SECONDARY MATHEMATICS · GRAPH INTELLIGENCE
Gradient is not just “rise over run”. It measures how much one quantity changes when another quantity changes by one unit.
Gradient is a rate of change
For a straight line through (2,5) and (6,13), the gradient is (13−5)/(6−2)=8/4=2. This means y increases by 2 units whenever x increases by 1 unit.
That interpretation is stronger than memorising a formula. It tells us what the number 2 means in the relationship.
Units belong to gradient
If y is distance in kilometres and x is time in hours, gradient has units kilometres per hour. If y is cost in dollars and x is number of items, gradient has units dollars per item. The graph may look identical while the meaning changes with the axes.
Positive, zero and negative gradient
A positive gradient means the output increases as the input increases. A zero gradient means the output remains constant. A negative gradient means the output decreases.
For y=20−3x, the gradient is −3. Every increase of 1 in x corresponds to a decrease of 3 in y.
Steepness is relative to the axis scales
A line can look steep or shallow depending on the graph’s horizontal and vertical scales. Numerical gradient is calculated from coordinate change, not visual appearance alone.
Gradient between two points
Use m=(y₂−y₁)/(x₂−x₁), provided x₂≠x₁. Reversing both subtraction orders gives the same result. Reversing only one changes the sign incorrectly.
A vertical line has x₂−x₁=0, so its gradient is undefined rather than zero.
Gradient connects graphs and equations
In y=mx+c, m is the gradient. For y=4x−7, each increase of 1 in x raises y by 4. The coefficient is not merely a symbol in a template; it encodes the line’s rate of change.
Worked checks
A. Through (1,4) and (5,12): gradient=(12−4)/(5−1)=2.
B. Through (−2,7) and (3,−3): gradient=(−3−7)/(3−(−2))=−10/5=−2.
C. y=6−0.5x has gradient −0.5, so y falls by 0.5 for each unit increase in x.
Continue
Continue to intercepts, reading change directly from graphs and recognising graph shape before calculation. Connect with One Relationship, Four Representations and return to the Secondary Mathematics Master Index.