Modelling with a linear function (Example on Celsius and Fahrenheit temperature scales). Expressing Celsius as a function of Fahrenheit and vice versa by getting the linear function and its inverse. Plotting and finding graphically the temperature that is the same in Celsius and Fahrenheit. Describing the relationship between the function and its inverse graphically.

1. Modelling with a linear function
(Example on Celsius and Fahrenheit temperature scales)
Prepared by
Ismail Mohammad El-Badawy
ismailelbadawy@gmail.com

2. Linear function
• A linear function refers to
equation of straight line.
y = mx + b

3. Calibration of a thermometer
(Celsius and Fahrenheit scales)
• Calibration of an instrument is putting a scale on it to provide reliable
measurements.
• The temperature scale is usually defined by two fixed points:
✓ The temperature at which pure ice melts (or pure water freezes).
→ 0°C (0 degrees Celsius) or 32°F (32 degrees Fahrenheit).
✓ The temperature at which pure water boils.
→ 100°C (100 degrees Celsius) or 212°F (212 degrees Fahrenheit).
• Once the two fixed points have been fixed on the thermometer scale, the
rest of the scale is made by dividing the distance between them into:
• 100 equal divisions or degrees in case of Celsius scale. (Why? 100 – 0 = 100)
• 180 equal divisions or degrees in case of Fahrenheit scale. (Why? 212 – 32 = 180)

4. Draw the straight line which passes
through these two points
Melting ice
Boiling water

5. Use your graph to convert and find …
o The freezing temperature of a solution of brine is 0°F. What is the value of this
temperature in degrees Celsius?
o What is the equivalent value of 30°F in degrees Celsius?
o What is the equivalent value of 34°C in degrees Fahrenheit?
o What is the temperature of boiling pure water (100 °C) in degrees Fahrenheit?

6. Use your graph to convert and find …
o The freezing temperature of a solution of brine is 0°F. What is the value of this
temperature in degrees Celsius?
✓ ~ − 18°C
o What is the equivalent value of 30°F in degrees Celsius?
✓ ~ − 1°C
o What is the equivalent value of 34°C in degrees Fahrenheit?
✓ ~93°F
o What is the temperature of boiling pure water (100 °C) in degrees Fahrenheit?
✓ ~212°F

7. Find the equation of the straight line
which passes through these two points
Write your answer in the form of
y = mx + b

8. 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 =
212 − 32
100 − 0
=
9
5
°F/°C
𝑦_𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 32°F
∴ 𝑇𝑒𝑚𝑝 °F =
9
5
𝑇𝑒𝑚𝑝 (°C) + 32
Fahrenheit as a linear function of Celsius
A rise of 1°C corresponds to a rise of 1.8°F
0°C corresponds to 32°F
Change in x Change in y
Gradient y-intercept
This linear function (i.e. equation
of the straight line) models the
relation between the Celsius and
Fahrenheit temperature scales.

9. Use the linear function to convert and find ..
o The freezing temperature of a solution of brine is 0°F. What is the value of this temperature in
degrees Celsius?
✓ 0°F =
9
5
𝑇𝑒𝑚𝑝 (°C) + 32 → 𝑇𝑒𝑚𝑝 (°C) =
5
9
0 − 32 = −17.8°C
o What is the equivalent value of 30°F in degrees Celsius?
✓ 30°F =
9
5
𝑇𝑒𝑚𝑝 (°C) + 32 → 𝑇𝑒𝑚𝑝 (°C) =
5
9
30 − 32 = −1.1°C
o What is the equivalent value of 34°C in degrees Fahrenheit?
✓ 𝑇𝑒𝑚𝑝 (°F) =
9
5
× 34 + 32 = 93.2°F
o What is the temperature of boiling pure water (100 °C) in degrees Fahrenheit?
✓ 𝑇𝑒𝑚𝑝 (°F) =
9
5
× 100 + 32 = 212°F
𝑻𝒆𝒎𝒑 °F =
𝟗
𝟓
𝑻𝒆𝒎𝒑 (°C) + 32

10. Use the linear function to convert and find ..
o The freezing temperature of a solution of brine is 0°F. What is the value of this temperature in
degrees Celsius?
✓ 0°F =
9
5
𝑇𝑒𝑚𝑝 (°C) + 32 → 𝑇𝑒𝑚𝑝 (°C) =
5
9
0 − 32 = −17.8°C
o What is the equivalent value of 30°F in degrees Celsius?
✓ 30°F =
9
5
𝑇𝑒𝑚𝑝 (°C) + 32 → 𝑇𝑒𝑚𝑝 (°C) =
5
9
30 − 32 = −1.1°C
o What is the equivalent value of 34°C in degrees Fahrenheit?
✓ 𝑇𝑒𝑚𝑝 (°F) =
9
5
× 34 + 32 = 93.2°F
o What is the temperature of boiling pure water (100 °C) in degrees Fahrenheit?
✓ 𝑇𝑒𝑚𝑝 (°F) =
9
5
× 100 + 32 = 212°F
𝑻𝒆𝒎𝒑 °F =
𝟗
𝟓
𝑻𝒆𝒎𝒑 (°C) + 32

11. Can we do the opposite ..?
o Fahrenheit as a linear function of Celsius
o 𝑇𝑒𝑚𝑝 °F =
9
5
𝑇𝑒𝑚𝑝 (°C) + 32
o What if we want to express Celsius as a linear function of Fahrenheit ??
✓ Yes, by getting the inverse function.

12. Celsius as a linear function of Fahrenheit
o Finding the inverse of 𝑓 𝑥 =
9
5
𝑥 + 32 𝑇𝑒𝑚𝑝 °F =
9
5
𝑇𝑒𝑚𝑝 (°C) + 32
✓ Step 1: y =
9
5
𝑥 + 32 rewrite the function replacing f(x) with y
✓ Step 2: x =
9
5
𝑦 + 32 interchange x and y
✓ Step 3: y =
5
9
𝑥 − 17.8 make y subject of the formula
✓ Step 4: 𝑓−1 x =
5
9
𝑥 − 17.8 replace y with f-1(x)
∴ 𝑇𝑒𝑚𝑝 °C =
5
9
𝑇𝑒𝑚𝑝 (°F) − 17.8

13. Celsius as a linear function of Fahrenheit
A rise of 1°F corresponds to a rise of 0.56°C
0°F corresponds to − 17.8°C
Change in x Change in y
Gradient y-intercept
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 =
100 − 0
212 − 32
=
5
9
°C/°F
𝑦_𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = −17.8°C
∴ 𝑇𝑒𝑚𝑝 °C =
5
9
𝑇𝑒𝑚𝑝 (°F) − 17.8

14. Now draw both lines on the same axes ..
o Find:
• A temperature that is the same in Celsius and Fahrenheit.
• A temperature in Fahrenheit where the same temperature in Celsius is a smaller number.
• A temperature in Fahrenheit where the same temperature in Celsius is a larger number.
o Describe the relationship between the two lines.

15. • A temperature that is the same
in Celsius and Fahrenheit.
✓ −40°C or −40°F
• A temperature in Fahrenheit
where the same temperature in
Celsius is a smaller number.
✓ 140°F (Any value > −40°F)
✓ 59.98°C
• A temperature in Fahrenheit
where the same temperature in
Celsius is a larger number.
✓ −200°F (Any value < −40°F)
✓ −128.9°C
𝑻𝒆𝒎𝒑 °C =
𝟓
𝟗
𝑻𝒆𝒎𝒑 (°F) − 17.8
𝑻𝒆𝒎𝒑 °F =
𝟗
𝟓
𝑻𝒆𝒎𝒑 (°C) + 32

16. o The graph of the inverse
function is a reflection of the
original function over the
line y = x.
o So, if you're asked to plot
a function and its inverse,
first plot the function and
then switch all x and y values
in each point to plot the
inverse function.
𝑻𝒆𝒎𝒑 °C =
𝟓
𝟗
𝑻𝒆𝒎𝒑 (°F) − 17.8
𝑻𝒆𝒎𝒑 °F =
𝟗
𝟓
𝑻𝒆𝒎𝒑 (°C) + 32
(212°F, 100°C)
(100°C, 212°F)