Linear relationships

How much will your phone bill be this month? Say your plan costs fifteen dollars, plus five dollars for every gigabyte of data you use. Let's turn that into a table, a graph, and an equation.

Start with a table of values. Zero gigabytes costs fifteen dollars. One gigabyte, twenty. Two, twenty five. Three, thirty. Every time the data goes up by one, the cost goes up by five. That's a constant rate of change.

Now plot each row as a point. Data goes across, cost goes up. Look, the points line up perfectly. Join them, and you get a straight line. That's why we call it a linear relationship.

How steep is the line? That's the slope. Go across one gigabyte: that's the run. The cost goes up five dollars: that's the rise. Slope is rise over run, five over one. Try a bigger step: up ten, across two. Still five!

Now look where the line starts, on the vertical axis. At zero gigabytes, you still pay fifteen dollars. That's the y-intercept: the starting value.

Put it together: y equals m x plus b. m is the slope, five. b is the y-intercept, fifteen. So the cost is five times the gigabytes, plus fifteen. Use six gigabytes? That's forty five dollars.

Now let's change the plan. Raise the base fee to twenty five dollars, and the whole line slides up. Drop the data price to three dollars, and the line gets flatter. So b moves the line up and down, and m changes how steep it is.

So which plan is cheaper? The two lines cross at five gigabytes, where both cost forty dollars. Use less than that, and the first plan wins. Use more, and the second plan is cheaper.

So remember: a constant rate of change makes a straight line. The slope, m, is rise over run. The y-intercept, b, is the starting value. And together, they give you y equals m x plus b.