The Quadratic Function Analyzer turns a, b and c into the roots, vertex, axis, intercept and a graph, with every step shown.
Enter a, b and c for y = ax² + bx + c. The analyzer works through the discriminant, roots, vertex and intercept, then graphs the parabola.
The discriminant is b² − 4ac. If it is positive there are two real roots, if it is zero there is one repeated root at the vertex, and if it is negative the roots are a complex pair and the parabola never touches the x-axis.
The quadratic formula gives x = (−b ± √(b² − 4ac)) / 2a. The analyzer shows the substitution and simplification, and marks the real roots on the graph as hollow dots on the x-axis.
The vertex sits at x = −b/2a, and substituting that x back in gives its y value. The vertical line through it is the axis of symmetry. The parabola opens up when a > 0 and down when a < 0, and it crosses the y-axis at c.