What Are Quadratic Functions?
In science and mathematics, we often model real-world phenomena — the path of a projectile, the area of a field, or the relationship between speed and stopping distance — using mathematical functions. One of the most powerful and versatile types is the quadratic function.
A quadratic function is a polynomial function of degree 2, meaning the highest power of the variable is 2. Its graph always forms a U-shaped (or inverted U-shaped) curve called a parabola.
The general rule is: if you see an term, you are dealing with a quadratic. These functions appear constantly in physics (projectile motion), biology (population dynamics — boom-bust cycles where a population rises then falls), and chemistry (simplified models of certain reaction processes).
Think of throwing a ball into the air. It rises, reaches a peak, and falls back down — tracing a curved path. That curve is a parabola, and the height of the ball at any moment in time can be described by a quadratic function.
Quadratic functions can be written in three different but equivalent algebraic forms, each of which highlights different features of the parabola. Understanding when to use each form — and how to convert between them — is a core skill at MYP 5 level.
Mathematical Models and Their Limitations
Before diving into the three forms, it is worth pausing to think about what a mathematical model actually is — because the unit theme is Thinking with Models.
A mathematical model is an equation or set of equations that describes a real-world situation. It is a simplification of reality, designed to capture the most important patterns so we can make predictions and draw conclusions.
Quadratic functions are widely used as models because they are relatively simple yet can capture curved, non-linear behaviour. However, every model has limitations:
- A model is only valid over a certain domain (range of input values). For example, a projectile height model is only physically meaningful from launch until the object hits the ground — negative values of are meaningless in context.
- Models make assumptions that may not hold perfectly in reality. A projectile model typically ignores air resistance; a population model may ignore migration.
- A quadratic model cannot capture all behaviour. Real population growth is better described by exponential or logistic functions; a quadratic boom-bust model is an approximation valid only over a restricted time window.
Whenever you use a quadratic model in a science context, always ask: Over what domain is this model valid? What assumptions does it make? What does it leave out? This kind of critical thinking is exactly what MYP Criterion D asks for.
A weather map is a model of atmospheric conditions. It is extremely useful, but it does not capture every gust of wind or local temperature variation. Similarly, a quadratic equation captures the key shape of a relationship without necessarily being a perfect description of reality.
In IB MYP, the key concept of Relationships underpins this unit — we are exploring how variables relate to one another, and how those relationships can be represented, analysed, and questioned using mathematical tools like quadratic functions.
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