Constructing Graphs from Descriptions

Imagine filling a straight-sided glass from a steady tap, and then filling a cone-shaped one standing on its point. Both take the same total time. Which one's water level rises fastest at the start, and which at the end? Say why in terms of how much water each layer needs.

To draw a graph from a description, break it into phases, then for each phase answer two questions: is the quantity rising, falling or steady, and is the rate constant, increasing or decreasing? Each answer names a shape, and the shapes are joined end to end.

Turning a description into a graph is a four-step process, and it is the reverse of the reading process used so far.

1. Choose and label the axes. Decide which quantity goes on each, and write both names with their units. Time nearly always goes across.

2. Break the description into phases. A phase is a stretch over which the behaviour does not change. Mark roughly where each begins and ends.

3. For each phase, answer two questions. Is the quantity rising, falling or steady? And is the rate constant, increasing or decreasing? The first answer gives the direction, the second decides between a straight line and one of the four curves from Lesson 3.

4. Join the phases and check. Draw each shape, join them end to end, and then read the finished graph back in words. If the description you produce does not match the one you were given, something is wrong.

Step 4 is not optional. Reading a graph back is a genuinely different mental operation from drawing it, and it catches errors that staring at the sketch does not.

The standard question of this type: water pours in at a constant rate, and you must sketch the height of the water against time for a container of a given shape.

The key idea is that a constant inflow means each second delivers the same volume. What that volume does to the height depends on how wide the container is at that level.

Narrow at that height: the same volume forms a tall thin layer, so the level jumps up quickly.
Wide at that height: the same volume spreads into a short wide layer, so the level creeps up slowly.

That single observation settles every case:

Straight sides. Every layer is the same width, so the level rises at a constant rate. The graph is a straight line.

Widening upward, like a bucket or a cone on its point. The layers get wider as it fills, so the rise slows. The graph is increasing at a decreasing rate: rising and flattening.

Narrowing upward, like a funnel or a cone on its base. The layers get narrower, so the rise speeds up. The graph is increasing at an increasing rate.

The width at a height controls the steepness at that height, and nothing else does.

How two phases join is itself information, and a good sketch gets it right.

A corner, where the graph bends sharply, means the rate changed suddenly. A tap turned off, a vehicle braking hard, a container's shape changing abruptly at a shoulder. The gradient jumps from one value to another with nothing in between.

A smooth join, with no sharp bend, means the rate changed gradually. A container whose walls curve continuously, a car easing to a stop, a temperature settling.

So a cylinder with a wider cylinder stacked on top gives two straight segments meeting at a corner: constant rate, then a slower constant rate, with the change happening the instant the water reaches the join. A smoothly curved vase gives a single curve with no corners anywhere.

Two more features worth drawing deliberately:

The starting value, which need not be zero. A container that already holds water starts partway up the vertical axis.

The end of the graph. If the container fills completely, the graph stops there; it does not continue rising, and drawing it flat afterwards would claim the water level stays put while the tap is still running, which is a different situation.

The same method handles any description, and questions use a wide range of contexts.

"The population grew slowly at first, then rapidly, then levelled off as resources ran short." Three phases, all increasing. First increasing at an increasing rate, then continuing to steepen, then increasing at a decreasing rate. The result is the stretched S shape mentioned in Lesson 3, with a point of inflection where the steepening turns into flattening.

"The car accelerated from rest, cruised at a steady speed, then braked sharply to a stop." For a graph of speed against time: rising, then horizontal, then falling steeply to zero. For a graph of distance against time the same journey looks quite different: increasing at an increasing rate, then a straight line, then increasing at a decreasing rate, and never falling. Read which quantity the question wants on the vertical axis before drawing anything.

"The medicine level dropped quickly at first and then more gradually." One phase, decreasing at a decreasing rate: a falling curve that flattens towards zero without reaching it.

The vocabulary in the question is usually a direct translation of the shape. "Levelled off" and "more gradually" mean a decreasing rate; "rapidly", "took off" and "accelerating" mean an increasing one; "steady" and "constant" mean a straight line.

Five items on constructing graphs. Work each one, then reveal the answer.

1. 1 A straight-sided glass fills from a steady tap. What shape is the height-time graph?

   Show answer
   Every layer is the same width.A straight line

2. 2 A bucket, wider at the top, fills from a steady tap. What shape?

   Show answer
   Higher layers are wider, so they take longer.Rising and flattening

3. 3 A funnel, narrower at the top, fills from a steady tap. What shape?

   Show answer
   Higher layers are narrower, so they fill faster.Rising and steepening

4. 4 A tap is turned off abruptly. What feature appears on the graph?

   Show answer
   The rate changes suddenly.A corner

5. 5 "It levelled off" describes which rate behaviour?

   Show answer
   The graph is flattening while still rising.A decreasing rate