Graphing Linear Relationships

Graphing a linear rule is a five-step routine: build a small table, plot the points, rule a straight line through them, extend it past the plotted points, and label it with its equation. Geometrically two points are enough to fix a line, but always plot at least three. The third costs nothing and acts as a free error check — if all three sit on one straight line your arithmetic is almost certainly right, and if one misses, you know immediately that a table entry is wrong. Extending the line matters too, because the relationship carries on beyond the x values you happened to choose.

There is a quicker route than a full table: find where the line crosses each axis. Every point on the y-axis has x = 0, so substituting x = 0 gives the y-intercept immediately — for y = −2x + 4 that is (0, 4). Every point on the x-axis has y = 0, so setting y = 0 gives 0 = −2x + 4, which solves to x = 2, giving (2, 0). Two points, join them, done. It is much faster than building a table, particularly when the numbers are awkward. Just plot one extra point as a check, since with only two points a slip is invisible.

Once a line is drawn you can read it in both directions. To find y for a given x, go up from the x-axis to the line and then across to the y-axis. To find x for a given y, go across from the y-axis to the line and then down. Draw those dashed construction lines rather than doing it by eye, because in exams they earn method marks. One important distinction: a graph gives an estimate, limited by how accurately you can read the scale, while the equation gives an exact answer. If the question says 'use the graph', read it off even when you could calculate it — that is what is being tested.

Linear graphs describe real situations all the time. A taxi charging a $4 flagfall plus $2 per kilometre gives fare = 2d + 4. The two numbers in the rule have plain meanings on the graph: the y-intercept of 4 is the fixed cost you pay before moving at all, and the steepness of 2 is the rate at which the fare climbs per kilometre. That is the real payoff of graphing — the shape of the line tells you about the situation. One difference from pure maths: in a real context negative values often make no sense. A taxi cannot travel minus three kilometres, so the graph starts at the y-axis and only goes right.

Drawing two lines on the same axes shows where they cross, and that point is the one pair of values satisfying both rules at once. Here y = 2x + 1 and y = −x + 4 meet at (1, 3), and you can verify it in both: 2 times 1 plus 1 is 3 ✓, and negative 1 plus 4 is 3 ✓. Always check the crossing point in both equations, since reading it off the graph is only an estimate. In a real context this point is the break-even — the moment two options cost the same. Before it one is cheaper, after it the other, which is exactly the question people actually want answered when comparing two plans.