6. Graphs of Functions Defined by Tables of Data

One important way to show the relationship between variables is by using a table of values obtained by observation and experimentation.

Such data values would indicate whether the variables are related (ie. have a formula that links them).

Such points are normally plotted with a smooth curve.

Exception:

When data values are taken only for certain intervals or are averaged over the intervals, then the intervals between the points have no real meaning.

Such points are connected by straight line segments.

Example 1

The electric energy usage (in MJ) for a particular house for each month of a certain year is given in the following table:

Month	Jan	Feb	Mar	Apr	May	Jun
Energy Usage	10 504	12 363	10 168	7 500	4 825	3 568

Month	Jul	Aug	Sep	Oct	Nov	Dec
Energy Usage	2 548	2 887	3 301	5 748	7 302	9 706

Plot these data

Answer

Example 2

Steam in a boiler was heated to 150° C. Its temperature was then recorded each minute as follows:

Time (min)	0.0	1.0	2.0	3.0	4.0	5.0
Temp (°C)	150.0	142.8	138.5	135.2	132.7	130.8

Plot the graph.

Answer

We can estimate values of one variable for given values of the other.

For instance, the temperature after 2.5 min can be estimated from the graph as 136.7° C.

Similarly, the time taken for the steam to cool down to 141.0° C is estimated to be 1.4 min.

Linear Interpolation

A more accurate way of estimating values from a graph is called linear interpolation.

Linear interpolation assumes that if a particular value lies between two of those listed in the table, then the corresponding value of the other variable is at the same proportional distance between the listed values.

Example (3)

Use linear interpolation to find the temperature of the cooling steam after 1.4 min:

Time (min)	1.0	1.4	2.0
Temp (°C)	142.8	???	138.5

We can use LiveMath to find this value for us.

LIVEMath

Answer

Exercise

The following table gives the fraction f of the total heating load of a system that will be supplied by a solar collector of area A:

f	0.22	0.30	0.37	0.44	0.50	0.56	0.61
A (m²)	20	30	40	50	60	70	80

By means of linear interpolation, for A = 36 m², find f.

Answer: 0.34

5. Graphing Using a Computer Algebra System

7. Continuous and Discontinuous Functions

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