Quadratic Function Analyzer
Analysis Results:
Enter coefficients and an X value, then click 'Analyze Function' to see the properties of your quadratic equation.
Analysis Results:
'; if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(x)) { resultDiv.innerHTML = 'Analysis Results:
Please enter valid numbers for all fields.'; return; } // Equation display var equation = "y = "; if (a !== 0) { equation += (a === 1 ? "" : (a === -1 ? "-" : a)) + "x²"; } if (b !== 0) { equation += (b > 0 && a !== 0 ? " + " : (b 0 && (a !== 0 || b !== 0) ? " + " : (c 0 ? " + " + c : (c < 0 ? " – " + Math.abs(c) : "")); } output += 'Equation: ' + equation + "; // Calculate y for the given x var yAtX = a * x * x + b * x + c; output += 'y-value at x = ' + x + ': ' + yAtX.toFixed(4) + "; // If 'a' is 0, it's not a quadratic function if (a === 0) { output += 'Note: Coefficient \'a\' is zero, so this is a linear function, not a quadratic.'; // For linear functions, vertex and discriminant are not applicable in the same way. // Y-intercept is still 'c'. output += 'Y-intercept (x=0): (0, ' + c.toFixed(4) + ')'; if (b !== 0) { var rootLinear = -c / b; output += 'X-intercept (y=0): (' + rootLinear.toFixed(4) + ', 0)'; } else if (c === 0) { output += 'X-intercept (y=0): All real numbers (y=0)'; } else { output += 'X-intercept (y=0): No x-intercept (y=' + c + ' where c ≠ 0)'; } resultDiv.innerHTML = output; return; } // Calculate Vertex var vertexX = -b / (2 * a); var vertexY = a * vertexX * vertexX + b * vertexX + c; output += 'Vertex: (' + vertexX.toFixed(4) + ', ' + vertexY.toFixed(4) + ')'; // Calculate Axis of Symmetry output += 'Axis of Symmetry: x = ' + vertexX.toFixed(4) + "; // Calculate Y-intercept (when x=0) var yIntercept = c; output += 'Y-intercept (x=0): (0, ' + yIntercept.toFixed(4) + ')'; // Calculate Discriminant var discriminant = b * b – 4 * a * c; output += 'Discriminant (Δ): ' + discriminant.toFixed(4) + "; // Calculate Roots (X-intercepts) if (discriminant > 0) { var root1 = (-b + Math.sqrt(discriminant)) / (2 * a); var root2 = (-b – Math.sqrt(discriminant)) / (2 * a); output += 'Real Roots (X-intercepts):'; output += '- x₁ = ' + root1.toFixed(4) + '
- x₂ = ' + root2.toFixed(4) + '
- x = ' + root.toFixed(4) + '
- x₁ = ' + realPart + ' + ' + imaginaryPart + 'i
- x₂ = ' + realPart + ' – ' + imaginaryPart + 'i
Understanding Quadratic Functions for Graphing
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 expressed as y = 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 for Graphing a Parabola:
To accurately graph a quadratic function, understanding its key properties is essential. Our Quadratic Function Analyzer helps you quickly determine these critical points:
-
Coefficients (a, b, c): These numbers dictate the shape, position, and orientation of the parabola.
- Coefficient 'a': Determines if the parabola opens upwards (a > 0) or downwards (a < 0). It also affects the "width" of the parabola – a larger absolute value of 'a' makes the parabola narrower. If 'a' is 0, the function becomes linear, not quadratic.
- Coefficient 'b': Along with 'a', 'b' influences the position of the vertex and the axis of symmetry.
- Coefficient 'c': This is the y-intercept of the parabola. When x = 0, y = c.
-
Y-value at a Specific X: This simply calculates the corresponding 'y' value for any given 'x' value you input into the function. It helps in plotting individual points on the graph.
-
Vertex: This is the most crucial point of a parabola. It's the turning point where the parabola changes direction. If the parabola opens upwards, the vertex is the minimum point; if it opens downwards, it's the maximum point. The x-coordinate of the vertex is given by the formula
x = -b / (2a), and the y-coordinate is found by substituting this x-value back into the original equation. -
Axis of Symmetry: This is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Its equation is
x = -b / (2a). -
Y-intercept: As mentioned, this is the point where the parabola crosses the y-axis. It always occurs when x = 0, so the y-intercept is (0, c).
-
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, 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 exactly one point (its vertex lies on the x-axis).
- If Δ < 0: There are no real roots (two complex conjugate roots), meaning the parabola does not cross or touch the x-axis.
-
Real Roots (X-intercepts): These are the points where the parabola crosses the x-axis (where y = 0). They are found using the quadratic formula:
x = [-b ± √(Δ)] / (2a). If the discriminant is negative, there are no real x-intercepts.
How to Use the Calculator:
Simply input the coefficients 'a', 'b', and 'c' from your quadratic equation y = ax² + bx + c. You can also provide a specific 'X Value' to see what 'y' would be at that point. Click 'Analyze Function' to get a detailed breakdown of the parabola's properties, which are crucial for sketching its graph.
Example:
Let's analyze the function y = x² - 3x + 2.
- Coefficient 'a': 1
- Coefficient 'b': -3
- Coefficient 'c': 2
- X Value to Evaluate: 0 (for example)
Using the calculator with these values, you would find:
- y-value at x = 0: 2
- Vertex: (1.5, -0.25)
- Axis of Symmetry: x = 1.5
- Y-intercept: (0, 2)
- Discriminant (Δ): 1 (since (-3)² – 4*1*2 = 9 – 8 = 1)
- Real Roots (X-intercepts): x₁ = 2, x₂ = 1
- Parabola Direction: Opens Upwards (since a = 1 > 0)
These points and properties provide a clear roadmap for sketching the graph of y = x² - 3x + 2.