1. Mathematical Modeling and Variation
Example 1

2. Mathematical Modeling and Variation

4. Mathematical Modeling and Variation
Direct Variation
We say that something is “directly proportional” to something else if it can be written in the form y = kx, where k is some nonzero constant.
For example, the length of an object’s shadow is directly proportional to the height of the object.
y can also be directly proportional to the nth power of x.
For example, the area of a circle is directly proportional to the 2nd power of it’s radius, 𝐴=𝜋 𝑟 2 . In this case, the constant k is equal to π.

5. Mathematical Modeling and Variation

6. Mathematical Modeling and Variation
Checkpoint

7. Mathematical Modeling and Variation

8. Mathematical Modeling and Variation
Checkpoint

9. Mathematical Modeling and Variation
Inverse Variation
We say that something is “inversely proportional” to something else if it can be written in the form 𝑦= 𝑘 𝑥 , where k is some nonzero constant.
For example, the hours left of sunlight is inversely proportional to the time of day.
y can also be inversely proportional to the nth power of x.
𝑦= 𝑘 𝑥 𝑛

10. Mathematical Modeling and Variation

11. Mathematical Modeling and Variation
Joint Variation
Joint variation is when a variable is directly proportional to the product of two other variables.
It can also be proportional to the nth power of the other variables.
𝑧=𝑘 𝑥 𝑛 𝑦 𝑚

12. Mathematical Modeling and Variation

13. Mathematical Modeling and Variation
Combined Variation
Combined variation is when a variable is directly proportional to another variable and inversely proportional to a third.
𝑧= 𝑘 𝑥 𝑛 𝑦 𝑚

14. Mathematical Modeling and Variation

15. Mathematical Modeling and Variation