Math

Quadratic Function Graph (y = ax² + bx + c)

Enter coefficients a, b, c to plot the parabola y = ax² + bx + c. Automatically calculates the vertex, axis of symmetry, discriminant, and real roots.

Tips

  • The quadratic coefficient a determines the shape of the parabola. If a > 0 it opens upward; if a < 0 it opens downward. The larger |a| is, the narrower the parabola.
  • The x-coordinate of the vertex is x = −b ÷ (2a). The vertex is the minimum point when a > 0 or the maximum point when a < 0, and is shown on the graph as a blue diamond.
  • The discriminant D = b² − 4ac determines the number of real roots. D > 0: the parabola crosses the x-axis at 2 points (green dots); D = 0: it touches the x-axis (double root); D < 0: no intersection.
  • Hover over the graph to see the (x, y) coordinates at any point. Enter an x value on the left to calculate the corresponding y value.

FAQ

When a = 0, the quadratic term vanishes and the function becomes y = bx + c (a linear function or constant). Since this tool is designed for quadratic functions, a = 0 is not accepted. Use the "Linear Function Graph" tool to plot straight lines.

The discriminant D = b² − 4ac determines the number of real solutions of ax² + bx + c = 0. D > 0: two distinct real roots; D = 0: a double root (the same value repeated); D < 0: no real roots (complex roots only). On the graph, D > 0 means the parabola crosses the x-axis at 2 points, D = 0 means it touches it, and D < 0 means it does not intersect.

Complete the square on y = ax² + bx + c to get vertex form y = a(x − p)² + q, where p = −b/(2a) and q = c − b²/(4a). This tool accepts the standard form and automatically computes and displays the vertex coordinates (p, q).

Yes. Changing a, b, and c updates the graph in real time, making it easy to see how each coefficient affects the shape. This tool covers the quadratic function topics taught from middle school through high school mathematics.

Side Note — The Origin of the Word "Parabola"

The name "parabola" was coined by the ancient Greek mathematician Apollonius of Perga (c. 262–190 BC) while studying conic sections. The Greek word "παραβολή" (parabole) means "placed side by side," referring to the geometric property that a parabola is the set of points equidistant from a focus and a directrix.

One of the most famous real-world appearances of parabolas is the trajectory of a projectile under gravity (ignoring air resistance), discovered experimentally by Galileo Galilei in the 17th century. Parabolic mirrors — formed by rotating a parabola around its axis — are used in flashlights and car headlights to convert a point light source at the focus into a parallel beam.

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