Enter the coefficients for the standard quadratic equation of a parabola: y = ax² + bx + c
Understanding Parabolas
A parabola is a symmetrical U-shaped curve defined by a quadratic equation. In its most common form, the equation of a parabola is written as:
y = ax² + bx + c
Where:
a, b, and c are coefficients that determine the parabola's shape, direction, and position.
The term ax² is the dominant term that gives the equation its quadratic nature.
x is the independent variable, and y is the dependent variable.
Key Features of a Parabola
The coefficients a, b, and c influence several key aspects of the parabola:
Direction of Opening:
If a > 0, the parabola opens upwards (like a smile).
If a < 0, the parabola opens downwards (like a frown).
If a = 0, the equation becomes linear (y = bx + c), resulting in a straight line, not a parabola.
Width/Stretchedness: The absolute value of a affects how narrow or wide the parabola is. A larger |a| results in a narrower parabola, while a smaller |a| results in a wider one.
Y-intercept: The constant term c directly represents the point where the parabola crosses the y-axis. This is because when x = 0, y = a(0)² + b(0) + c = c. So, the y-intercept is always at the point (0, c).
Vertex: The vertex is the turning point of the parabola. It's either the minimum point (if opening upwards) or the maximum point (if opening downwards). The x-coordinate of the vertex is given by the formula: x_vertex = -b / (2a). To find the y-coordinate of the vertex, substitute this x_vertex value back into the original equation: y_vertex = a(x_vertex)² + b(x_vertex) + c.
Axis of Symmetry: This is a vertical line that passes through the vertex, dividing the parabola into two mirror images. The equation of the axis of symmetry is x = -b / (2a).
How This Calculator Works
This calculator takes your input for the coefficients a, b, and c from the equation y = ax² + bx + c. It then performs the following calculations:
Vertex Calculation: It computes the coordinates of the vertex using x_vertex = -b / (2a) and y_vertex = a(x_vertex)² + b(x_vertex) + c.
Y-intercept: It identifies the y-intercept as (0, c).
Axis of Symmetry: It determines the equation of the axis of symmetry as x = -b / (2a).
Direction: It infers whether the parabola opens upwards or downwards based on the sign of a.
The results are presented clearly below, summarizing the key graphical features of the parabola defined by your coefficients.
Example Usage
Let's consider the equation y = 1x² - 2x + 1.
Inputs: a = 1, b = -2, c = 1
Calculation:
Vertex x: -(-2) / (2 * 1) = 2 / 2 = 1
Vertex y: 1(1)² - 2(1) + 1 = 1 - 2 + 1 = 0. So, the vertex is at (1, 0).
Y-intercept: (0, 1)
Axis of Symmetry: x = 1
Direction: Since a = 1 (which is > 0), it opens upwards.
This calculator helps visualize these properties without needing to manually plot points or perform complex calculations.
function calculateParabola() {
var coeffA = parseFloat(document.getElementById("coeffA").value);
var coeffB = parseFloat(document.getElementById("coeffB").value);
var coeffC = parseFloat(document.getElementById("coeffC").value);
var resultDiv = document.getElementById("result");
// Clear previous results
resultDiv.innerHTML = ";
// Input validation
if (isNaN(coeffA) || isNaN(coeffB) || isNaN(coeffC)) {
resultDiv.innerHTML = "Please enter valid numbers for all coefficients.";
return;
}
if (coeffA === 0) {
resultDiv.innerHTML = "Coefficient 'a' cannot be zero for a parabola. This equation represents a line.";
return;
}
// Calculate Vertex
var vertexX = -coeffB / (2 * coeffA);
var vertexY = coeffA * Math.pow(vertexX, 2) + coeffB * vertexX + coeffC;
// Determine Direction
var direction = coeffA > 0 ? "Upwards" : "Downwards";
// Format results for display
var formattedVertexX = vertexX.toFixed(3);
var formattedVertexY = vertexY.toFixed(3);
var formattedCoeffC = coeffC.toFixed(3);
var formattedAxisX = (-coeffB / (2 * coeffA)).toFixed(3);
// Display results
var resultHTML = "