AI Tutor
Imagine checking your phone battery: at 1:00 PM it is at 80%, at 2:00 PM it is at 65%, and at 3:00 PM it is at 50%. How can you tell if your battery is draining at a perfectly steady rate just by looking at a list of numbers?

The Constant Rate of Change

A function is linear if its graph forms a straight line, which happens only when it has a constant rate of change.
We calculate this rate by comparing the change in output values (Ξ”y\Delta y) to the change in input values (Ξ”x\Delta x) using the ratio Ξ”yΞ”x\frac{\Delta y}{\Delta x}.
What does this look like in practice when all input values increase by the exact same amount?

Tables with Equal Steps

When the xx-values increase by a fixed step size, determining if the function is linear is straightforward: just check if the yy-values also change by a constant amount.
Because Ξ”x=+1\Delta x = +1 every time and Ξ”y=+3\Delta y = +3 every time, the ratio Ξ”yΞ”x=31=3\frac{\Delta y}{\Delta x} = \frac{3}{1} = 3 is constant throughout the table.
But what happens if the table skips some input values and the xx-values do not increase evenly?

Tables with Uneven Steps

If the xx-values jump by different amounts, checking only Ξ”y\Delta y can easily mislead you.