Graph a Quadratic Function (Parabola)

Q: How do I find the vertex of a parabola?

For y = ax² + bx + c, the vertex is at x = −b ÷ (2a). Substitute that x back into the equation to get y. For y = x² − 6x + 5, x = 3 and y = −4, so the vertex is (3, −4).

Q: How do I graph y = x²?

Type y = x^2 into Plotline. It is a parabola with its vertex at the origin, passing through (1, 1), (2, 4) and (−2, 4).

Q: Is it free?

Yes. The graphing calculator is free with no account and no sign-up: plot as many equations as you like, use sliders, save graphs in your browser and share them by link. It installs as an app and keeps working offline.

A quadratic y = ax² + bx + c graphs as a U-shaped parabola with its vertex at x = −b ÷ 2a; it opens upward when a > 0 and downward when a < 0.

Graph of the parabola y = x² − 4, opening upward with its vertex at (0, −4) and crossing the x-axis at −2 and 2. — y = x² − 4: vertex (0, −4), roots x = −2 and x = 2.

At a glance

What it is	Graph quadratic functions y = ax² + bx + c online. Find the vertex, roots and axis of symmetry, and drag sliders for a, b and c. Free.
Type	Interactive guide with a calculator
Level	High school and college
Price	Free, with no account or sign-up
Works offline	Yes, after the first visit; installs as an app
Made by	Aalok (aalok.builds@gmail.com)
Last updated	1 November 2026

Key facts

Standard form	y = ax² + bx + c
Vertex	x = −b ÷ (2a), then substitute to find y
Axis of symmetry	The vertical line x = −b ÷ (2a)
Roots (x-intercepts)	x = (−b ± √(b² − 4ac)) ÷ (2a)
y-intercept	(0, c)
Example: y = x² − 4	Vertex (0, −4), roots x = −2 and x = 2

What does the discriminant tell you?

The discriminant b² − 4ac tells you how many times the parabola crosses the x-axis: two real roots if it is positive, one (the vertex touches the axis) if it is zero, and none if it is negative.

What is vertex form?

Written as y = a(x − h)² + k, the vertex is simply (h, k). Changing h slides the parabola left and right, k moves it up and down, and a makes it narrower or wider.

How to use it in Plotline

The graph of y = ax² + bx + c with sliders for a, b and c is already loaded above. Edit the equation in the list to change it.
Drag the graph to move around, scroll or pinch to zoom, and tap the crosshair to reset the view.
Hover over or tap the curve to read the coordinates of any point.
Press Enter to add another equation and compare graphs in different colors.

Frequently asked questions

How do I find the vertex of a parabola?

For y = ax² + bx + c, the vertex is at x = −b ÷ (2a). Substitute that x back into the equation to get y. For y = x² − 6x + 5, x = 3 and y = −4, so the vertex is (3, −4).

How do I graph y = x²?

Type y = x^2 into Plotline. It is a parabola with its vertex at the origin, passing through (1, 1), (2, 4) and (−2, 4).

Is it free?