Quadratic Function Grapher & Calculator — Vertex and Roots
Graph y = ax² + bx + c free by entering a, b, and c. Instantly calculate the vertex, axis of symmetry, discriminant, and real roots, then inspect coordinates on the parabola.
What the graph of a quadratic function is
A quadratic function is written y = ax² + bx + c, and its graph is a parabola — a curve symmetric about a vertical line. The a settles how the curve opens: positive and it opens upward, negative and it opens downward, and the larger the magnitude the narrower it becomes. If a is zero the x² term vanishes and what remains is a straight line, so it can no longer be treated as a quadratic.
This tool draws the parabola from those three coefficients and works out the vertex, the axis of symmetry, the discriminant D, the nature of the real roots and the y-intercept together. The x-coordinate of the vertex comes from x = −b ÷ (2a), and that point is the minimum when a is positive and the maximum when it is negative. The sign of the discriminant D = b² − 4ac settles whether the graph meets the x-axis at two points, touches it, or misses it altogether. The range of x on display can be changed, so you can zoom in on the vertex or look at what happens further out.
How to use the quadratic graph
- Enter the three coefficients Put in a, b and c and the parabola is drawn. Setting a to zero leaves no quadratic at all, and an error says so.
- Set the range of x Give xMin and xMax to show only the stretch you want. If xMin is not below xMax there is no range, and an error follows.
- Read the vertex and the axis The coordinates of the vertex and the equation of the axis of symmetry appear on their own. The vertex is marked on the graph, so you can see where the minimum or maximum sits.
- Learn about the roots from the discriminant A positive D reads as two distinct real roots, zero as a repeated root, and a negative D as no real roots — telling you at once how the graph stands in relation to the x-axis.
- Find y for a particular x Type a number into the x field and the matching y is calculated, more exactly than it could be read off the curve.
Tips for getting more out of it
- 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.
Where the quadratic graph helps
Studying and checking school mathematics
Redraw the curve while changing the coefficients a little at a time and you can see how each of a, b and c bears on its shape and position. It also serves for checking a vertex or a root obtained by completing the square or by the quadratic formula.
Seeing how the discriminant relates to the crossings
Draw the curve while changing the sign of D and the three cases — meeting the x-axis twice, touching it, missing it — appear as shapes you can actually see.
Problems about a maximum or a minimum
For the familiar questions about the price at which profit is greatest or the side length at which area is greatest, the position of the vertex is what you are after.
Describing projectile motion
The path of an object under gravity alone traces a parabola. The changing height of something thrown upward can be written as a quadratic with time as x.
Comparing with other curves
To set it against a straight line, use the linear graph; to go a degree higher, the cubic graph.
Terms used with quadratic functions
- Quadratic function
- A function of the form y = ax² + bx + c with a not zero. It takes its name from the highest power of x being two, and its graph is a parabola.
- Parabola
- The symmetric curve traced by the graph of a quadratic function. It is named for the path an object follows when thrown.
- Vertex
- The turning point of the parabola. Its x-coordinate is x = −b ÷ (2a), and it gives the minimum of the function when a is positive and the maximum when a is negative.
- Axis of symmetry
- The vertical line that divides the parabola into two mirror halves and passes through the vertex. Its equation is x = −b ÷ (2a).
- Discriminant
- The value D = b² − 4ac. When D is greater than zero there are two distinct real roots; when it is zero, a repeated root; when it is less than zero there are no real roots and the graph never meets the x-axis.
- Repeated root
- The case in which the two roots fall together on the same value. It arises when the discriminant is zero, and the graph then touches the x-axis.
- Completing the square
- Rewriting the function in the form y = a(x − p)² + q. In that form the vertex is plainly (p, q).
FAQ
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.