1. 8.1: Direct, Inverse, and Joint Variations
Grade Distribution
3rd
6th
7th
9th A 3 2 B 9 13 10 C 7 6 D 5 F
No Show 4
Range
100+
Avg.
81.25
78.93
79.95
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8.1: Direct, Inverse, and Joint Variations

2. Direct and Inverse Variations
Section 8-1
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8.1: Direct, Inverse, and Joint Variations

3. 8.1: Direct, Inverse, and Joint Variations
Direct Variation
Direct Variation: A linear function which two variables are which one is a constant multiple of the other; when one variable changes the other changes in proportion to the first y x
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8.1: Direct, Inverse, and Joint Variations

4. 8.1: Direct, Inverse, and Joint Variations
Direct Variation
B. Direct Variation: (y/x) = k OR y = kx
K = Constant of Variation
X = Independent Variable
Y = Dependent Variable
If X goes up, Y has to go up and vice versa
C. KEYWORD: VARY(ies) DIRECTLY or DIRECTLY PROPORTIONAL
Divide the variables
Y varies directly with X
D. NOTE
k is not always going to be used in all questions
It is used only when the problem is comparing itself
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8.1: Direct, Inverse, and Joint Variations

5. 8.1: Direct, Inverse, and Joint Variations
Example 1
The weekly salary a woman earns, S, varies directly as the number of hours, h, which she works.  Express this relation as a formula.
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8.1: Direct, Inverse, and Joint Variations

6. 8.1: Direct, Inverse, and Joint Variations
Example 2
The values of y and x vary directly. Determine the constant variation if x = 3 when y = 12.
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8.1: Direct, Inverse, and Joint Variations

7. Example 3
If a varies directly as b and a = 3 when b = 24, find b when a = 10.
Would k be necessary into this equation?
Another way of finding out this problem is through proportions.
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8.1: Direct, Inverse, and Joint Variations

8. 8.1: Direct, Inverse, and Joint Variations
Your Turn
Suppose y varies directly with x and x = 27 when y = –51, find x when y = –17
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8.1: Direct, Inverse, and Joint Variations

9. 8.1: Direct, Inverse, and Joint Variations
Example 4
The values of y and x vary directly. Determine the constant variation if x = 0 when y = 3.
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8.1: Direct, Inverse, and Joint Variations

10. 8.1: Direct, Inverse, and Joint Variations
Inverse Variation
Inverse Variation: A quadratic function which in two variables, the product is constant; when one variable increases the other decreases in proportion so that the product is unchanged. y x
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8.1: Direct, Inverse, and Joint Variations

11. 8.1: Direct, Inverse, and Joint Variations
Inverse Variation
B. Inverse Variation: xy = k OR y = k/x
K = Constant of Variation
X = Independent Variable
Y = Dependent Variable
If X goes up, Y has to go down and vice versa
KEYWORD: VARY(ies) INVERSELY
Multiply the variables
Y varies inversely with X
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8.1: Direct, Inverse, and Joint Variations

12. 8.1: Direct, Inverse, and Joint Variations
Example 5
The variables of x and y vary inversely. Solve for the constant of variation that relates the variables if x = 4 and y = –3.
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8.1: Direct, Inverse, and Joint Variations

13. 8.1: Direct, Inverse, and Joint Variations
Example 6
If y varies inversely as x, and x = 3 when y = 4, find y when x = 8.
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8.1: Direct, Inverse, and Joint Variations

14. 8.1: Direct, Inverse, and Joint Variations
Your Turn
If p varies inversely as the square root of q, and p = 10 when q = 36, find p when q = 16.
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8.1: Direct, Inverse, and Joint Variations

15. 8.1: Direct, Inverse, and Joint Variations
Example 7
Is this relationship between the values in each table as a direct variation, inverse variation, or neither? Then, write the equation. x
0.5 2 6
f(x)
1.5 18
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8.1: Direct, Inverse, and Joint Variations

16. 8.1: Direct, Inverse, and Joint Variations
Example 8
Is this relationship between the values in each table as a direct variation, inverse variation, or neither? Then, write the equation. x
0.2
0.6
1.2
f(x) 12 4 2
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8.1: Direct, Inverse, and Joint Variations

17. 8.1: Direct, Inverse, and Joint Variations
Joint Variation: z = kxy OR (z/xy) = k
K = Constant of Variation
X = Independent Variable
Y = Dependent Variable
Z = Comparison Variable
KEYWORD: VARY(ies) JOINTLY
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8.1: Direct, Inverse, and Joint Variations

18. 8.1: Direct, Inverse, and Joint Variations
Example 9
The variable z varies jointly with the product of x and y. Find an equation which relates the variables if x = 4, y = –3, and z = 12.
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8.1: Direct, Inverse, and Joint Variations

19. 8.1: Direct, Inverse, and Joint Variations
Example 10
If a varies jointly as c and d and a = 20 when c = 2 and d = 4, find d when a = 25 and c = 8
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8.1: Direct, Inverse, and Joint Variations

20. 8.1: Direct, Inverse, and Joint Variations
Your Turn
If a varies jointly as c and d squared, and a = 10 when c = 5 and d = 4, find d when a = 25 and c = 8
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8.1: Direct, Inverse, and Joint Variations

21. 8.1: Direct, Inverse, and Joint Variations
Example 11
The power P in watts of an electrical circuit varies jointly as the resistance R and the square of the current I. For a 600-watt microwave oven that draws a current of 5.0 amperes, the resistance is 24 ohms. What is the resistance of a 200-watt refrigerator that draws a current 2.5 amperes?
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8.1: Direct, Inverse, and Joint Variations

22. 8.1: Direct, Inverse, and Joint Variations
Your Turn
The volume V of a tetrahedron varies jointly with its altitude h and the base area B. The shape has a 5 cm altitude and 6 in2 base has volume of 10 cm3. Write an equation.
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8.1: Direct, Inverse, and Joint Variations

23. 8.1: Direct, Inverse, and Joint Variations
Assignment
Worksheet
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8.1: Direct, Inverse, and Joint Variations