Introduction to Quadratic Functions

A quadratic function is any function that can be written with an \(x^2\) term as its highest power. This one detail changes everything about how the function behaves and how its graph looks. Instead of a straight line, a quadratic function always graphs as a curved shape called a parabola.

The most common way to write a quadratic function is in standard (or general) form:

\(y = ax^2 + bx + c\)

Here, \(a\), \(b\), and \(c\) are numbers (constants), and \(a\) cannot equal \(0\). If \(a = 0\), the \(x^2\) term disappears and the function becomes linear instead of quadratic. You can explore this structure in more depth in the lesson on quadratic functions in general form.

Before graphing anything, it helps to recognize a quadratic function just by looking at its equation. Compare these two functions:

\(y = 3x + 2\) is linear because the highest power of \(x\) is \(1\).

\(y = 3x^2 + 2\) is quadratic because the highest power of \(x\) is \(2\).

Even a small change like adding an exponent of \(2\) transforms a straight-line relationship into a curved one. That single squared term is what defines the entire quadratic family of functions.

Every quadratic function produces a U-shaped curve called a parabola when graphed. Consider the function:

\(y = x^2 - 4x + 3\)

If you plot several \((x, y)\) points from this equation, they form a smooth, symmetric curve rather than a straight line.

Notice a few important features in this graph:

• The curve opens upward because the coefficient of \(x^2\) is positive.
• The lowest point of the curve is called the vertex. Here, the vertex sits at \(x = 2\).
• A vertical line through the vertex, called the axis of symmetry, splits the parabola into two mirror-image halves.
• The points where the curve crosses the x-axis, at \(x = 1\) and \(x = 3\), are the roots (or zeros) of the function.

For a full walkthrough of plotting parabolas from an equation, see graphing parabolas for given quadratic functions.

The leading coefficient \(a\) tells you which way the parabola opens.

• If \(a > 0\), the parabola opens upward, like a smile, and has a lowest point (a minimum).
• If \(a < 0\), the parabola opens downward, like a frown, and has a highest point (a maximum).

The size of \(a\) also affects how narrow or wide the parabola looks: a larger value of \(|a|\) makes the curve narrower, while a smaller value makes it wider.

Standard form is not the only way to describe a quadratic function. Sometimes it is more useful to write the function in vertex form, which shows the vertex coordinates directly. You can learn how this form works, and how to convert between forms, in the lesson on quadratic functions in vertex form.

Try deciding whether each function below is quadratic:

\(y = 5x^2 + x - 7\) — quadratic, since the highest power of \(x\) is \(2\).

\(y = 2x + 9\) — not quadratic, since there is no \(x^2\) term.

\(y = -x^2 + 4\) — quadratic, since \(a = -1\) and the highest power of \(x\) is \(2\).

Being able to identify a quadratic function at a glance, and knowing what its graph should look like, is the foundation for everything else you will study about parabolas, vertices, and solving quadratic equations.

Related lessons

• Algebra 2