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Quadratic Graph Calculator

Graph a quadratic function to find its vertex, roots, axis of symmetry, and maximum or minimum value.

Quadratic equation

Coefficient explorer · y = ax^2 + bx + c

opening and width
vertex and axis position
y-intercept

x [-10, 10]

Drag to pan · scroll or pinch to zoom · keyboard: arrows pan, + / - zoom, 0 reset

Quadratic overview

Enter a quadratic to see its vertex, roots, and axis here.

What is a quadratic graph?

A quadratic graph is the U-shaped parabola drawn by a quadratic graph equation like y = ax² + bx + c. The graph of a quadratic function is called a parabola. If you are studying parabolas in class, finding the vertex, roots, or axis of symmetry, this page draws the curve and labels each feature so you can check your work at a glance.

Use the coefficient explorer to feel how each number reshapes the parabola, or jump straight to the vertex, roots, and discriminant in the overview.

Try an example

How to graph a quadratic function

  1. Identify the coefficients a, b, and c in y = ax^2 + bx + c.
  2. Find the vertex at x = -b / (2a), then substitute to get y.
  3. Draw the axis of symmetry, the vertical line through the vertex.
  4. Calculate the roots with the discriminant when real roots exist.
  5. Plot the y-intercept at (0, c).
  6. Sketch the parabola through these points, opening upward if a is positive.

Quadratic graph example: y = x^2 - 4x + 3

Here a = 1, b = -4, c = 3. The vertex is at x = -(-4) / (2 · 1) = 2, and y = 2^2 - 4 · 2 + 3 = -1, so the vertex is (2, -1) and the axis of symmetry is x = 2.

The discriminant is D = (-4)^2 - 4 · 1 · 3 = 4. Since D is positive there are two real roots: x = 1 and x = 3. The y-intercept is (0, 3). Because a is positive the parabola opens upward, the minimum is -1 at x = 2, and the range is y ≥ -1. In vertex form this is y = (x - 2)^2 - 1.

Graph this example to see every point highlighted.

What to know about parabolas

Finding the vertex

Use x = -b / (2a), then substitute that x back into the equation to find y. The vertex is the turning point of the curve.

Reading the discriminant

D = b^2 - 4ac decides the roots: two real roots when D is positive, one repeated root when D is zero, and no real roots when D is negative.

Standard form vs. vertex form

Standard form y = ax^2 + bx + c shows the y-intercept c. Vertex form y = a(x - h)^2 + k shows the vertex (h, k) directly. Use whichever reveals what you need.

How a and the vertex shape the graph

A positive quadratic graph has a greater than 0 and opens upward, while a negative a opens downward. A larger absolute value makes a narrower parabola. The value of b moves the vertex and axis sideways, and c sets the y-intercept, sliding the graph up or down.

Quadratic graphing FAQ

How to graph quadratic equation?

Type the quadratic graph equation, for example x^2 - 4x + 3, and press Graph. Read the vertex, roots, and axis in the overview, then drag to pan and scroll to zoom. No account needed.

How to graph quadratic function step by step?

Identify a, b, and c, find the vertex at x = -b / (2a), draw the axis of symmetry, then plot the roots and y-intercept. That is how to graph quadratic function examples by hand; this tool draws the same parabola instantly so you can check each step.

What is the graph of a quadratic function called?

The graph of a quadratic function is called a parabola: a symmetric U-shape that opens upward when a is positive and downward when a is negative.

How do I find the vertex of a parabola?

For y = ax^2 + bx + c, compute x = -b / (2a) and substitute back for y. For x^2 - 4x + 3 the vertex is (2, -1), and the axis of symmetry is the vertical line x = 2 through it.

How do I find the roots of a quadratic function?

Use the discriminant D = b^2 - 4ac. Two real roots when D is positive, one repeated root when D is zero, none when D is negative. For x^2 - 4x + 3 the roots are 1 and 3.

What does the discriminant tell you?

It tells how many real roots exist: two when positive, one repeated when zero, none when negative.

How do I know whether a parabola opens upward or downward?

Check the sign of a. Positive opens upward, negative opens downward. -2x^2 + 8x - 5 opens downward with maximum 3.

Can a quadratic have no real roots?

Yes, when the discriminant is negative the parabola never touches the x-axis. x^2 + 4x + 8 has no real roots.

How do I find the maximum or minimum of a quadratic?

It sits at the vertex. Upward-opening parabolas have a minimum, downward-opening ones a maximum. For x^2 - 4x + 3 the minimum is -1 at x = 2.