Single Line Graphs

How is one point read accurately?

Start at the horizontal label requested by the question, move to the plotted point and then read its vertical position using the numerical scale. An original tank-volume series has 120 litres at 08:00, 90 at 09:00, 90 at 10:00 and 150 at 12:00. These four coordinate pairs are the data. Reading 10:00 as the third hour after a different origin would change the problem unnecessarily.

Check how much each grid interval represents. If two labelled horizontal lines are 100 and 200 with four equal spaces between them, each space represents 25 units. Count spaces rather than the number of boundary lines. A graph with an omitted origin still has readable values, but bar-like visual intuition about height from the page edge becomes unreliable.

What do rising, falling and flat segments mean?

In the tank example, volume falls by 30 litres from 08:00 to 09:00, stays at the same recorded level from 09:00 to 10:00 and rises by 60 litres from 10:00 to 12:00. The final rise occurs across two hours, giving an average net increase of 30 litres per hour.

The flat portion does not prove that no water entered or left the tank. Inflow and outflow could have balanced. It establishes only that the recorded volume did not change between the endpoints. If the question explicitly says the connecting segments describe continuous linear change, the flat segment represents constant volume throughout that interval. Without that convention, intermediate fluctuations remain unobserved.

How are maximum and minimum distinguished from fastest change?

The maximum recorded tank volume is 150 litres at noon; the minimum is 90 litres at two observation times. Neither fact alone identifies the greatest rate of change. Rates require differences and durations. In the example, the first decline is −30 litres per hour, while the last increase is +30 litres per hour on average.

When the stem asks for the greatest increase, a negative rate is not a candidate even if its magnitude is large. When it asks for the greatest absolute rate of change, both rising and falling intervals may matter. Read the requested sign and comparison carefully rather than treating steepness as an unqualified answer.

How is a value between observations estimated?

Linear interpolation assumes a constant change between the two surrounding observations. Under that explicit assumption, the tank's volume at 11:00 is halfway between 90 at 10:00 and 150 at 12:00, so it is 120 litres. The arithmetic uses the fraction of elapsed time, not merely the midpoint of numbers regardless of when they were measured.

At 10:30, one quarter of the two-hour interval has elapsed, giving 90 + 60/4 = 105 litres under the same assumption. Label an interpolated value as model-based. A line connecting sparse measurements can be a visual aid without proving that the physical process followed a constant rate between them.

What changes when the vertical axis is distance?

An original distance-from-start graph shows 0 km at 08:00, 20 km at 09:00 and 10 km at 10:00. The object is closer to the starting point at the final observation than at the middle one. The downward line does not mean negative distance; it means the displayed distance from the start decreased.

If the problem specifies travel along one straight route without additional reversals, the journey covers 20 km outward and 10 km back, totalling 30 km. Its final displacement from the start is 10 km in the outward direction. Without the route assumption, distance-from-start observations alone do not determine every part of the path travelled.

How should missing and repeated observations be treated?

A missing point is not a zero. If observations are at 08:00, 10:00 and 12:00, there may be no measured value for 09:00. A repeated vertical value at two times likewise does not prove that the process remained unchanged at every intervening instant. The question must specify whether interpolation is intended.

If several observations share the maximum value, report all requested times rather than selecting the first one automatically. If a question asks for the first time a threshold is reached, distinguish the first recorded point at or above it from an interpolated crossing. Those can differ even in a well-drawn graph.

How can the solution be checked?

Create a small table containing time, value, change and duration. For the tank data, the changes are −30, 0 and +60, and durations are one, one and two hours. Their sum is +30 litres, matching final minus initial volume: 150 − 120. This telescoping check catches a missed interval or reversed subtraction.

End by returning to the axis wording. Litres per hour is a rate; litres is a level or change. A final value, a net change and total movement answer different questions. Reading one line well means preserving these distinctions while using only the measurements and assumptions actually supplied.