In a straight-line graph of y against x, the gradient is the change in y for each one-unit increase in x, and the y-intercept is the value of y when x is zero. In context questions, both must be described using the real quantities and units. This skill appears in cost, distance, temperature and growth questions.
It follows plotting with a sensible scale in graphs and transformations, and it prepares you for modelling work later.
How do you turn a line into a sentence?
Name the two quantities first. The gradient is always “vertical unit per horizontal unit”: “RM per km”, “litres per minute”, “cm per week”.
Then calculate the gradient as change in y divided by change in x, using two clear points on the line. Read the intercept from where the line meets the y-axis.
Finally, write the sentence: “For every extra [x unit], [y quantity] increases (or decreases) by [gradient] [y unit]. When [x quantity] is zero, [y quantity] is [intercept].”
Worked example
A taxi fare F (in RM) is plotted against distance d (in km). The line passes through (0, 5) and (10, 20). Find the equation and interpret it.
Step 1, gradient: change in F is 20 − 5 = 15. Change in d is 10 − 0 = 10. So the gradient is 15 ÷ 10 = 1.5.
Step 2, intercept: the line crosses the vertical axis at F = 5.
Step 3, equation: F = 1.5d + 5.
Step 4, meaning: the fare rises by RM1.5 for every extra kilometre. The RM5 is the starting charge before any distance is travelled.
Check: substitute d = 10: 1.5 × 10 + 5 = 20, which matches the second point. To predict a 14 km trip, F = 1.5 × 14 + 5 = 21 + 5 = RM26.
The mistake to watch for
A common slip is to divide the wrong way, or to state the gradient without saying what it measures.
Mistaken answer: “The gradient is 10 ÷ 15 = 0.67, so the fare is RM0.67.”
The student divided change in d by change in F, and also gave no unit.
The correction is to start with the quantity on the vertical axis: “change in fare over change in distance”. The answer 1.5 then reads as “RM per km”. The reversed value 0.67 would mean kilometres per ringgit, which is a different question.
Check yourself
Try these without a calculator, then open each answer.
1. The volume V (litres) of water in a tank is plotted against time t (minutes). The line passes through (0, 200) and (8, 120). Find the gradient and explain it.
Show answer
Gradient = (120 − 200) ÷ 8 = −80 ÷ 8 = −10. The water decreases by 10 litres every minute. The intercept, 200, is the starting volume. The equation is V = 200 − 10t, and the tank is empty when 200 − 10t = 0, so after 20 minutes.
2. A phone plan costs C = 30 + 0.5g, where C is in RM and g is data in GB. Interpret 30 and 0.5.
Show answer
30 is the fixed charge, RM30, when no data is used. 0.5 is the extra cost, RM0.50 for each additional GB.
3. A plant’s height is y = 12 + 3x, where y is in cm and x is the number of weeks. State the starting height, the weekly growth and the height after 6 weeks.
Show answer
Starting height 12 cm, growth 3 cm per week. After 6 weeks, y = 12 + 3 × 6 = 12 + 18 = 30 cm.
Where this leads next
Once you can describe a line in words, move on to translating a simple graph, then use the graphs and transformations practice set. The non-calculator working trainer helps with the fraction and decimal division inside a gradient.
Some students calculate correctly but leave the meaning unstated or misstated. Our teachers look for that pattern in online one-to-one Mathematics tuition.