# 1.4.5 Graphical conversions: How would you go about drawing a graph to convert from one scale to the other?

First you need some data about corresponding temperatures on each scale. In the case of Celsius and Fahrenheit, there are two fixed points of reference: the freezing and boiling points of water. On the Celsius scale, the freezing point is defined to be 0°C; on the Fahrenheit scale, the freezing point is 32°F. So if you plot degrees Celsius on the horizontal axis and degrees Fahrenheit on the vertical axis of a graph, the freezing point of water is represented by a point with the coordinates (0,32).

You can also relate the corresponding values at which water boils. On the Celsius scale, 100°C is defined to be the temperature at which boiling occurs; on the Fahrenheit scale, boiling occurs at 212°F. So on the graph the boiling point of water is represented by the point (100, 212).

The two fixed reference points are plotted in Figure 12 and joined by a straight line. This graph enables you to convert any temperature value between 0 and 100°C on the Celsius scale to the corresponding temperature value between 32 and 212°F on the Fahrenheit scale, and vice versa.

## Activity 6: Temperature conversion

Use the graph in Figure 12 to find the values corresponding to 15 °C and 200 °F.

### Discussion

15 °C is equivalent to 59°F.

200 °F corresponds to about 90°C.

If you extend the graph above and below the reference points you can convert from any temperature value on one scale to the corresponding value on the other. This is useful if you want to extend the range of the graph, so that you can use it for temperatures that are below the freezing point, or above the boiling point, of water. Extending a graph beyond the known values is called ‘extrapolation’. How do you know that the extrapolated graph will continue to be a straight line for all temperatures on the two scales? It will be so, because the relationship between the Celsius and Fahrenheit scales is not a matter for experiment, it is *defined* to be a straight line for all temperatures.

Figure 12 shows the range of temperatures appropriate for the liquid state of water. However, would this be a good range for the graph if you were drawing a conversion graph for normal air temperatures in Europe?

Air temperatures in Europe rarely go above 50°C, so there is no point in going as high as the boiling point of water. However, they do go below freezing point. So a range of -50°C to 50°C might be a good starting point.

## Activity 7: Changing the range

By extending the straight line on Figure 12 below 0°C, draw a graph that you can use to convert from Celsius to Fahrenheit over the range -50°C to 50°C. What is the corresponding range of the Fahrenheit scale?

Is there any point for which the numerical values on each scale are the same? If you extended the line indefinitely in both directions, how many other such points do you think you would find?

### Discussion

Figure 13 shows the conversion graph extended to cover the range −50 °C to 50 °C. The corresponding range on the Fahrenheit scale is −58 °F to 122 °F.

A temperature of −40 °C corresponds to −40 °F. This is the only point over the entire range at which a single temperature reading on the two scales are numerically equal.

A temperature conversion graph is different from the earlier conversion graphs. The line does not start at the point (0,0) and the relationship between degrees Celsius and degrees Fahrenheit is not a direct proportionality. For example, a temperature of 15°C corresponds to 59°F, but a temperature 30°C does not correspond to 2 × 59°F = 118°F. In fact, it corresponds to 86°F. Doubling the temperature on the Celsius scale, therefore, is not equivalent to doubling the temperature on the Fahrenheit scale, and vice versa. In other words, you *cannot*express the relationship between the scales in the form

Fahrenheit = some number x degrees Centigrade