Name: _____________________________________ Period: __________ Date: ______________ID: A
Algebra II Chapter 9 Practice Test 1. Is the relationship between the variables in the table a direct variation, an inverse variation, or neither? If it is a direct or inverse variation, write a function to model it. x
6
10
11
15
y
84
140
154
210
9. Simplify the complex fraction. 2 3 − 4y y
1 y
–5
–1
4
7
y
16
8
–2
–8
3. Suppose that x and y vary inversely and that y =
n+3 11. Solve the equation. Check the solution. −2 4 = x+4 x+3
1 6
12. Solve the equation. Check the solution. a 2 1 + = 2 a − 36 a − 6 a+6
4. Suppose that y varies directly with x and inversely with z, and y = 25 when x = 35 and z = 7. Write the equation that models the relationship. Then find y when x = 12 and z = 4.
13. Solve the equation. Check the solution. 5 1 + = −4 6w w
5. Sketch the asymptotes and graph the function. 2 y = −3 x+2
4 x
2y
n 2 + 11n + 24 n+1
when x = 3. Write a function that models the inverse variation and find y when x = 10.
6. Write an equation for the translation of y =
3
10. Simplify the complex fraction. n−6
2. Is the relationship between the variables in the table a direct variation, an inverse variation, or neither? If it is a direct or inverse variation, write a function to model it. x
+
14. The volume V of a cylinder varies jointly with the height h and the radius squared r2, and V = 157.00 cm3 when h = 2 cm and r 2 = 25 cm2. Find V when h = 3 cm and r 2 = 36 cm2. Round your answer to the nearest hundredth.
that
25 5x 3 ÷ 9 . Assume that all expressions 2 3x y 3y are defined.
has the asymptotes x = 7 and y = 6.
15. Divide
7. Simplify the rational expression. State any restrictions on the variable. p 2 − 4p − 32
16. Solve
p+4 8. Add or subtract. Simplify if possible. b 2 − 2b − 8 6 − b2 + b − 2 b−1
x 2 + x − 30 = 11. Check your answer. x−5
17. Solve the equation
6x 4x + 6 = . x−3 x−3
18. Simplify the rational expression. −4x 3 3 x − 2x 4
1
ID: A 19. Simplify the rational expression. x 2 + 4x − 21 x 2 + x − 42
25. Suppose that y varies jointly with w and x and inversely with z and y = 360 when w = 8, x = 25 and z = 5. Write the equation that models the relationship. Then find y when w = 4, x = 4 and z = 3.
20. Multiply. y2 − 9 −5y ⋅ −2y y+3
26. Multiply or divide. State any restrictions on the variables. x 2 + 5x + 4 x 2 − 16 ÷ x 2 + 5x + 6 x 2 − 2x − 8
21. Divide. s−5 s 2 − 2s ÷ s 2 + 3s − 10 s + 5 22. Suppose that x and y vary inversely, and x = 7 when y = 11. Write the function that models the inverse variation.
27. Compare the graphs of the inverse variations 13 −13 y = and y = x x
23. Simplify the rational expression. State any restrictions on the variable. q 2 + 11q + 24
28. Graph the function. −2 y = x
q 2 − 5q − 24
29. Add or subtract. 7 x − x 2 − 49 x 2 − 49
24. Describe the combined variation that is modeled by the formula or equation. w y = 2x 2 30. Multiply or divide. State any restrictions on the variables. x+2 x+4 ÷ 2 x − 1 x + 4x − 5 31. Add or subtract. Simplify if possible. 7 7 + 2 a + 8 a − 64
32. Find the least common multiple of x 2 + x − 12 and x 2 + 2x − 15 . 33. Add or subtract. Simplify if possible. a 2 − 2a − 3 a 2 − 5a − 6 − a 2 − 9a + 18 a 2 + 9a + 8 34. Multiply or divide. State any restrictions on the variables. 4a 5 2b 2 ⋅ 7b 4 2a 4
2
ID: A
Algebra II Chapter 9 Practice Test Answer Section 1. ANS: direct variation; y = 14x PTS: 1 DIF: L2 OBJ: 9-1.1 Using Inverse Variation KEY: rational function | direct variation 2. ANS: neither
REF: 9-1 Inverse Variation TOP: 9-1 Example 2
PTS: 1 DIF: L2 OBJ: 9-1.1 Using Inverse Variation KEY: rational function | no variation 3. ANS: 1 1 y = ; 2x 20
REF: 9-1 Inverse Variation TOP: 9-1 Example 2
PTS: 1 DIF: L2 REF: 9-1 Inverse Variation OBJ: 9-1.1 Using Inverse Variation TOP: 9-1 Example 3 KEY: rational function | inverse variation 4. ANS: 5x y = ; 15 z PTS: 1 DIF: L2 REF: 9-1 Inverse Variation OBJ: 9-1.2 Using Combined Variation TOP: 9-1 Example 5 KEY: direct variation | combined variation 5. ANS:
PTS: 1 DIF: L2 REF: 9-2 The Reciprocal Function Family OBJ: 9-2.2 Graphing Translations of Reciprocal Functions TOP: 9-2 Example 4 KEY: graphing | asymptote 1
ID: A 6. ANS:
y =
4 x−7
+6
PTS: 1 DIF: L2 REF: 9-2 The Reciprocal Function Family OBJ: 9-2.2 Graphing Translations of Reciprocal Functions TOP: 9-2 Example 5 KEY: asymptote | translation 7. ANS: p − 8; p ≠ −4 PTS: 1 DIF: L2 REF: 9-4 Rational Expressions OBJ: 9-4.1 Simplifying Rational Expressions STA: CA A2 7.0 TOP: 9-4 Example 1 KEY: rational expression | simplifying a rational expression | restrictions on a variable 8. ANS: b − 10
b−1 PTS: OBJ: TOP: KEY: 9. ANS: 1 − 2
1 DIF: L2 REF: 9-5 Adding and Subtracting Rational Expressions 9-5.1 Adding and Subtracting Rational Expressions STA: CA A2 7.0 9-5 Example 4 simplifying a rational expression | subtracting rational expressions
PTS: OBJ: TOP: KEY: 10. ANS: n
1 DIF: L2 REF: 9-5 Adding and Subtracting Rational Expressions 9-5.2 Simplifying Complex Fractions STA: CA A2 7.0 9-5 Example 5 complex fraction | simplifying a rational expression | simplifying a complex fraction
−6
(n + 1)(n + 8) PTS: 1 DIF: L2 REF: 9-5 Adding and Subtracting Rational Expressions OBJ: 9-5.2 Simplifying Complex Fractions STA: CA A2 7.0 TOP: 9-5 Example 5 KEY: dividing rational expressions | simplifying a complex fraction 11. ANS: 11 − 3 PTS: 1 DIF: L2 OBJ: 9-6.1 Solving Rational Equations KEY: rational equation
REF: 9-6 Solving Rational Equations STA: CA A2 7.0 TOP: 9-6 Example 1
2
ID: A 12. ANS: –9 PTS: 1 DIF: L2 OBJ: 9-6.1 Solving Rational Equations KEY: rational equation | no solutions 13. ANS: 11 − 24
REF: 9-6 Solving Rational Equations STA: CA A2 7.0 TOP: 9-6 Example 2
PTS: 1 DIF: L2 REF: 9-6 Solving Rational Equations OBJ: 9-6.1 Solving Rational Equations STA: CA A2 7.0 TOP: 9-6 Example 2 KEY: rational equation 14. ANS: 339.12 cm3 V varies jointly as h and r2. V = khr 2 157.00 = k(2)(25) Substitute 157.00 for V, 2 for h and 25 for r2. 3.14 = k Solve for k.
V = (3.14)hr 2 V = 3.14(3)(36) V = 339.12 PTS: 1 NAT: 12.5.4.c 15. ANS: xy 8 5
Replace k in the function. Substitute 3 for h and 36 for r2.
DIF: Average REF: Page 570 TOP: 8-1 Variation Functions
OBJ: 8-1.3 Solving Joint Variation Problems KEY: joint variation | volume
5x 3 25 ÷ 3x 2 y 3y 9
Rewrite as multiplication by the reciprocal.
9 5x 3 3y = 2 ⋅ 3x y 25
Simplify by canceling common factors.
xy 8 = 5 PTS: 1 DIF: Basic REF: Page 579 OBJ: 8-2.4 Dividing Rational Expressions NAT: 12.5.3.c STA: 2A7.0 TOP: 8-2 Multiplying and Dividing Rational Expressions KEY: divide rational expressions | simplify
3
ID: A 16. ANS: There is no solution because the original equation is undefined at x = 5 . x 2 + x − 30 Note that x ≠ 5. = 11 x−5 (x − 5 ) (x + 6 ) Factor. = 11 x−5 x + 6 = 11 The factor (x − 5 ) cancels. x=5 Because the left side of the original equation is undefined when x = 5 , there is no solution. PTS: 1 DIF: Average NAT: 12.5.3.c STA: 2A7.0 KEY: solving rational equations 17. ANS: There is no solution. 6x 4x + 6 (x − 3) = (x − 3) x−3 x−3 6x = 4x + 6 2x = 6 x=3
REF: Page 579 OBJ: 8-2.5 Solving Simple Rational Equations TOP: 8-2 Multiplying and Dividing Rational Expressions
Multiply each term by the LCD, (x – 3). Simplify. Note that x g 3. Solve for x.
The solution x = 3 is extraneous because it makes the denominators of the original equation equal to 0. Therefore the equation has no solution. PTS: 1 DIF: Average NAT: 12.5.4.a STA: 2A7.0 KEY: solving rational equations 18. ANS: −4 1 − 2x
REF: Page 601 OBJ: 8-5.2 Extraneous Solutions TOP: 8-5 Solving Rational Equations and Inequalities
PTS: 1 DIF: L2 REF: 11-1 Simplifying Rational Functions OBJ: 11-1.1 Simplifying Rational Expressions STA: CA A1 12.0 TOP: 11-1 Example 2 KEY: rational expression 19. ANS: x−3 x−6 PTS: 1 DIF: L2 REF: 11-1 Simplifying Rational Functions OBJ: 11-1.1 Simplifying Rational Expressions STA: CA A1 12.0 TOP: 11-1 Example 2 KEY: rational expression 20. ANS: −5(y − 3) −2 PTS: 1 DIF: L3 REF: 11-2 Multiplying and Dividing Rational Expressions OBJ: 11-2.1 Multiplying Rational Expressions STA: CA A1 2.0 | CA A1 13.0 TOP: 11-2 Example 2 KEY: rational expression 4
ID: A 21. ANS: s s−5 PTS: 1 DIF: L2 REF: 11-2 Multiplying and Dividing Rational Expressions OBJ: 11-2.2 Dividing Rational Expressions STA: CA A1 2.0 | CA A1 13.0 TOP: 11-2 Example 4 KEY: rational expression 22. ANS: 77 y = x PTS: 1 DIF: L2 REF: 9-1 Inverse Variation OBJ: 9-1.1 Using Inverse Variation TOP: 9-1 Example 1 KEY: rational function | inverse variation 23. ANS: q+8 ; q ≠ −3, q ≠ 8 q−8 PTS: 1 DIF: L2 REF: 9-4 Rational Expressions OBJ: 9-4.1 Simplifying Rational Expressions STA: CA A2 7.0 TOP: 9-4 Example 1 KEY: rational expression | simplifying a rational expression | restrictions on a variable 24. ANS: y varies directly as w and inversely as the square of x. PTS: 1 DIF: L3 REF: 9-1 Inverse Variation OBJ: 9-1.2 Using Combined Variation TOP: 9-1 Example 4 KEY: direct variation | inverse variation | combined variation 25. ANS: 9wx y = ; 48 z PTS: 1 DIF: L3 REF: 9-1 Inverse Variation OBJ: 9-1.2 Using Combined Variation TOP: 9-1 Example 5 KEY: direct variation | combined variation | joint variation 26. ANS: (x − 4) 2 ; x ≠ − 4, − 3, − 2, − 1, 4 (x + 3)(x + 1) PTS: 1 DIF: L3 REF: 9-4 Rational Expressions OBJ: 9-4.2 Multiplying and Dividing Rational Expressions STA: CA A2 7.0 TOP: 9-4 Example 4 KEY: restrictions on a variable | dividing rational expressions
5
ID: A 27. ANS: Answers may vary. Sample: The axes are asymptotes for both graphs. Both graphs are symmetric with respect to y = x and y = –x. The x- and y-axes are lines of reflection. PTS: 1 DIF: L3 REF: 9-2 The Reciprocal Function Family OBJ: 9-2.1 Graphing Reciprocal Functions TOP: 9-2 Example 2 KEY: graphing | inverse variation 28. ANS:
PTS: 1 DIF: L2 REF: 9-2 The Reciprocal Function Family OBJ: 9-2.1 Graphing Reciprocal Functions TOP: 9-2 Example 1 KEY: graphing | inverse variation 29. ANS: 1 x+7 PTS: 1 DIF: L3 REF: 11-4 Adding and Subtracting Rational Expressions OBJ: 11-4.1 Rational Expressions With Like Denominators STA: CA A1 13.0 TOP: 11-4 Example 2 KEY: rational expression 30. ANS: (x + 2)(x + 5) , x ≠ 1, − 4 x+4 PTS: OBJ: TOP: 31. ANS: 7a
1 DIF: L2 REF: 9-4 Rational Expressions 9-4.2 Multiplying and Dividing Rational Expressions STA: CA A2 7.0 9-4 Example 4 KEY: restrictions on a variable | dividing rational expressions
− 49
(a − 8)(a + 8) PTS: 1 DIF: L2 REF: 9-5 Adding and Subtracting Rational Expressions OBJ: 9-5.1 Adding and Subtracting Rational Expressions STA: CA A2 7.0 TOP: 9-5 Example 3 KEY: simplifying a rational expression | adding rational expressions
6
ID: A 32. ANS: (x + 4)(x − 3)(x + 5) PTS: 1 DIF: L2 REF: 9-5 Adding and Subtracting Rational Expressions OBJ: 9-5.1 Adding and Subtracting Rational Expressions STA: CA A2 7.0 TOP: 9-5 Example 2 KEY: least common multiple 33. ANS: 21a − 28
(a − 6)(a + 8) PTS: OBJ: TOP: KEY: 34. ANS: 4a , 7b 2 PTS: OBJ: TOP: KEY:
1 DIF: L3 REF: 9-5 Adding and Subtracting Rational Expressions 9-5.1 Adding and Subtracting Rational Expressions STA: CA A2 7.0 9-5 Example 4 simplifying a rational expression | subtracting rational expressions
a ≠ 0, b ≠ 0 1 DIF: L2 REF: 9-4 Rational Expressions 9-4.2 Multiplying and Dividing Rational Expressions STA: CA A2 7.0 9-4 Example 3 simplifying a rational expression | restrictions on a variable | multiplying rational expressions
7