Plant height vs weeks: y = 2x + 2
A scientist measured a plant's height each week. Can we write an equation that predicts how tall it will be later?
Here are the measurements plotted as points — weeks across the bottom, height in centimeters going up.
The points climb in a steady pattern. We draw a line of best fit through the middle of them.
To write the equation, we pick two clear points that sit ON the line. Let's use (2, 6) and (6, 14).
Slope tells us how steep the line is. It's the rise — change in y — over the run — change in x.
Rise is fourteen minus six, which is eight. Run is six minus two, which is four. So the slope, m, is two.
Every line can be written as y equals m x plus b, where m is the slope and b is the y-intercept.
We know m is two. Plug in one of our points — let's use (2, 6) — to find b.
Six equals two times two plus b. Six equals four plus b, so b is two.
Our linear model is y equals two x plus two.
The slope of two means the plant grows two centimeters each week. The intercept of two means it started at two centimeters.
Let's predict week ten. y equals two times ten plus two — twenty-two centimeters.
Careful, though! Predicting far past our data — called extrapolation — may not stay accurate. Plants don't grow forever.
We wrote y equals two x plus two and used it to make a prediction. That's how a linear model works!