Straight-Line Graphs Explained: y = mx + c

Which phone plan is cheaper? Careful: the lower starting price can cost you more. Imagine you have just arrived at Shibuya Crossing in Tokyo, and your phone needs mobile data. Two made up offers are on the table. Plan A charges five credits for the month, plus three credits per gigabyte. Plan B charges seventeen credits, plus one per gigabyte. A credit is just our imaginary money unit. Which would you choose? The answer is hiding where two straight lines cross.

Chapter one: build the bill. First, turn the bill into a picture. Put data used on the horizontal axis, called x, and monthly cost on the vertical axis, called y. Both plans charge for the exact amount of data used, with no extra fees in our model. A point tells you two things. At two gigabytes, Plan A costs eleven credits: five plus three times two. So the point has coordinates two comma eleven. Read across first, then up. The graph is a map of possible bills, not a picture of a hill.

Now look at zero data. Plan A still costs five credits. Its line crosses the y axis where x is zero and y is five. That starting value is the y intercept. In y equals m times x plus c, c is the y intercept value, and here c equals five. Some books use b instead of c. Same job, different letter. The starting fee shifts the line up. It does not tell you how fast your bill grows. For that, we need the other number.

Chapter two: how fast does it grow? Every extra gigabyte adds three credits to Plan A. That constant rate is the slope, also called the gradient. We write it as m. Slope equals change in y divided by change in x: vertical rise over horizontal run. From two gigabytes to four, the bill rises from eleven to seventeen. The rise is six. The run is two. Six divided by two is three credits per gigabyte. Here is the trap: counting squares only works if you check what each square represents. Always read the axis scales.

Put the two parts together. Plan A is y equals three times x plus five. The three tells you the rate. The five tells you the starting value. To draw it, plot zero comma five. Move one gigabyte right and three credits up. That gives one comma eight. Repeat to get two comma eleven. Join the points with a straight line. It stays straight because the rate stays constant. We only use the part where x is zero or greater: you cannot use negative mobile data.

Chapter three: the crossover. Plan B is y equals x plus seventeen. Its starting fee is higher, but its slope is only one. Draw both plans on the same axes. At two gigabytes, A costs eleven and B costs nineteen. A wins. At eight gigabytes, A costs twenty nine and B costs twenty five. Now B wins. A starts cheaper, but climbs faster. Somewhere between those amounts, the prices match. Can you spot the crossing before we calculate it?

At the crossing, both plans have the same x value and the same y value. So set their cost expressions equal: three times x plus five equals x plus seventeen. Subtract x from both sides. Two times x plus five equals seventeen. Subtract five. Two times x equals twelve. Divide by two. X equals six. Now substitute six into either plan. Three times six plus five equals twenty three. The crossing is six comma twenty three. Check Plan B: six plus seventeen is also twenty three. Both bills agree.

Now you can make the decision. Below six gigabytes, Plan A is cheaper. Above six, Plan B is cheaper. At exactly six, both cost twenty three credits. Imagine your next month: mostly messages, or lots of Tokyo crossing videos? Estimate your data, then read which line is lower. Lower on this graph means a smaller bill. A graph turns two pricing rules into a choice you can see. Of course, these are simplified offers; a real purchase also depends on the actual terms.

Chapter four: your turn. A different line has equation y equals negative two times x plus twelve. Pause and find its slope, its y intercept, and y when x equals three. Ready? The slope is negative two: as x increases by one, y falls by two. The intercept is zero comma twelve. At x equals three, y equals negative two times three plus twelve, which is six. A negative slope goes downhill from left to right. A zero slope would give a horizontal line.

Back at the crossing, your phone decision is solved. Remember four useful phrases: y intercept, the value when x is zero. Slope, change in y over change in x. Linear equation, a model with a constant rate, written here as y equals m times x plus c. Intersection, a point that satisfies both equations. These connect starting value, rate of change, and equal cost to the symbols. Next time two offers compete, sketch the lines. The cheapest beginning does not always make the cheapest bill.