Straight lines–you have met them before!
A linear relationship means that as one thing increases or decreases, the other valuable increases or decreases by the same proportion. If you are paid an hourly rate in your job you are acutely aware of this relationship because:
If you work:
• twice as many hours in a week, your pay is twice as much.
• only 50 % of the hours, you get 50% of your usual pay.
• no hours, you don’t get paid.
The relationship between the hours worked (independent variable) and your pay (dependent variable) is linear and if graphed would give you a straight line. The steeper the graph, the higher your hourly rate. Have a look at the graph to the right.
In all cases, the person’s pay is directly proportional to the work hours – double the hours, double the pay. But some people are accumulating money a lot faster than other people. That is reflected in the steeper slope/gradient of some lines (like Bill Gates).
In fact, you can look at a graph such as the one on the right and work out how much each person makes per hour. (In other words, “the change in money per time” or “rate of pay”.)
The PC tech makes about…
The pizza guy makes about…
You should recognise that the “change in money per time” as the same as the change in y value divided by the change in x, which is the classic formula for slope or gradient of a line.
If you can engrave this on your brain, you’ll have a head start interpreting lots of graphs, whether they are straight lines or not. The slope at any part of a graph tells you how fast the process is happening right then.
TEST This graph represents sales of pizza slices over a period of 4 hours. At what point in time is pizza income increasing at the fastest rate?
Click the buttons to see how slope and rate are related: