document.getElementById('natureOfRoots'); var xIntercept1Elem = document.getElementById('xIntercept1'); var xIntercept2Elem = document.getElementById('xIntercept2'); // Clear previous results and errors errorMessage.textContent = "; vertexXElem.textContent = 'N/A'; vertexYElem.textContent = 'N/A'; axisOfSymmetryElem.textContent = 'N/A'; yInterceptElem.textContent = 'N/A'; discriminantValueElem.textContent = 'N/A'; natureOfRootsElem.textContent = 'N/A'; xIntercept1Elem.textContent = 'N/A'; xIntercept2Elem.textContent = 'N/A'; if (isNaN(a) || isNaN(b) || isNaN(c)) { errorMessage.textContent = 'Please enter valid numbers for all coefficients.'; return; } if (a === 0) { errorMessage.textContent = 'Coefficient "a" cannot be zero for a quadratic function. This would be a linear function.'; return; } // Calculate Vertex var vertexX = -b / (2 * a); var vertexY = a * Math.pow(vertexX, 2) + b * vertexX + c; // Calculate Discriminant var discriminant = Math.pow(b, 2) – 4 * a * c; // Determine Nature of Roots and X-intercepts var natureOfRoots; var xIntercept1 = 'N/A'; var xIntercept2 = 'N/A'; if (discriminant > 0) { natureOfRoots = 'Two distinct real roots'; xIntercept1 = (-b + Math.sqrt(discriminant)) / (2 * a); xIntercept2 = (-b – Math.sqrt(discriminant)) / (2 * a); } else if (discriminant === 0) { natureOfRoots = 'One real root (repeated)'; xIntercept1 = -b / (2 * a); xIntercept2 = xIntercept1; // Same root } else { natureOfRoots = 'Two complex conjugate roots'; var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); xIntercept1 = realPart.toFixed(4) + ' + ' + imaginaryPart.toFixed(4) + 'i'; xIntercept2 = realPart.toFixed(4) + ' – ' + imaginaryPart.toFixed(4) + 'i'; } // Display results vertexXElem.textContent = vertexX.toFixed(4); vertexYElem.textContent = vertexY.toFixed(4); axisOfSymmetryElem.textContent = 'x = ' + vertexX.toFixed(4); yInterceptElem.textContent = '(0, ' + c.toFixed(4) + ')'; discriminantValueElem.textContent = discriminant.toFixed(4); natureOfRootsElem.textContent = natureOfRoots; xIntercept1Elem.textContent = xIntercept1; xIntercept2Elem.textContent = xIntercept2; }

Understanding Quadratic Functions and Their Graphs

A quadratic function is a polynomial function of degree two. Its graph is a U-shaped curve called a parabola. These functions are fundamental in mathematics, physics, engineering, and economics, describing phenomena from projectile motion to the shape of satellite dishes.

The Standard Form

The standard form of a quadratic function is given by:

f(x) = ax² + bx + c

Where:

  • a, b, and c are real numbers.
  • a ≠ 0 (If a were 0, the function would be linear, not quadratic).
  • The value of a determines the direction and width of the parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards. A larger absolute value of a makes the parabola narrower.
  • The value of c is the y-intercept, where the graph crosses the y-axis (i.e., when x = 0, f(0) = c).

Key Properties of a Quadratic Graph

1. Vertex

The vertex 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 (h, k) can be found using the formulas:

h = -b / (2a)
k = f(h) = a(h)² + b(h) + c

2. Axis of Symmetry

This is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is simply:

x = -b / (2a)

3. Y-intercept

The point where the parabola intersects the y-axis. This occurs when x = 0. For f(x) = ax² + bx + c, the y-intercept is always (0, c).

4. X-intercepts (Roots/Zeros)

These are the points where the parabola intersects the x-axis, meaning f(x) = 0. They are also known as the roots or zeros of the quadratic equation. They can be found using the quadratic formula:

x = [-b ± sqrt(b² - 4ac)] / (2a)

5. Discriminant (Δ)

The term b² - 4ac from the quadratic formula is called the discriminant (Δ). It tells us about the nature and number of the x-intercepts (roots):

  • If Δ > 0: There are two distinct real roots, meaning 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 its vertex.
  • If Δ < 0: There are no real roots, meaning the parabola does not intersect the x-axis. In this case, there are two complex conjugate roots.

How to Use the Calculator

Simply input the coefficients a, b, and c from your quadratic function f(x) = ax² + bx + c into the respective fields. The calculator will instantly provide you with the vertex coordinates, axis of symmetry, y-intercept, discriminant, and the nature and values of the x-intercepts (roots).

Examples:

Example 1: Two Real Roots
Consider the function: f(x) = x² - 4x + 3
Here, a = 1, b = -4, c = 3.

  • Vertex: x = -(-4)/(2*1) = 2; y = (2)² – 4(2) + 3 = 4 – 8 + 3 = -1. Vertex is (2, -1).
  • Axis of Symmetry: x = 2.
  • Y-intercept: (0, 3).
  • Discriminant: (-4)² – 4(1)(3) = 16 – 12 = 4. (Δ > 0, so two real roots).
  • X-intercepts: x = [4 ± sqrt(4)] / (2*1) = [4 ± 2] / 2. So, x1 = 3, x2 = 1.

Example 2: One Real Root
Consider the function: f(x) = 2x² + 4x + 2
Here, a = 2, b = 4, c = 2.

  • Vertex: x = -4/(2*2) = -1; y = 2(-1)² + 4(-1) + 2 = 2 – 4 + 2 = 0. Vertex is (-1, 0).
  • Axis of Symmetry: x = -1.
  • Y-intercept: (0, 2).
  • Discriminant: (4)² – 4(2)(2) = 16 – 16 = 0. (Δ = 0, so one real root).
  • X-intercepts: x = [-4 ± sqrt(0)] / (2*2) = -4 / 4 = -1. So, x1 = x2 = -1.

Example 3: No Real Roots
Consider the function: f(x) = x² + x + 1
Here, a = 1, b = 1, c = 1.

  • Vertex: x = -1/(2*1) = -0.5; y = (-0.5)² + (-0.5) + 1 = 0.25 – 0.5 + 1 = 0.75. Vertex is (-0.5, 0.75).
  • Axis of Symmetry: x = -0.5.
  • Y-intercept: (0, 1).
  • Discriminant: (1)² – 4(1)(1) = 1 – 4 = -3. (Δ < 0, so no real roots).
  • X-intercepts: x = [-1 ± sqrt(-3)] / (2*1) = [-1 ± i*sqrt(3)] / 2. Complex roots.

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