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9.4 : Direct, Joint, and Inverse Variation

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**1. **9.4 : Direct, Joint, and Inverse Variation Translating from words to equation
Proportion approach
“Diagonal” hyperbolas

**2. **Direct Variation If a variable (y) is equal to another variable (x) times a constant, we can express this relationship with the phrase:
“y varies directly as x”
This means y = kx, where k is called the constant of variation (k does not equal zero)
The graph of y = kx is a line, with y-intercept of 0 and slope of k
One implication of this relationship is that (y1 / x1) = (y2 / x2
So you can form a proportion to solve for a missing value

**3. **Example 4-1a

**4. **Example 4-1b

**5. **Example 4-1c

**6. **JOINT VARIATION Joint variation occurs when one quantity varies directly as the product of two or more other quantities
Formally, we say y varies jointly as x and z if there is some number k such that y = kxz where neither k,x, nor z equals 0
We call also take a proportion approach:
Y1 / (x1z1) = y2 / (x2z2)
Note that we can’t graph joint variation on the x-y plane since there are more than 2 variables

**7. **Example 4-2a

**8. **Example 4-2b

**9. **Example 4-2c

**10. **Inverse Variation With inverse variation, as one quantity increases the other decreases proportionately
More formally, we say that y varies inversely as x if y = k / x (where x and k do not equal 0)
We can also write this last equation as xy = k
Notice that with inverse variation, the product of x and y is always the same number
The graph is interesting.. We’ll look at an example in a minute
You can also solve for a missing value using
X1y1 = x2y2

**11. **Example 4-3a

**12. **Example 4-3b

**13. **Example 4-3c

**14. **Example 4-4a

**15. **Example 4-4b

**16. **Example 4-4c

**17. **Example 4-4d

**18. **Example 4-4e

**19. **You can combine variations.. Wow! For example, the force exerted between two objects varies jointly with the masses of the two objects and varies inversely with the square of the distance between them (r)
We say F = (k * m1 * m2 ) / r2

**20. **Homework: Pg. 496
14 – 38 evens,
50 – 53 all
Also pg. 489, #s 24, 26, 32