Quadratic Function Grapher
Enter coefficients for y = ax² + bx + c
Function Properties
Understanding Quadratic Functions
A quadratic function is a polynomial function of degree two. The general form is y = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola.
Key Features of the Graph
Using the calculator above, you can visualize how changing the coefficients affects the shape and position of the parabola:
- Coefficient a: Determines the direction and width of the parabola. If a > 0, the parabola opens upwards. If a < 0, it opens downwards. Larger absolute values of a make the parabola narrower.
- Coefficient b: Affects the position of the vertex along the x-axis and the axis of symmetry.
- Constant c: Represents the y-intercept, the point where the graph crosses the y-axis.
How to Find the Vertex
The vertex is the highest or lowest point of the parabola. The x-coordinate of the vertex can be found using the formula x = -b / (2a). Once you have the x-coordinate, substitute it back into the original equation to find the y-coordinate.
Calculating the Roots
The roots (or zeros) of the function are the points where the graph intersects the x-axis (where y = 0). They can be calculated using the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
The term inside the square root, b² – 4ac, is called the discriminant. It tells you how many real roots exist. If the discriminant is positive, there are two real roots. If it is zero, there is exactly one real root. If it is negative, there are no real roots (the parabola does not touch the x-axis).