8 Chapter Test

Do you know HOW?

Write a function that models each variation.

1. x = 2 x equals 2  when y = − 8 , y equals negative 8 comma  and y varies inversely with x.
2. x = 0.2 x equals 0.2  and y = 3 y equals 3  when z = 2 , z equals 2 comma  and z varies jointly with x and y.
3. x = 1 3 , y = 1 5 , x equals , 1 third , comma y equals , 1 fifth , comma  and r = 3 r equals 3  when z = 1 2 , z equals , 1 half , comma  and z varies directly with x and inversely with the product of r 2 r squared  and y.

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model any direct or inverse variations.

4. x	y
   3	6
   5	8
   7	10
   9	12

5. x	y
   4	32
   8	16
   16	8
   32	4

Write and graph an equation of the translation of y = 7 x y equals , 7 over x  that has the given asymptotes.

6. x = 1 ; y = 2 x equals 1 semicolon y equals 2
7. x = − 3 ; y = − 2 x equals negative 3 semicolon y equals negative 2

For each rational function, identify any holes or horizontal or vertical asymptotes of the graph.

8. y = x + 1 x − 1 y equals . fraction x plus 1 , over x minus 1 end fraction
9. y = x + 3 x + 3 y equals . fraction x plus 3 , over x plus 3 end fraction
10. y = x − 2 ( x + 1 ) ( x − 2 ) y equals . fraction x minus 2 , over open , x plus 1 , close . open , x minus 2 , close end fraction
11. y = 2 x 2 x 2 − 4 x y equals . fraction 2 , x squared , over x squared , minus 4 x end fraction
12. y = 1 x + 2 − 3 y equals . fraction 1 , over x plus 2 end fraction . minus 3
13. y = x 2 + 5 x − 5 y equals . fraction x squared , plus 5 , over x minus 5 end fraction

Simplify each complex fraction.

14. 2 x 1 − 1 y fraction fraction 2 , over x end fraction , over 1 minus , 1 over y end fraction
15. 3 − 3 x 1 2 − 1 x fraction 3 minus , 3 over x , over 1 half , minus , 1 over x end fraction

Simplify each rational expression. State any restrictions on the variable.

16. x 2 + 7 x + 12 x 2 − 9 fraction x squared , plus 7 x plus 12 , over x squared , minus 9 end fraction
17. ( x + 3 ) ( 2 x − 1 ) x ( x + 4 ) ÷ ( − x − 3 ) ( 2 x + 1 ) x fraction open , x plus 3 , close . open , 2 x minus 1 , close , over x . open , x plus 4 , close end fraction . divides . fraction open , negative x minus 3 , close . open , 2 x plus 1 , close , over x end fraction
18. x 2 − 1 x 2 + 2 x − 3 − x + 1 x + 3 fraction x squared , minus 1 , over x squared , plus 2 x minus 3 end fraction . minus . fraction x plus 1 , over x plus 3 end fraction
19. x ( x + 4 ) x − 2 + x − 1 x 2 − 4 fraction x . open , x plus 4 , close , over x minus 2 end fraction . plus . fraction x minus 1 , over x squared , minus 4 end fraction

Solve each equation. Check your solutions.

20. x 2 = x + 1 4 x over 2 , equals . fraction x plus 1 , over 4 end fraction
21. 3 x − 1 = 4 3 x + 2 fraction 3 , over x minus 1 end fraction . equals . fraction 4 , over 3 x plus 2 end fraction
22. 3 x x + 1 = 0 fraction 3 x , over x plus 1 end fraction . equals 0
23. 3 x + 1 = 1 x 2 − 1 fraction 3 , over x plus 1 end fraction . equals . fraction 1 , over x squared , minus 1 end fraction
24. 1 x + 1 3 = 6 x 2 1 over x , plus , 1 third , equals , fraction 6 , over x squared end fraction
25. 1 x + x x + 2 = 1 1 over x , plus . fraction x , over x plus 2 end fraction . equals 1
26. Your neighbor can seal your driveway in 4 hours. Working together, you and your neighbor can seal it in 2.3 hours. How long would it take you to seal it working alone?

Do you UNDERSTAND?

27. Vocabulary Describe a situation that represents an inverse variation.
28. Compare and Contrast How is simplifying rational expressions similar to simplifying fractions? How is it different?
29. Writing When does a discontinuity result in a vertical asymptote? When does it result in a hole in the graph?
30. Open-Ended Write a function whose graph has a hole, a vertical asymptote, and a horizontal asymptote.

31. Reasoning State any restrictions on the variable in the complex fraction.

    x − 3 x + 4 x 2 − 1 x fraction fraction x minus 3 , over x plus 4 end fraction , over fraction x squared , minus 1 , over x end fraction end fraction

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