1. An Introduction to Direct and Inverse Proportion
Slideshow 26, Mathematics
Mr Richard Sasaki, Room 307

2. Objectives
Recall the meaning of proportional and learn two common forms
See how the rate of change differs between both forms
Practice finding the rate of change for functions where there is no constant

3. Proportion
Two things are proportional when there is no constant added to them (b = 0).
𝑓 𝑥 =𝑎𝑥 ⇒
𝑓 𝑥 α 𝑥
Here, 𝑓 𝑥 and 𝑥 are directly proportional.
(Their relation is only affected by a multiple.)
Let’s look at an example.

4. Example (Direct Proportion)
Let’s look at an example.
A boy bakes cakes at a rate of 8 cakes per hour. Represent this as a function where 𝑥 is the number of hours after he starts.
𝑓 𝑥 = 8𝑥
As b (the constant) = 0, 𝑓 𝑥 α 𝑥.
8𝑥 means that for every hour he works, he makes 8 cakes.

5. Example (Direct Proportion)
Let’s look at this in the form of a table.
𝑓 𝑥 = 8𝑥 𝒙 1 2 3
𝑓(𝑥) 8 16 24
We can see that the way that 𝑥 and 𝑓(𝑥) are increasing relate to each other. They also start at (0, 0) because of no b constant. This shows that they are directly proportional.
The rate of change will remain the same. (𝒂 = 𝟖)

6. Example (Inverse Proportion)
Let’s look at another example.
A girl bakes 24 cakes and needs to split them between 𝑥 people (𝑥 is the number of people). Represent this as a function where 𝑓 𝑥 is the amount of cake that each person gets.
24 𝑥
𝑓 𝑥 =
As b (the constant) = 0, 𝑓 𝑥 α 1 𝑥 .
This shows that 𝑓 𝑥 and 𝑥 are inversely proportional (𝑓 𝑥 α 1 𝑥 ).

7. Example (Inverse Proportion)
Try the worksheet!
Let’s look at this in the form of a table.
24 𝑥
𝑓 𝑥 = 𝒙 1 2 3
𝑓(𝑥) ∞ 24 12 8
We can see that the way that 𝑥 and 𝑓(𝑥) are changing relate to each other. As one goes up, the other goes down. There is no b constant but they do not start at (0,0) because the function is not linear. This implies that they are inversely proportional.
What is happening with the rate of change? It is changing…

8. 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙
𝑓 𝑥 =2500𝑥
𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝒙 1 2 3
𝑓(𝑥)
0 Yen
2500 Yen
5000 Yen
7500 Yen
𝑓 𝑥 = 𝑥
𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝒙 1 2 3 4
𝑓(𝑥)
48000 Yen
24000 Yen
16000 Yen
12000 Yen
𝑓 𝑥 =− 1 2 𝑥
𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙