Desmos Calculator Graphing

Quadratic Function Analyzer for Desmos Graphing

Enter the coefficients of your quadratic equation in the form ax² + bx + c = 0 to analyze its properties for graphing in Desmos.

Results:

Enter coefficients and click 'Calculate Properties' to see the vertex, roots, and axis of symmetry.

function calculateQuadraticProperties() { var a = parseFloat(document.getElementById('coeffA').value); var b = parseFloat(document.getElementById('coeffB').value); var c = parseFloat(document.getElementById('coeffC').value); var resultDiv = document.getElementById('quadraticResult'); var output = '

Results:

'; if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = output + 'Please enter valid numbers for all coefficients.'; return; } if (a === 0) { resultDiv.innerHTML = output + 'Coefficient \'a\' cannot be zero for a quadratic equation. This would be a linear equation.'; return; } // Vertex Calculation var vertexX = -b / (2 * a); var vertexY = a * Math.pow(vertexX, 2) + b * vertexX + c; output += 'Vertex: (' + vertexX.toFixed(4) + ', ' + vertexY.toFixed(4) + ')'; // Axis of Symmetry output += 'Axis of Symmetry: x = ' + vertexX.toFixed(4) + "; // Roots Calculation (using discriminant) var discriminant = Math.pow(b, 2) – 4 * a * c; output += 'Discriminant: ' + discriminant.toFixed(4) + "; if (discriminant > 0) { var root1 = (-b + Math.sqrt(discriminant)) / (2 * a); var root2 = (-b – Math.sqrt(discriminant)) / (2 * a); output += 'Real Roots: x₁ = ' + root1.toFixed(4) + ', x₂ = ' + root2.toFixed(4) + "; } else if (discriminant === 0) { var root = -b / (2 * a); output += '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); output += 'Complex Roots: x₁ = ' + realPart + ' + ' + imaginaryPart + 'i, x₂ = ' + realPart + ' – ' + imaginaryPart + 'i'; } resultDiv.innerHTML = output; }

Understanding Quadratic Functions for Desmos Graphing

Desmos is an incredibly powerful and intuitive online graphing calculator that makes visualizing mathematical functions simple and engaging. While Desmos handles the plotting, understanding the fundamental properties of the functions you input can greatly enhance your ability to interpret and manipulate your graphs.

One of the most common and foundational types of functions you'll graph is the quadratic function. A quadratic function is a polynomial function of degree two, meaning the highest exponent of the variable (usually 'x') is 2. Its standard 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 U-shaped curve called a parabola.

Key Properties of a Parabola:

1. Vertex: This is the most crucial point on a parabola. It's the turning point – either the lowest point (if the parabola opens upwards, i.e., a > 0) or the highest point (if the parabola opens downwards, i.e., a < 0). The coordinates of the vertex can be calculated using the formula:

   x-coordinate = -b / (2a)
y-coordinate = f(-b / (2a))

   Knowing the vertex helps you quickly locate the center of your parabola in Desmos.

2. Axis of Symmetry: This is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Its equation is simply:

   x = -b / (2a)

   Graphing this line in Desmos alongside your parabola can help confirm its symmetry.

3. Roots (x-intercepts): These are the points where the parabola crosses the x-axis, meaning f(x) = 0. They are also known as the zeros of the function. The number and type of roots depend on the discriminant, which is calculated as:

   Discriminant (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). The parabola touches the x-axis at its vertex.
   • If Δ < 0: There are no real roots. The parabola does not intersect the x-axis (it's entirely above or below it). In this case, there are two complex conjugate roots.

   The roots themselves can be found using the quadratic formula:

   x = [-b ± sqrt(b² - 4ac)] / (2a)

   Identifying the roots helps you understand where the function's value is zero, which is often critical in problem-solving.

Using the Calculator with Desmos:

This calculator helps you quickly determine these key properties for any quadratic equation. Once you have the vertex, axis of symmetry, and roots, you can:

• Verify your Desmos graph: Plot the calculated vertex and roots as points in Desmos to ensure your function's graph passes through them correctly.
• Understand transformations: See how changing 'a', 'b', or 'c' affects these properties and, consequently, the shape and position of your parabola. For example, a larger absolute value of 'a' makes the parabola narrower, while 'c' shifts it vertically.
• Solve problems: Many real-world problems can be modeled with quadratic functions (e.g., projectile motion, optimizing area). Analyzing these properties helps in finding maximum/minimum values or specific intercepts.

By combining the analytical power of this calculator with the visual feedback of Desmos, you can gain a deeper and more intuitive understanding of quadratic functions.