You’ve probably seen a parabola before, even if you didn’t know what it was called. The arc of water from a fountain, the path of a basketball in the air, the curve of a suspension bridge cable, or the shape of a satellite dish; they’re all examples of the same type of mathematical curve: the quadratic function!

The general form is:

f(x) = ax² + bx + c.  Here, a, b, and c are constants, and a ≠ 0.

What makes it quadratic is the x² term. That squared term is what gives the graph its curve. Instead of moving in a straight line, the rate of change itself is changing, which creates that familiar bend. If you’re taking physics, it’s interesting to know that any time you’re taking the area under a straight-line graph, the result is quadratic. For example, distance from the area under a linearly growing velocity graph, or spring energy from the area under the graph F = kx.

Another important property of parabolas is that they’re symmetrical! This symmetry happens because the x² term grows the same way for positive and negative values. For example, (2)² and (−2)² both equal 4, so the shape is identical on either side of the vertex.

Standard form

Let’s take a closer look at the standard form, f(x) = ax² + bx + c, to understand what it tells us.

The a value tells us the direction and steepness of the parabola. If a > 0, it opens upward (like a U-shape). If a < 0, it opens downward (like an arch). Larger |a| values make it narrower and steeper, while smaller |a| values make it wider and flatter.

The c value is the y-intercept, the point where the graph crosses the vertical axis (x = 0).

The vertex

The vertex is the turning point of the parabola; its highest point if it opens downward, or its lowest point if it opens upward.

From the standard form, you can find it using:

x = −b / (2a)

Then plug this x-value into the equation to get the y-coordinate.

This formula comes from a process called completing the square, which rewrites the equation so the vertex is easy to identify. The vertex is especially important in real-world problems that involve maximizing or minimizing something, like finding the highest point of a jump or the lowest cost for a project. In fact, this idea is explored much more in calculus. Check out my notes on optimization if you’d like to read more about this topic!

Vertex form

In vertex form, the equation looks like:

f(x) = a(x − h)² + k

Here, (h, k) is the vertex.

h shifts the parabola left or right
k shifts it up or down
a still controls the direction and steepness

This form is often the easiest for visualizing the graph before plotting it.

Factored form

When a quadratic factors nicely, it can be written as:

f(x) = a(x − r₁)(x − r₂)

The values r₁ and r₂ are the roots, aka the x-values where the graph touches or crosses the x-axis.

This is helpful for figuring out when something reaches zero in a real situation, like when an object hits the ground, when profit reaches break-even, or when velocity becomes zero.

There are three possible scenarios:

Two distinct real roots: the parabola crosses the x-axis twice.

One repeated root: the parabola just touches the x-axis.

No real roots: the parabola stays entirely above or below the x-axis.

Whether a quadratic has zero, one, or two real roots can be determined using the discriminant. Check out my post on solving quadratic equations if you’d like to learn about this in more detail!