Online Calculator Graphing – Loan calculator

Understanding Quadratic Equation Graphs with Our Calculator

A quadratic equation is a polynomial equation of the second degree, typically written in the form y = ax² + bx + c, where 'a', 'b', and 'c' are coefficients, and 'a' is not equal to zero. The graph of a quadratic equation is a U-shaped curve called a parabola. Understanding the key features of this parabola is crucial for various applications in mathematics, physics, and engineering.

Key Features of a Parabola

Every parabola has several defining characteristics that help in sketching its graph:

Vertex: This is the highest or lowest point on the parabola. It represents the turning point of the graph. For a parabola opening upwards (a > 0), the vertex is the minimum point. For a parabola opening downwards (a < 0), it's the maximum point.
Y-intercept: This is the point where the parabola crosses the y-axis. It occurs when x = 0.
X-intercepts (Roots): These are the points where the parabola crosses the x-axis. They occur when y = 0. A parabola can have two, one, or no real x-intercepts.
Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
Direction of Opening: Determined by the sign of the 'a' coefficient. If 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards.

How Our Quadratic Graphing Calculator Works

Our online calculator simplifies the process of finding these critical points for any quadratic equation. Simply input the coefficients 'a', 'b', and 'c' from your equation y = ax² + bx + c. The calculator will then instantly provide you with:

The coordinates of the Vertex.
The coordinates of the Y-intercept.
Any real X-intercepts (roots).
The direction in which the parabola opens.

This information is invaluable for accurately sketching the graph of your quadratic function without tedious manual calculations.

Formulas Used in the Calculation

The calculator uses standard algebraic formulas:

Vertex x-coordinate: x = -b / (2a)
Vertex y-coordinate: Calculated by substituting the vertex x-coordinate back into the original equation y = ax² + bx + c.
Y-intercept: Simply (0, c).
X-intercepts (Quadratic Formula): x = [-b ± sqrt(b² - 4ac)] / (2a). The term b² - 4ac is known as the discriminant, which determines the number of real roots.

Example Calculation

Let's consider the quadratic equation: y = 1x² - 4x + 3

Here, a = 1, b = -4, and c = 3.

Vertex x-coordinate: x = -(-4) / (2 * 1) = 4 / 2 = 2
Vertex y-coordinate: y = 1*(2)² - 4*(2) + 3 = 4 - 8 + 3 = -1. So, the Vertex is (2, -1).
Y-intercept: (0, 3).
X-intercepts:
- Discriminant D = (-4)² - 4*(1)*(3) = 16 - 12 = 4
- x1 = (4 + sqrt(4)) / (2 * 1) = (4 + 2) / 2 = 6 / 2 = 3
- x2 = (4 - sqrt(4)) / (2 * 1) = (4 - 2) / 2 = 2 / 2 = 1
So, the X-intercepts are (1, 0) and (3, 0).
Direction of Opening: Since a = 1 (positive), the parabola opens upwards.

Using these points, you can easily sketch the graph of y = x² - 4x + 3.

Quadratic Equation Graphing Calculator

Enter the coefficients for your quadratic equation y = ax² + bx + c to find its key graphing features.

Coefficient 'a' (for x²):

Coefficient 'b' (for x):

Coefficient 'c' (constant):

/* Basic Styling for the Calculator – Can be customized */ .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; font-family: Arial, sans-serif; } .calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; } .calculator-input-group { margin-bottom: 15px; } .calculator-input-group label { display: block; margin-bottom: 5px; font-weight: bold; color: #555; } .calculator-input-group input[type="number"] { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calculator-container button { background-color: #007bff; color: white; padding: 12px 20px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; width: 100%; margin-top: 10px; } .calculator-container button:hover { background-color: #0056b3; } .calculator-result { background-color: #e9ecef; border: 1px solid #ced4da; padding: 15px; border-radius: 4px; margin-top: 20px; min-height: 50px; color: #333; } .calculator-result p { margin: 5px 0; } .calculator-result strong { color: #0056b3; } .calculator-article { font-family: Arial, sans-serif; line-height: 1.6; color: #333; max-width: 600px; margin: 20px auto; padding: 0 15px; } .calculator-article h2, .calculator-article h3 { color: #007bff; margin-top: 25px; margin-bottom: 15px; } .calculator-article ul { list-style-type: disc; margin-left: 20px; margin-bottom: 15px; } .calculator-article code { background-color: #e9ecef; padding: 2px 4px; border-radius: 3px; font-family: 'Courier New', Courier, monospace; } function calculateQuadraticGraph() { 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("graphingResult"); resultDiv.innerHTML = ""; // Clear previous results if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = "Please enter valid numbers for all coefficients."; return; } if (a === 0) { resultDiv.innerHTML = "Coefficient 'a' cannot be zero for a quadratic equation (it would be a linear equation)."; return; } // 1. Direction of Opening var direction = (a > 0) ? "Upwards" : "Downwards"; resultDiv.innerHTML += "Direction of Opening: " + direction + ""; // 2. Vertex var vertexX = -b / (2 * a); var vertexY = a * Math.pow(vertexX, 2) + b * vertexX + c; resultDiv.innerHTML += "Vertex: (" + vertexX.toFixed(4) + ", " + vertexY.toFixed(4) + ")"; // 3. Y-intercept var yIntercept = c; // When x = 0, y = a*0^2 + b*0 + c = c resultDiv.innerHTML += "Y-intercept: (0, " + yIntercept.toFixed(4) + ")"; // 4. X-intercepts (Roots) var discriminant = b * b – 4 * a * c; resultDiv.innerHTML += "X-intercepts (Roots):"; if (discriminant > 0) { var x1 = (-b + Math.sqrt(discriminant)) / (2 * a); var x2 = (-b – Math.sqrt(discriminant)) / (2 * a); resultDiv.innerHTML += " x1 = " + x1.toFixed(4) + ""; resultDiv.innerHTML += " x2 = " + x2.toFixed(4) + ""; } else if (discriminant === 0) { var x = -b / (2 * a); resultDiv.innerHTML += " x = " + x.toFixed(4) + " (one real root)"; } else { resultDiv.innerHTML += " No real x-intercepts (complex roots)"; } }