Chapter Contents

• Functions and Graphs
• 1. Introduction to Functions
• 2. Functions from Verbal Statements
• 3. Rectangular Coordinates
• 4. The Graph of a Function
• Domain and Range of a Function
• Comparison calculator BMI - BAI
• 5. Graphing Using a Computer Algebra System
• Online graphing calculator (1): Plot your own graph (JSXGraph)
• Online graphing calculator (2): Plot your own graph (SVG)
• 6. Graphs of Functions Defined by Tables of Data
• 7. Continuous and Discontinuous Functions
• 8. Split Functions
• 9. Even and Odd Functions
• Functions and Graphs Problem Solver

• Comments, Questions?

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6. Graphs of Functions Defined by Tables of Data

When performing an experiment, or obsservations, we'll have a table of values. If we graph the data, we can get a better insight into the relationship between our variables.

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

We have 2 possible situations:

1. The data points can be joined by a smooth curve, because the values in between the given data points have meaning. An example would be temperatures taken at the beginning of each month. We would join these with a smooth curve because the temperature rises (or falls) during the month. Such data arises from measurement and is called "continuous" data.
2. We need to use a bar graph (histogram) since the data is not "smooth". That is, the values between the given data points would have no meaning. An example of this case would be if we count a number of objects produced monthly by a factory. Such data is called "discrete".

Example 1

The number of ice creams produced by an Australian factory in each month is given in the following table (The hottest month is February):

Month	Jan	Feb	Mar	Apr	May	Jun
Factory output	10 490	12 325	10 201	7 496	4 816	3 678

Month	Jul	Aug	Sep	Oct	Nov	Dec
Factory output	2 532	2 890	3 312	5 754	7 312	9 690

Plot these data.

Answer

Example 2

Oil is collected from an engine and allowed to cool. Its temperature was recorded each minute as follows:

Time (min)	0.0	1.0	2.0	3.0	4.0	5.0
Temp (°C)	150.0	143.5	137.9	135.1	132.6	131.5

Plot the graph.

Answer

Estimating values

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 137° C (dark red arrows below).

Similarly, the time taken for the oil to cool down to 141.0° C is estimated to be 1.4 min (dark green arrows).

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.

Let's see how this works in an example.

Example 3

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

Time (min)	1.0	1.4	2.0
Temp (°C)	143.5	???	137.9

Answer

Exercise

In a biology lab, an experimenter observes the rate of growth of a bacteria colony at various temperatures, as shown in the table.

Temp (°C)	20	30	40	50	60	70	80
Rate (mg/min)	0.21	0.30	0.37	0.45	0.52	0.57	0.62

By means of linear interpolation, for temp = 36°, find the rate.

Answer