Construct Graphs from Tables of Values or Descriptions of Situations

By drawing a graph, we can visualise the behaviour of a mathematical function or of a physical process.

Imagine water flowing into a cylindrical tank at a constant rate. To answer the question of how deep the water will be at any given time after the water has begun to flow, we might depict the situation as a graph showing time on one axis and depth of water on the other axis.

In this case, the depth of water increases by the same amount for every unit of elapsed time. Therefore, the graph will appear as a straight line with a slope that depends on the flow rate of the water entering the tank.

Example 1

Now, suppose the tank is conical in profile. If water is poured in at a constant rate, the depth increases rapidly at first but at a decreasing rate because the width of the tank increases with depth. The following diagram illustrates this situation.

Without going into the details of the mathematics, we can state that the volume of water in the tank is proportional to the cube of the depth, and so, the depth is proportional to the cube root of the volume. But the volume is proportional to the elapsed time because the water is flowing in at a constant rate.

It follows that the depth of water in the tank is proportional to the cube root of the elapsed time. The graph, in this scenario, will have a shape like the following, although the scale could be different. We might have guessed the shape of the graph given that the rate of increase of depth decreases with time.

 

 

Example 2

We can visualise the behaviour of a function by constructing a table of values of the function, plotting the points from the table, and joining the plotted points with a smooth curve.

Suppose a function is given by $y(x)=x^2+3x-4$y(x)=x2+3x−4. To make a table of values, we choose a few values of the domain variable $x$x and calculate the corresponding function values $y(x)$y(x).

$x$x	$-5$−5	$-4$−4	$-3$−3	$-2$−2	$-1$−1	$0$0	$1$1	$2$2
$y(x)$y(x)	$6$6	$0$0	$-4$−4	$-6$−6	$-6$−6	$-4$−4	$0$0	$6$6

The plot of these points looks like this:

Next, we draw a smooth curve that passes through the plotted points. We are assuming that the function behaves predictably and smoothly between the points we have calculated. This is a safe assumption for the kinds of functions that we usually deal with: linear functions, quadratics and other polynomial functions, and exponentials, provided that the chosen values of the domain variable are close enough together. 

 

 

 

Situations occur in which measurements are made of two variable quantities that seem to be related in a regular way. It may not be immediately obvious how the relationship can be described algebraically but we can make progress by plotting the pairs of measurements as a graph. 

For example, it may be noticed that the height of the water at a particular coastal location varies with time. The tidal variation can be displayed graphically for a reasonably long period so that future tides can be predicted on the assumption that the observed pattern repeats at regular time intervals. 

In this case, the graph represents the functional relation between time and water height even though an algebraic model may not have been discovered.

 

Worked Examples

Question 1

Question 2

Question 3