What Are Linear Functions?
A linear function is a relationship between two variables that produces a straight line when graphed. It has the general form , where is the gradient (slope) and is the y-intercept.
Linear functions are one of the most powerful models in science and mathematics. They describe situations where one quantity changes at a constant rate relative to another — like distance travelled at constant speed, or the cooling of an object over short time periods.
The two key features of any linear function are:
- Gradient (): How steep the line is, and whether it goes up or down. It represents the rate of change.
- Y-intercept (): Where the line crosses the vertical axis. It represents the starting value when .
Think of a linear function like a taxi fare. The y-intercept is the fixed starting charge just for getting in the cab, and the gradient is the cost per kilometre. Every kilometre adds the same amount to your total — that constant rate is what makes it linear.
Before we can understand transformations, we need to be confident reading and interpreting the equation .
What Is a Transformation?
A transformation is a change applied to a function (or shape) that moves, flips, stretches, or otherwise alters its position or appearance on a graph. The original function is called the parent function.
In MYP Sciences, we use the idea of transformations as part of thinking with models — when the world around us changes (e.g., a new starting condition, a reversed process), our mathematical model changes in a predictable way.
For linear functions, the two most important basic transformations are:
- Translation — sliding the line up, down, left, or right without changing its slope
- Reflection — flipping the line across an axis, which changes the direction of the gradient
Transformations are not random changes — they follow strict mathematical rules. Understanding these rules means you can quickly sketch a new function from an old one, or write a new equation from a graph. This is a key modelling skill in science.
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