Graphing Quadratic Functions Calculator

What Is a Quadratic Function?

A quadratic function is a function with x squared as its highest power. Its standard form is f(x) = ax² + bx + c, where a is not zero. The graph is always a parabola — a smooth U-shaped curve.

The sign of a tells you which way the parabola opens. When a is positive, the parabola opens up, like a cup. When a is negative, it opens down, like a hill. The size of a tells you how wide or narrow the curve is: a large |a| makes a narrow parabola, and a small |a| makes a wide one.

The Vertex and Axis of Symmetry

The vertex is the highest or lowest point of the parabola. For a parabola that opens up, the vertex is the lowest point. For one that opens down, it is the highest point.

To find the vertex, use the formula x = -b / (2a). That gives the x-value. Then put that x back into the function to get the y-value. For f(x) = x² - 4x + 3, we have a = 1 and b = -4, so x = 4 / 2 = 2. Then f(2) = 4 - 8 + 3 = -1. The vertex is (2, -1).

The axis of symmetry is the vertical line through the vertex. It splits the parabola into two mirror-image halves. Its equation is x = -b / (2a) — the same x-value as the vertex. For the example above, the axis of symmetry is x = 2. If you know a point on one side of the axis, you know its mirror point on the other side.

Roots and the Discriminant

The roots of a quadratic are the x values where the graph crosses the x-axis — where f(x) = 0. A quadratic can have two roots, one root, or no real roots at all.

The discriminant tells you which case you have before you solve anything. It is D = b² - 4ac:

• D > 0: two different real roots. The parabola crosses the x-axis twice.
• D = 0: one root, a double root. The vertex sits exactly on the x-axis.
• D < 0: no real roots. The parabola never touches the x-axis.

For f(x) = x² - 4x + 3, D = 16 - 12 = 4, which is positive, so there are two roots: x = 1 and x = 3. You can see both on the graph. Try the same function in the calculator above and check that the graph crosses the x-axis at exactly those points.

Standard Form, Vertex Form and Factored Form

The same quadratic can be written in three useful forms. Each form shows you something different.

• Standard form: f(x) = ax² + bx + c. The y-intercept is c, and the vertex formula works from a and b.
• Vertex form: f(x) = a(x - h)² + k. The vertex is (h, k) — you can read it straight off. For f(x) = (x - 2)² - 1, the vertex is (2, -1).
• Factored form: f(x) = a(x - r1)(x - r2). The roots are r1 and r2. For f(x) = (x - 1)(x - 3), the roots are 1 and 3.

How to Graph a Quadratic Function

You can sketch any parabola by hand in five steps:

1. Find the vertex with x = -b / (2a).
2. Find the y-intercept — it is c, the value f(0).
3. Find the roots by factoring or with the quadratic formula.
4. Plot one or two more points on each side of the vertex.
5. Draw a smooth curve through the points. A parabola is curved, not a set of straight lines.

Quadratics in Real Life

Quadratics describe many real situations. When you throw a ball, its height over time follows a parabola that opens down. The vertex is the highest point of the throw. When a shop sets a price, its profit often follows a parabola too — too cheap and the profit per item is small, too expensive and nobody buys. The vertex gives the best price.

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