Straight-Line Graphs: Cracking the y = mx + c Code

Imagine a machine: you put in a number, it spits out a different number. You try it three times and record: put in 1, get out 5; put in 2, get out 7; put in 3, get out 9. Can you spot the pattern? Every time the input goes up by 1, the output goes up by 2. And when the input was 0, the output would have been 3. This rule — a constant rate of change starting from a fixed value — is exactly what a linear graph shows.

A linear equation is an equation whose graph is a perfectly straight line. Every straight-line graph can be written in the form y = mx + c. Here is what each letter means: • y is the output value (shown on the vertical axis). • x is the input value (shown on the horizontal axis). • m is the gradient — how steep the line is. It tells you how much y changes each time x increases by 1. A positive m means the line slopes upward; a negative m means it slopes downward. • c is the y-intercept — the value of y when x = 0. It is the point where the line crosses the y-axis. So for the mystery machine above, the rule was y = 2x + 3. Here m = 2 (the line goes up 2 units for every 1 unit across) and c = 3 (the line crosses the y-axis at 3). How to plot a straight-line graph in three steps: 1. Choose three values of x (small numbers work best, e.g. 0, 1, 2). 2. Substitute each x into the equation to find y. Record your results in a table of values. 3. Plot each (x, y) point on a coordinate grid and draw a straight line through all three points. Two points are enough to fix a line, but a third acts as a check — if all three are in line, you know you have made no arithmetic error.