What is a Quadratic Function?

Quadratic Function

A function of the form $f (x) = a x^{2} + b x + c$ , where $a \neq = 0$ . The highest power of $x$ is 2, which gives the function its characteristic curved shape called a parabola.

Quadratic functions are everywhere in the real world. When you throw a ball through the air, the path it follows is a parabola. The reflectors in car headlights and satellite dishes are designed using quadratic relationships. Even the cables of a parabolic arch bridge (where the load is spread evenly along the horizontal) form a true parabolic curve.

Note

You may have heard that suspension bridge cables hang in a parabola — but this is actually a common misconception! Freely hanging cables form a catenary curve, which is described by a different mathematical function. However, when a bridge deck hangs from cables with a uniformly distributed horizontal load, the cable shape becomes very close to a parabola. It's a great reminder that real-world models are always approximations!

The standard form of a quadratic function is:
$f (x) = a x^{2} + b x + c$

where:

$a$ controls the width and direction of the parabola
$b$ interacts with $a$ to influence the position of the curve — its effect is complex and best understood through vertex form (introduced next)
$c$ is the y-intercept (where the curve crosses the y-axis)

Note

In MYP Sciences and Mathematics, we often use quadratic models to describe motion, area, and other physical phenomena. Understanding how to transform these functions helps us adapt our models to fit real data.

What is a Quadratic Function?

The Vertex Form — A Powerful Way to See Transformations

Vertex Form

The vertex form of a quadratic function is $f (x) = a (x - h)^{2} + k$ , where $(h, k)$ is the vertex (the turning point) of the parabola.

Vertex form is the most useful form for understanding transformations because each parameter directly controls a specific visual feature of the graph.

Parameter	Role
$a$	Stretches/compresses and reflects the parabola
$h$	Shifts the parabola left or right
$k$	Shifts the parabola up or down

Analogy

Think of vertex form like adjusting a camera. The parameter $h$ moves the camera left or right, $k$ moves it up or down, and $a$ zooms in or out (and can flip the image upside down). Each control does exactly one job — making it easy to predict the result.

The vertex $(h, k)$ is the most important point on a parabola — it is either the minimum (lowest point) when $a > 0$ , or the maximum (highest point) when $a < 0$ .

The Vertex Form — A Powerful Way to See Transformations

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