8-1 Inverse Variation

Quick Review

An equation in two variables of the form y = k x y equals , k over x  or x y = k , x y equals k comma  where k ≠ 0 , k not equal to 0 comma  is an inverse variation with a constant of variation k. Joint variation describes when one quantity varies directly with two or more other quantities.

Example

Suppose that x and y vary inversely, and x = 10 when y = 15. Write a function that models the inverse variation. Find y when x = 6.

y = k x 15 = k 10 , so  k = 150. The inverse variation is  y = 150 x . When  x = 6 , y = 150 6 = 25. table with 4 rows and 1 column , row1 column 1 , y equals , k over x , row2 column 1 , 15 equals , k over 10 , comma so k equals , 150. , row3 column 1 , cap theinversevariationis . y equals , 150 over x , . , row4 column 1 , cap when , x equals 6 comma y equals , 150 over 6 , equals 25. , end table

Exercises

6. Suppose that x and y vary inversely, and x = 30 x equals 30  when y = 2. y equals 2.  Find y when x = 5. x equals 5.

Write a direct or inverse variation equation for each relation.

7. x	3	4	8
   y	24	18	9

8. x	5	7	9
   y	30	42	54

Write the function that models each relationship. Find z when x = 4 and y = 8.

9. z varies jointly with x and y. When x = 2 x equals 2  and y = 2 , z = 7 . y equals 2 comma z equals 7 .
10. z varies directly with x and inversely with y. When x = 5 x equals 5  and y = 2 , z = 10 . y equals 2 comma z equals 10 .

8-2 The Reciprocal Function Family

Quick Review

The graph of a reciprocal function has two parts called branches. The graph of y = k x − b + c y equals . fraction k , over x minus b end fraction . plus c  is a translation of y = k x y equals , k over x  by b units horizontally and c units vertically. It has a vertical asymptote at x = b x equals b  and a horizontal asymptote at y = c . y equals c .

Example

Graph the equation y = 3 x − 2 + 1 . y equals . fraction 3 , over x minus 2 end fraction . plus 1 .  Identify the x- and y-intercepts and the asymptotes of the graph.

b = 2 , b equals 2 comma  so the vertical asymptote is x = 2. x equals 2.

c = 1 , c equals 1 comma  so the horizontal asymptote is y = 1. y equals 1.

Translate y = 3 x y equals , 3 over x  two units to the right and one unit up.

When y = 0 , x = − 1 . y equals 0 comma x equals negative 1 .

The x-intercept is ( − 1 , n 0 ) . negative 1 comma n 0 close .

When x = 0 , y = − 1 2 . x equals 0 comma y equals negative , 1 half , .

The y-intercept is (0, − 1 2 ) . negative , 1 half , close .

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Exercises

Graph each equation. Identify the x- and y-intercepts and the asymptotes of the graph.

11. y = 1 x y equals , 1 over x
12. y = − 2 x 2 y equals . fraction negative 2 , over x squared end fraction
13. y = − 1 x − 4 y equals , negative 1 over x , minus 4
14. y = 2 x + 3 − 1 y equals . fraction 2 , over x plus 3 end fraction . minus 1

Write an equation for the translation of y = 4 x y equals , 4 over x  that has the given asymptotes.

15. x = 0 , y = 3 x equals 0 comma y equals 3
16. x = 2 , y = 2 x equals 2 comma y equals 2
17. x = − 3 , y = − 4 x equals negative 3 comma y equals negative 4
18. x = 4 , y = − 3 x equals 4 comma y equals negative 3

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