Mathematics Year 7: Graphs: describing and modelling situations
Mathematics, Year 7 (Portugal): Reading a graph from left to right tells the story of two variables; modelling goes the other way, from a situation to the function that describes it.
A graph is the story of two variables
A graph shows how one variable depends on another. On the horizontal axis goes the variable that changes freely, such as time; on the vertical axis, the one that depends on it. Each point is a pair of values. If the variable can take any value in an interval, like time, the points are joined by a line; if it only takes separate values, like the number of tickets, only the points are drawn. To describe a graph, read it from left to right: does the line go up (increases), go down (decreases) or stay horizontal (does not change)?
From a situation to a function
A graph shows the variation at a glance; a model lets you predict. Modelling means starting from a situation, choosing the two variables and finding the function that links them. A square panel with n tiles along each side uses n × n tiles. Table: n = 1, 2, 3, 4, 5 gives 1, 4, 9, 16, 25 tiles. The expression is y = n × n, and the graph is only separate points, because there is no panel with 2.5 tiles on a side. The points rise faster and faster: the differences are 3, 5, 7, 9. A panel with 10 tiles per side needs 100 tiles.
Reading Tomás's morning walk
The graph of Tomás's distance from home (km) against time (h) joins the points (0, 0), (1, 3), (2, 3) and (4, 0) with segments. In the first hour he goes away from home, 3 km, at 3 km per hour. From 1 h to 2 h the line is horizontal: the distance stays at 3 km, so he is not getting closer or farther (he is resting, for example). From 2 h to 4 h the distance falls from 3 km to 0: he comes home at 3 ÷ 2 = 1.5 km per hour, slower than on the way out, which is why that segment is less steep. At 3 h he is 3 − 1.5 = 1.5 km from home.
The graph is not a drawing of the path
Looking at Tomás's graph, many people say: «he climbed a hill, walked on the flat and came down». It is a natural first impression, because the line goes up, along and down. But the vertical axis is the distance from home, not the height. Up means farther from home, horizontal means the distance does not change, and down means closer to home. He may have walked along a perfectly flat, straight road the whole time. Always read the axes first, then the line.
Filling a container and boiling water
Water flows into a container at a steady rate and you record the height of the water over time. In a cylinder the graph is a straight segment. In a vase that widens towards the top, the water rises more and more slowly, so the graph gets less steep: the graph tells you about the shape of the container. In Physics and Chemistry, the graph of the temperature of water being heated has a horizontal stretch while it boils: the temperature stays the same as long as the water boils.
Test what you learned
A graph has time in hours on the horizontal axis and the temperature of a room in °C on the vertical axis. What does the point (3, 18) say?
- a) The room warmed up by 18 °C in 3 hours
- b) After 3 hours the temperature is 18 °C
- c) After 18 hours the temperature is 3 °C
Correct answer: b) After 3 hours the temperature is 18 °C — The first number is read on the horizontal axis (time: 3 h) and the second on the vertical axis (temperature: 18 °C).
The volume of water in a tank (litres) against time (minutes) is a graph of segments through (0, 20), (4, 60), (6, 60) and (10, 20). Which description is correct?
- a) It gains 40 L per minute, then stays at 60 L, then loses 40 L per minute
- b) It gains 10 L per minute for 4 minutes, then the tank is empty for 2 minutes, then it loses 10 L per minute
- c) It gains 10 L per minute for 4 minutes, stays at 60 L for 2 minutes, then loses 10 L per minute
Correct answer: c) It gains 10 L per minute for 4 minutes, stays at 60 L for 2 minutes, then loses 10 L per minute — Up: (60 − 20) ÷ 4 = 10 L per minute. Then a horizontal stretch at 60 L. Down: (20 − 60) ÷ 4 = −10 L per minute.
In a graph of distance from home (vertical axis) against time (horizontal axis), one stretch of the line goes down. What does it mean?
- a) The person is getting closer to home
- b) The person is walking down a slope
- c) The person has stopped
Correct answer: a) The person is getting closer to home — The vertical axis measures distance from home, so a falling line means that distance is getting smaller.
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