Linear Functions

Experimenting with a linear function

A digital graphing tool turns a linear function into something you can play with. You type a rule of the form y = mx + c, and the tool draws the matching line at once; change the value of m or c and the line moves in response. Working this way, by trying a case, looking at the result and then trying another, is called experimenting, and it is a powerful way to learn. Rather than being told in advance what m and c do, a student can discover their roles by changing one number at a time and watching carefully. The line on the screen becomes a kind of laboratory, and each new rule is another small experiment whose result is there to be read.

How m changes the steepness

To see what m does, hold c still and change only m. Start with y = 1x + 1, then try y = 2x + 1: the second line is steeper, climbing more quickly as you move to the right. A larger value of m makes a steeper line, and a smaller value makes a flatter one. Something new happens when m becomes negative. The line y = -1x + 1 does not climb at all; it falls from left to right, sloping downward. So m carries two pieces of information at once, the steepness of the line and its direction, whether it rises or falls. All three of these lines pass through the same point, because they share the same c, yet each has its own slope set by m.

How c slides the line up and down

The number c has a different job, which becomes clear when you hold m still and change only c. Take y = 1x + 0, then y = 1x + 2, then y = 1x + 4. Every line has the same steepness, so they all tilt at the same angle, but each sits higher than the last. Increasing c lifts the whole line upward and decreasing it lowers the line, without ever changing how steep it is. The value of c is the y-intercept, the height at which the line crosses the y-axis, so a line written as y = 1x + 4 meets that axis at 4. Because these lines share the same m, they never cross one another; lines with equal slope are parallel.

Making and testing a conjecture

After watching a few cases like these, a pattern starts to suggest itself, and you might propose that lines with the same m are parallel. A statement like this, offered as a careful guess from the evidence so far, is called a conjecture. A conjecture is not yet a fact, and the next step is to test it rather than simply trust it. Choose fresh cases the conjecture has not yet seen, such as y = 2x + 1 and y = 2x - 3, draw them, and check whether they behave as predicted. Here the two lines stay the same distance apart and never meet, which supports the conjecture. Testing can also work the other way: a single case where the lines did meet would refute the conjecture at once.

Generalising the pattern

Once a conjecture has survived several honest tests, it is reasonable to state it as a general pattern, one expected to hold for every line of the form y = mx + c. Gathering the findings together gives a tidy summary: m controls the steepness and the direction of a line, c controls the y-intercept, two lines with equal m are parallel, and changing c produces a vertical shift of the whole line. This act of pulling many tested cases into one statement is called generalising. It is worth remembering that a generalisation always remains open to challenge, because a single counterexample, just one line that disobeyed the rule, would overturn it. That is exactly why the testing step matters so much before any pattern is announced as general.

Why this matters

Experimenting, conjecturing, testing and generalising is not just a classroom routine; it is close to how mathematics is actually discovered. Linear functions make an ideal first playground for these habits, because each line responds instantly and the effect of every change is easy to see. Understanding how m and c shape a line also underpins a great deal of later work, from reading graphs to building models and studying functions more formally. Digital tools help by making the drawing fast, so that attention can stay on the thinking, on the pattern itself rather than on the plotting. The aim throughout this unit is exploration of how a linear function behaves as its numbers change, rather than drill in solving equations, and that spirit of investigation is the part most worth carrying forward.