Characteristics of Linear Functions

In science, we use mathematical models to describe and predict the behaviour of the natural world. Linear functions are among the most powerful and commonly used models — from the relationship between distance and time at constant speed, to how temperature changes as you add heat energy.

A linear function can be written in the general form:

$y=mx+c$

Where:

• $y$ = the dependent variable (what you measure or observe)
• $x$ = the independent variable (what you control or change)
• $m$ = the gradient (or slope) — how steeply the line rises or falls
• $c$ = the y-intercept — the value of $y$ when $x=0$

The gradient is calculated using:

$m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

Where $\Delta$ (delta) means "change in".

Interpreting the sign of the gradient:

• Positive gradient ($m>0$): The line slopes upward from left to right. As $x$ increases, $y$ increases. Example: a car travelling at 60 km/h covers 60 km more for every additional hour driven.
• Negative gradient ($m<0$): The line slopes downward from left to right. As $x$ increases, $y$ decreases. Example: a cooling object loses temperature over time.
• Zero gradient ($m=0$): The line is horizontal. $y$ does not change as $x$ changes. Example: an object at rest — distance stays constant over time.