📐 Quadratic Axis And Vertex And Roots All In One 計算機

This Quadratic Function Analyzer takes any quadratic in standard form, f(x) = ax² + bx + c, and instantly returns its complete profile: the vertex, the axis of symmetry, the discriminant, the real roots (x-intercepts), and the y-intercept. It is a universal math tool that works for parabolas in algebra, precalculus, physics projectile problems, and optimization.

Enter the three coefficients a, b, and c. The coefficient a must be nonzero for a true quadratic (if a = 0 the equation is linear). Press calculate and read off every key feature of the parabola in one view.

The axis of symmetry and the vertex share the same x-value:

Plug that x back into the function to get the vertex y-value. The roots come from the quadratic formula, The quantity under the root, b^{2} - 4ac, is the discriminant: if it is positive there are two real roots, if it is zero there is one repeated root, and if it is negative there are no real roots (the parabola never crosses the x-axis). The y-intercept is simply c, since f(0) = c.

For f(x) = x^{2} - 3x + 2 (a = 1, b = −3, c = 2): axis

vertex so the vertex is (1.5, -0.25). Discriminant = 9 - 8 = 1, giving roots The y-intercept is (0, 2).