Quadratic Equation Graphing Calculator
Graph Features:
Enter the coefficients a, b, and c to find the key features of your quadratic equation's graph.
Graph Features:
'; 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; } // Direction of Opening var direction = (a > 0) ? 'Upwards' : 'Downwards'; output += 'Direction of Opening: ' + direction + "; // Vertex var vertexX = -b / (2 * a); var vertexY = a * Math.pow(vertexX, 2) + b * vertexX + c; output += 'Vertex: (' + vertexX.toFixed(3) + ', ' + vertexY.toFixed(3) + ')'; // Axis of Symmetry output += 'Axis of Symmetry: x = ' + vertexX.toFixed(3) + "; // Y-intercept output += 'Y-intercept: (0, ' + c.toFixed(3) + ')'; // X-intercepts (Roots) var discriminant = (b * b) – (4 * a * c); output += 'X-intercepts (Roots): '; if (discriminant > 0) { var x1 = (-b + Math.sqrt(discriminant)) / (2 * a); var x2 = (-b – Math.sqrt(discriminant)) / (2 * a); output += '(' + x1.toFixed(3) + ', 0) and (' + x2.toFixed(3) + ', 0)'; } else if (discriminant === 0) { var x = -b / (2 * a); output += '(' + x.toFixed(3) + ', 0) (one real root)'; } else { output += 'No real X-intercepts (complex roots)'; } resultDiv.innerHTML = output; }Understanding Quadratic Equations and Their Graphs
A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. It is typically written in the standard form:
ax² + bx + c = 0
where 'a', 'b', and 'c' are coefficients, and 'a' cannot be zero. If 'a' were zero, the equation would become linear (bx + c = 0).
The Parabola: Graph of a Quadratic Equation
When you graph a quadratic equation, the resulting shape is a U-shaped curve called a parabola. The characteristics of this parabola are determined by the values of its coefficients 'a', 'b', and 'c'.
Key Features of a Parabola:
- Direction of Opening: The coefficient 'a' dictates whether the parabola opens upwards or downwards.
- If
a > 0, the parabola opens upwards (like a smile). - If
a < 0, the parabola opens downwards (like a frown).
- If
- Vertex: This is the turning point of the parabola. It's either the lowest point (minimum) if the parabola opens upwards, or the highest point (maximum) if it opens downwards. The coordinates of the vertex (h, k) can be found using the formulas:
h = -b / (2a)k = a(h)² + b(h) + c(substitute 'h' 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 always
x = h, where 'h' is the x-coordinate of the vertex. - Y-intercept: This is the point where the parabola crosses the y-axis. It occurs when
x = 0. By substitutingx = 0into the standard equation, we find that the y-intercept is always(0, c). - X-intercepts (Roots/Zeros): These are the points where the parabola crosses the x-axis. At these points,
y = 0. Finding the x-intercepts involves solving the quadratic equationax² + bx + c = 0for 'x'. This can be done using the quadratic formula:
The termx = [-b ± √(b² - 4ac)] / (2a)(b² - 4ac)is called the discriminant.- If
discriminant > 0, there are two distinct real x-intercepts. - If
discriminant = 0, there is exactly one real x-intercept (the vertex touches the x-axis). - If
discriminant < 0, there are no real x-intercepts (the parabola does not cross the x-axis).
- If
How to Use the Calculator
Our Quadratic Equation Graphing Calculator simplifies the process of finding these key features. Simply input the coefficients 'a', 'b', and 'c' from your quadratic equation ax² + bx + c = 0 into the respective fields. For example, if your equation is y = x² - 2x - 3, you would enter:
- Coefficient 'a':
1 - Coefficient 'b':
-2 - Coefficient 'c':
-3
Click "Calculate Graph Features," and the calculator will instantly provide you with the direction of opening, the vertex coordinates, the axis of symmetry, and any real x-intercepts, helping you visualize and understand the graph of your quadratic equation.