Straight Line Graphs (y = mx + c)

La méthode, étape par étape

1. 1

   Read the intercept straight from the equation

   y = 2x + 3 → crosses the y-axis at (0, 3)

   The constant is where the line meets the y-axis, because substituting x = 0 leaves y = c. This gives a guaranteed point with no calculation, which is why sketching should always start here.

2. 2

   Use the gradient as a step instruction

   m = 2 → from (0,3), go 1 right and 2 up to (1,5)

   Gradient is change in y over change in x, so m = 2 means two up for every one right. Plotting a second point by stepping is faster and less error-prone than computing a table of values.

3. 3

   Find a gradient from two points

   (1, 4) and (3, 10) → (10 − 4)/(3 − 1) = 6/2 = 3

   Subtract the y values and divide by the difference in x values. The order must match on both — taking y from the first point and x from the second inverts the sign, which is the commonest error here.

4. 4

   Rearrange into the standard form before reading anything

   2y = 6x + 8 → y = 3x + 4 → m = 3, c = 4

   The gradient is only visible when y is alone with a coefficient of 1. Reading m = 6 from 2y = 6x + 8 is wrong by a factor of two, and it is the single most common misreading in the topic.

5. 5

   Sketch from two points and a check

   Plot (0, 3) and (1, 5), draw the line, check a third point fits

   Two points define a line but give no way to detect an error. A third point is the check, and it costs one substitution — well worth it when the sketch carries several marks.

What the two letters actually describe

The gradient is a rate: how much y changes for each unit increase in x. On a distance-time graph that rate is speed; on a cost graph it is the price per item. Reading m as a rate rather than as steepness is what makes graphs useful outside the maths classroom.

The intercept is the starting value — where the line is when x is zero. On a cost graph it is the fixed charge before any items are bought, on a distance-time graph the distance already travelled at the start. Both letters mean something, and questions increasingly ask what.

• m = gradient = change in y ÷ change in x
• c = y-intercept, the value when x = 0
• Positive m rises left to right, negative m falls
• Parallel lines share m
• Perpendicular gradients multiply to −1

Parallel and perpendicular

Parallel lines have equal gradients — they climb at the same rate, so they never meet. Any line parallel to y = 3x + 1 has the form y = 3x + k, and only the intercept distinguishes them.

Perpendicular gradients multiply to −1, so the perpendicular to a gradient of 3 has gradient −⅓: flip the fraction and change the sign. Students frequently do one of those two operations and not the other, which produces a line that is neither parallel nor perpendicular.

Sketching without a table of values

A table of values is slow and offers many chances for arithmetic slips. Since the intercept is readable directly and the gradient gives a step, two points can be plotted in about ten seconds with no calculation beyond addition.

This matters under exam pressure, where a graph is often worth two or three marks and the time saved goes into a question worth more. Students who learn only the table method spend disproportionate time on cheap marks.

How we teach straight line graphs

We insist that any equation is rearranged into y = mx + c form before a student reads a gradient off it. The misreading of 2y = 6x + 8 as gradient 6 is so common that we make the rearrangement an explicit written step rather than something done mentally.

We also teach gradient as a rate from the first lesson, using distance-time and cost graphs rather than abstract lines. It costs nothing and it is what makes the topic connect to the interpretation questions that carry the most marks.