1. Model Inverse & Joint Variation Multiply & Divide Rational Expressions
8.1: Variation; 8.4: Mult/Divide Rational Expressions
Model Inverse & Joint Variation Multiply & Divide Rational Expressions
Objectives:
Write a mathematical model involving direct, inverse, or joint variation
To multiply and divide rational expressions

2. Warm-Up 1
Find a mathematical model to represent this little know statement from classical mechanics:
The force of gravity F between two masses m1 and m2 is directly proportional to the product of the two masses and inversely proportional to the square of the distance between them r.
Newton’s Law of Universal Gravitation

3. Warm-Up 2
Find a mathematical model to represent this little know statement from classical mechanics:
In a physical sense, what does this this equation mean?
What happens when one of the masses increase?
Newton’s Law of Universal Gravitation

4. Warm-Up 2
Find a mathematical model to represent this little know statement from classical mechanics:
In a physical sense, what does this this equation mean?
What happens if the distance between the masses increases?
Newton’s Law of Universal Gravitation

5. 8.1: Variation; 8.4: Mult/Divide Rational Expressions
Objective 1
You will be able to write a mathematical model involving direct, inverse, or joint variation

6. This type of line shows direct variation.
The simplest kind of line is one that has a slope and intersects the 𝑦-axis at the origin.
This type of line shows direct variation.
Equation: 𝑦=𝑚𝑥

7. Direct Variation
When a line shows direct variation between 𝑥 and 𝑦, we write 𝑦=𝑎𝑥, where 𝑎 is the constant of variation.
𝑦 is said to vary directly with 𝑥.
𝑎 is the same thing as slope!

8. Direct Variation, Linear
The following are equivalent:
𝑦 varies directly as (or with) 𝑥
𝑦=𝑎𝑥, for some nonzero constant 𝑎, called the constant of variation or constant of proportionality
𝑦 is directly proportional to 𝑥

9. Exercise 1
The variable 𝑦 varies directly as 𝑥. Write a direct variation equation that has the given ordered pair as a solution. Then find each constant of variation.
(3, −9)
(−7, 4)

10. Directly Proportional
Since 𝑦=𝑎𝑥 can be rewritten as 𝑎=𝑦/𝑥, a set of ordered pairs shows direct variation if 𝑦/𝑥 is constant. y = 2x y = 2x + 1 x 1 2 3 4 y 6 8 x 1 2 3 4 y 5 7 9

11. Exercise 2
Great white sharks have triangular teeth. The table below gives the length of a side of a tooth and the body length for each of six great white sharks. Tell whether tooth length and body length shows direct variation. If so, write an equation that relates the quantities.

12. Exercise 3
The variable y is directly proportional to the square of x such that y = 18 when x = 6. Write an equation that relates x and y. What is the constant of variation? Find y when x = 12.

13. Exercise 4
The variable y varies directly as the square root of x such that y = 18 when x = 9. Write an equation that relates x and y. What is the constant of variation? Find y when x = 16.

14. More Direct Variation
Sometimes it’s boring just being directly proportional to 𝑥. So we can spice things up a bit by letting 𝑦 be directly proportional to…
The square of 𝒙:
𝒚=𝒂 𝒙 𝟐
The square root of 𝒙:
𝒚=𝒂 𝒙
The cube of 𝒙:
𝒚=𝒂 𝒙 𝟑
Et Cetera

15. Direct Variation, Nonlinear
The following are equivalent:
𝑦 varies directly as (or with) the 𝑛th power of 𝑥
𝑦 is directly proportional to the 𝑛th power of 𝑥
𝑦=𝑎𝑥𝑛, for some nonzero constant 𝑎

16. Exercise 5
A company has found that the demand for its product varies inversely as the price of the product. Write an equation relating demand d and price p. Interpret the meaning of your model.

17. Inverse Variation
The following are equivalent: Your model could be inversely proportional to the 𝑛th power of 𝑥
𝑦 varies inversely as (or with) 𝑥
𝑦 is inversely proportional to 𝑥
𝑦= 𝑎 𝑥 , for some nonzero constant 𝑎

18. Exercise 6
The variable y is inversely proportional to x such that y = 12 when x = 1/2. Write an equation that relates x and y. What is the constant of variation? Find y when x = 12.

19. Inversely Proportional
Since 𝑦= 𝑎 𝑥 can be rewritten as 𝑎=𝑥𝑦, a set of ordered pairs shows inverse variation if 𝑥𝑦 is constant. 𝑦=2/𝑥 x 1 2 3 4 y
2/3
1/2

20. Exercise 7
The table compares the area A (in mm2) of a computer chip with the number c of chips that can be obtained from a silicon wafer.
Area (mm2), A 58 62 66 70
Number of chips, c
448
424
392
376
Write a model that gives c as a function of A.
Predict the number of chips per wafer when the area of a chip is 81 mm2.

21. Joint Variation
The following are equivalent: Your model could be jointly proportional to the 𝑛th power of 𝑥 and the 𝑚th power of 𝑦
𝑧 varies jointly as (or with) 𝑥 and 𝑦
𝑧 is jointly proportional to 𝑥 and 𝑦
𝑧=𝑎𝑥𝑦, for some nonzero constant 𝑎

22. Joint Variation
The following are equivalent: Your model could be jointly proportional to more than 2 variables
𝑧 varies jointly as (or with) 𝑥 and 𝑦
𝑧 is jointly proportional to 𝑥 and 𝑦
𝑧=𝑎𝑥𝑦, for some nonzero constant 𝑎

23. Exercise 8
The variable z is jointly proportional to x and y such that z = 7 when x = 1 and y = 2. Write an equation that relates x, y, and z. What is the constant of variation? Find z when x = −2 and y = 5.

24. Exercise 9
The maximum load that can be safely supported by a horizontal beam is jointly proportional to the width of the beam and the square of its depth, and inversely proportional to the length of the beam. Determine the change in the maximum safe load under the following conditions:
The width of the beam doubles
The depth of the beam doubles
The length of the beam doubles

25. Exercise 10
Write a sentence using variation terminology for the volume of a cone.

26. 8.1: Variation; 8.4: Mult/Divide Rational Expressions
You will be able to multiply and divide rational expressions
Objective 2

27. Rational Expressions
Recall that a rational number is one that can be written as the ratio of two integers. Likewise, a rational expression is a type of expression that is written as the ratio of two expressions.

28. Simplifying Rational Expressions
A rational expression is said to be in simplified form if the numerator and denominator contain no nontrivial common factors.

29. Simplifying Rational Expressions
A rational expression is said to be in simplified form if the numerator and denominator contain no nontrivial common factors.

30. Simplifying Rational Expressions
A rational expression is said to be in simplified form if the numerator and denominator contain no nontrivial common factors. To simplify expressions:
Factor the numerator and denominator
Cancel out factors that are common to the numerator and denominator

31. Exercise 11
Simplify each expression.

32. Exercise 12
Determine what is wrong with the following simplification.

33. Multiplying Rational Expressions
Multiplying rational expressions is just like multiplying any other fractions: Of course, you’ll need to factor and simplify as needed. One more thing:

34. Multiplying Rational Expressions
Multiplying rational expressions is just like multiplying any other fractions:

35. Exercise 13
Multiply:

36. Dividing Rational Expressions
Dividing rational expressions is just like dividing any other fractions. Just multiply the first by the reciprocal of the second: Again, you’ll need to factor and simplify as needed.

37. Dividing Rational Expressions
Dividing rational expressions is just like dividing any other fractions. Just multiply the first by the reciprocal of the second:

38. Exercise 14
Divide:

39. Exercise 15
Perform the indicated operation.

40. Model Inverse & Joint Variation Multiply & Divide Rational Expressions
Objectives:
Write a mathematical model involving direct, inverse, or joint variation
To multiply and divide rational expressions