# 6. Graphs of Functions Defined by Tables of Data

When performing an experiment, or observations, 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:

- 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. - 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.

### 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.

## 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 `136°\ "C"` (magenta arrows below).

*t*(min)

Estimating the temperature after 2.5 minutes.

Similarly, the time taken for the oil to cool down to `141.0°\ "C"` is estimated to be `1.4` min (gray arrows).

*t*(min)

Estimating the time to cool to `141°C`.

## 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.

*x*

_{1},

*y*

_{1})

*a*,

*b*)

*x*

_{2},

*y*

_{2})

Linear interpolation: Find `b` for some value `a` such that `x_1 < a < x_2`.

If we have 2 known points `(x_1, y_1)` and `(x_2, y_2)` and we want to find the interpolated value `b` at `x = a` (this value `a` is between `x=x_1` and `x=x_2`), we make use of the formula for slope of a line, and the fact the 2 triangles in the diagram above are in proportion, as follows:

`(y_2 - y_1)/(x_2 - x_1) = (b - y_1)/(a - x_1)`

Solving for `b` gives:

`b = y_1 + (y_2 - y_1)(a - x_1)/(x_2 - x_1)`

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 |

### 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 find the rate for temp = 36°.

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