Quadratic Graph Calculator
Enter the coefficients a, b, and c for the quadratic equation y = ax² + bx + c.
Equation Coefficients
Graph Properties
Enter coefficients above to see graph properties.
Understanding Quadratic Equations and Their Graphs
A quadratic equation is a polynomial equation of the second degree. The standard form of a quadratic equation is:
y = ax² + bx + c
where 'a', 'b', and 'c' are constants, and 'a' cannot be zero (if a=0, it becomes a linear equation). The graph of a quadratic equation is a parabola, a distinctive U-shaped curve.
Key Components of a Quadratic Graph:
- Parabola Shape: The coefficient 'a' determines the direction and width of the parabola.
- If
a > 0, the parabola opens upwards (U-shaped). - If
a < 0, the parabola opens downwards (∩-shaped). - The larger the absolute value of 'a' (e.g.,
a = 5vsa = 0.5), the narrower the parabola.
- If
- Y-intercept: This is the point where the graph crosses the y-axis. It occurs when x = 0. By substituting x = 0 into the equation, we get
y = a(0)² + b(0) + c, which simplifies toy = c. So, the y-intercept is always at the point(0, c). - Axis of Symmetry: This is a vertical line that divides the parabola into two mirror images. The equation for the axis of symmetry is
x = -b / (2a). - Vertex: This is the highest or lowest point on the parabola. It lies on the axis of symmetry. The x-coordinate of the vertex is the same as the axis of symmetry:
x_vertex = -b / (2a). To find the y-coordinate, substitute this x-value back into the original quadratic equation:y_vertex = a(x_vertex)² + b(x_vertex) + c. - X-intercepts (Roots): These are the points where the graph crosses the x-axis (where y = 0). Finding the x-intercepts involves solving the quadratic equation
ax² + bx + c = 0. This can be done using the quadratic formula:
The term inside the square root,x = [-b ± sqrt(b² - 4ac)] / (2a)Δ = b² - 4ac, is called the discriminant.- If
Δ > 0, there are two distinct real roots (two x-intercepts). - If
Δ = 0, there is exactly one real root (the vertex touches the x-axis). - If
Δ < 0, there are no real roots (the parabola does not cross the x-axis).
- If
How This Calculator Works:
This calculator takes the coefficients a, b, and c of your quadratic equation (y = ax² + bx + c) and computes the following properties:
- Opening Direction: Determined by the sign of 'a'.
- Y-intercept: Simply the value of 'c'.
- Axis of Symmetry: Calculated using
x = -b / (2a). - Vertex Coordinates: Calculated by finding the x-coordinate first, then substituting it back to find the y-coordinate.
- Discriminant (Δ): Calculated using
b² - 4acto determine the nature and number of x-intercepts. - X-intercepts (if real): Calculated using the quadratic formula based on the discriminant's value.
Use Cases:
Quadratic equations and their graphs are fundamental in many areas of mathematics, science, and engineering:
- Physics: Describing projectile motion (the path of a ball thrown in the air is often parabolic).
- Engineering: Designing parabolic reflectors, antennas, and satellite dishes.
- Economics: Modeling cost and revenue functions.
- Optimization Problems: Finding maximum or minimum values (e.g., maximizing profit or minimizing material usage).
" + x1.toFixed(4) + " and x = " + x2.toFixed(4) + "";
} else if (discriminant === 0) {
var x = -coeffB / (2 * coeffA);
xIntercepts.push(x.toFixed(4));
xInterceptsHTML = "X-intercept: One real root (vertex on x-axis) at x = " + x.toFixed(4) + "";
} else {
xInterceptsHTML = "X-intercepts: No real roots (the parabola does not cross the x-axis).";
}
// Display results
var htmlOutput = "";
htmlOutput += "Opening Direction: " + openingDirection + "";
htmlOutput += "Y-intercept: (0, " + yIntercept + ")";
htmlOutput += "Axis of Symmetry: x = " + axisOfSymmetryX.toFixed(4) + "";
htmlOutput += "Vertex: (" + vertexX.toFixed(4) + ", " + vertexY.toFixed(4) + ")";
htmlOutput += "Discriminant (Δ): " + discriminant.toFixed(4) + "";
htmlOutput += xInterceptsHTML;
htmlOutput += "
";
resultDiv.innerHTML = htmlOutput;
}