Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter)

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1 Name: Per.: Date: Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter) Your Algebra 1 Final will be on at. You will need to bring your textbook and number 2 pencils with you to the final exam. The final exam will cover the entire year. Re-review the material from the fall semester as well. Do not lose this packet. Replacement packets will cost $$$. 1

2 What You Must Memorize For Final 1) Quadratic Formula: x = b ± b2 4ac 2a 2) Standard Form: ax 2 + bx + c = 0 3) Perfect Squares from 0 to 169: 0, 1, 4, 16, 4) Discriminant: b 2-4ac ) Complete the Square: x 2 x + 6) Direct Variation: y=kx ) Inverse Variation: 8) Vertex: 9) Pythagorean Theorem: a 2 + b 2 = c 2 10) X and Y Intercepts: To find y-intercept, set x s equal to zero. To find, x-intercepts, set y equal to zero and solve for x. 2

3 Chapter *** show your work wherever applicable for full credit *** Match each of the following polynomials with its special term name. 1. 4x x 8 2. x 3 y x + x 3 a) trinomial b) binomial c) monomial Add or subtract the following polynomials. 4. (8t 2 10t + 2) + (8t + 13). (x 2 + x 1) (7x 2 + 2) 6. (x 4 + 7x 3 + 7) (2x 4 4x 3 + 1) 7. (3n 3 + n 2 n 4) + (n 3 4n ) Write in decimal form (standard notation) Write in scientific notation ,

4 Simplify the following. Leave all answers with positive exponents. 16. x 2 x (2w 2 x 4 ) (a 4 ) 12 28x y (3b) 3 14 x x x y d d 26. df f 21. 6x 8 y t 4 v 3 21t 2 v 6 Multiply or Divide. Express your answers in scientific notation. 28. ( )( )

5 Multiply x(4x 9) 31. 4x 2 (x + 6) 32. 2x 2 (1x 3 10) Multiply the following. Use your choice of methods, but show your work!! 33. (x + 9)(x 6) 36. (x 6)(x 8) 34. (x + 3)(4x + ) 37. (7x + 3)(7x 2) 3. (3x 1)(8x + 1) 38. (3x 3 2x 2 + 6)(x + )

6 Chapter 6 Factor out the largest possible monomial. 39. x a + 10b c 4 6c 2 1c Factor completely (remember they are not always ready to go ). 42. x 2 + 9x y 2 1y t 2 + 8t m m x 2 x x 2 + xy 42y 2 Factor the following differences of squares completely. 48. x a m

7 Factor the following perfect square trinomials completely. 1. x x m m y 3 16y y Factor completely (remember they are not always ready to go ). 4. 3y 2 20y x 2 13x n n 2 6n Solve for the given variable. 7. (a )(a + 2) = 0 8. x(x 3) = 0 9. y y 24 = 0 7

8 Chapter 10 Multiply or Divide x 2 x x 2 x 1 Add or Subtract (x 7) (8x 12) 67. (12x + 4) (4x + 3) Simplify Completely. 12x 4 y x 7 y x + 9 3x 8

9 70. 3a + 9b 12a y 2 + 3y 3y 2 + 6y a 2 14b 2 21a 21b 73. b 2 10b + 21 b 2 11b + 28 Multiply x 4x x 2 x 2x 76. 4x 2x + 2 4x + 4 8x 77. m 2 4 m 4m 2 m + 2 9

10 Divide. 2x x x 3 x 2x3 y 6x x x x 6 6x 9 2 Add or Subtract. 4x x x + 3 x x + 7 x w 2 + w w 3 9 w 3 8. (4x + 2) (7x 6) (3x + 1) (3x + 1) 10

11 3x 2 + 2x x2 x + 6 x + 1 x x x 88. m m 1 3 m(m 1) 89. 2u uv 3 v u 2 v a a a x 4 x + 4 x 2 x x x 2 + 4x b b 6 11

12 Chapter Estimate which two integers each square root is between: a) 73 is between and b) 13 is between and 9. Simplify (show work): a) 180 b) 22 c) x 2 y 3 d) 196b 96. Simplify to find (show work): a) 81 2 b) c) d) e) 3 f) ( + ) 97. Find the missing side length for each triangle (show your steps): a) b) u v 12

13 98. Solve (show work): a) 4x + 7 = 1 b) x = To hang a math poster in his office, Mr. Zito leaned a 10 foot ladder against the wall, placing the bottom of the ladder on the floor 3 feet away from the wall. How high up on the wall was the ladder? (Show work. A labeled picture is also required) Chapter Write the quadratic formula. Solve the following using the QUADRATIC FORMULA. Complete all blanks x = 8x (standard form) a = substitution b = c = solution(s) for x: 13

14 102. 6x + = -2x 2 (standard form) a = substitution b = c = solution(s) for x: 103. Write the formula for the discriminant For questions 7, the blanks provided are for the following information: a) Substitute values for a, b, & c into the discriminant formula. b) Find the discriminant. c) Tell how many solutions the quadratic has x = 3x a) d = (substitution) b) d = (simplified) c) (# of solutions) 10. 4x + 2 = 3x 2 a) d = (substitution) b) d = (simplified) c) (# of solutions) 106. x 2 = 8x 16 a) d = (substitution) b) d = (simplified) c) (# of solutions) 14

15 107. Solve each of the following using the zero-product property. SHOW WORK. a) (x + 10)(x + 24) = 0 b) (2x 6)(x + 2) = 0 c) x(x + 18) = 0 d) x(4x 16)(x + 9) = Solve each of the following equations by factoring: SHOW WORK. a) x 2 8x + 16 = 0 b) x 2 + 4x 21 = Complete the square for the following: a) x x b) u 2 26u 110. Solve using any method you want show work!! a) x 2 49 = 0 b) x x = 4 1

16 Chapter 12 Identify the domain and range for each. Tell if the relation is a function h: {(3, 1), (2, 4), (3, ), (4, 8)} 112. j: {(2, 7), (3, 6), (4, ), (, 4), (6, 3)} a) domain: a) domain: b) range: b) range: c) function / not a function c) function / not a function 113. k: {(1, 2), (2, 3), (3, 2), (4, 1)} 114. m: {(4, ), (4, 2), (4, 1), (4, 3), (1, 6)} a) domain: a) domain: b) range: b) range: c) function / not a function c) function / not a function Find the indicated outputs for the following functions. 11. f(x) = 4x g(x) = x f(1) = g( 2) = f( 3) = g( 1) = f(0) = g(4) = 16

17 Determine which of the following graphs represent functions y y y x x x function / not a function function / not a function function / not a function State the domain and range for the following graphs y 121. y 122. y x x x Domain: Domain: Domain: Range: Range: Range: Graph the following functions f(x) = x 124. g(x) = x h(x) = 2 3 x + 2 y y y x x x 17

18 Graph each of the following quadratic functions, finding all indicated information f(x) = x 2 b Vertex 2a y-intercept x-intercept(s) 127. f(x) = x 2 b 2a Vertex y-intercept x-intercept(s) 128. f(x) = x 2 4x 12 b 2a Vertex y-intercept x-intercept(s) 18

19 129. Find an equation of variation where y varies directly as x for each pair of values given. a) y = 3 when x = 24 b) y = 0 when x = Find an equation of variation where y varies inversely as x for each pair of values given. a) y = 2 3 when x = 27 b) y = 4 when x = 8 19

20 131. Find the Vertex: f(x) = 4x 2 + 8x Write the equation in Standard Form and then identify a, b, and c. 3x 2 = -2x Solve using the quadratic formula: 3x 2-7x +4 = Find the x and y intercepts: f(x) = x 2 3x y 136. What is the Domain and the Range of a relation defined by: {(,2), (,3), (6,2), (3,1)} x Function? Yes or No Domain: Domain: Range: Range: 137. Find the indicated outputs for the function: f(x) = 3x 2 1 for f(-2) 138. Find the equation of variation where y varies directly as x, and y = 14 and x = 2. 20

21 139. Find the equation of variation where y varies inversely with x, and y = 14 and x = x 1 = ( x + 3) 2 = What number should be added to complete the square? X 2 + 6x = 146. ( 2 ) 147. Find the length of side a. a

22 ANSWER KEY 17 a b or w 6 x x 2 y 9 or 28y 9 x 2 24 X x x x 2 27x 31 4x 3 +24x x 20x 2 33 x 2 +3x 4 Page x a 3 24x 2 +x 1 2 c 36 X 2 14x=48 3 b 37 4x 2 +7x 6 4 8t 2 2t x 4 +13x 3 10x 2 +6x+30 6x 2 +x 3 39 (x 2 3) 6 x 4 +11x (4a+b 8) 7 8n 3 3n 2 n c(c 3 2c ) 8 80, (x+2)(x+7) 9 982, (y 6)(8 9) (t+3)(t+) (m 1)(m+24) 12 4 x (x+3)(x 4) x (x 6y)(x+7y) 14.6 x (x 11)(x+11) x (10a 12)(10a+12) 16 X 9 0 (m 2)(m+2) 22

23 ANSWER KEY 1 (x+9) (m+3) y(y 2) (3y 2)(y 6) 84 (2x+)(x 9) 8 6 3n(6n 1)(n+2) 86 7 a= ; a= X=0 ; x= Y=1 ; y= a) 8 and 9 b) 3 and 4 a) 6 ; b) 6 7 ; c) xy ; d) x+ 96 a) 4 ; b) 10 ; c) 11 7 ; d) 6 6 ; e) a) b) 29 or a) 16 b) or x = 2 & x = 72 2 NO SOLUTION; can t have a negative inside radical b) 39 ; c) zero solutions b) 40 ; c) 2 solutions b) 0 ; c) 1 solution a) 10 & 24 ; b) 3 & ; c) 0 & d) 0 & 4 & a) 4 ; b) 7 & a) +100 ; b) a) 7 & 7 ; b) 21 23

24 ANSWER KEY 111 a) {2,3,4} ; b) {1,4,,8} ; c) not a function 112 a) {2,3,4,,6} ; b) {3,4,,6,7} ; c) function 113 a) {1,2,3,4} ; b) {1,2,3} ; c) function 114 a) {1,4} ; b) {1,2,3,,6} ; c) not a function 11 6 ; 38 ; ; 2 ; 117 Not a function 118 Not a function 119 function 120 Domain: {x x 2} ; Range: {y all real numbers} 121 Domain: {x 1 x } ; Range: {y 1 y 4} 122 Domain: all real numbers ; Range: all real numbers 123 Graph should form a v: 124 Graph should form a v: 12 Graph should be a negative sloped line: ; vertex (0,0) ; y intercept (0,0) ; x intercepts (0,0) ; vertex (0, ) ; y intercept (0, ) ; x intercepts =,0 &, ; vertex (2, 16) ; y intercept (0, 12) ; x intercepts = 6 & 2 or (6,0) & ( 2,0) 129 a) 130 a) 131 ( 1, 3) ; b) y=2x ; b) 3x 2 + 2x 1 = 0 a=3 ; b=2 ; c= 1 & & ; or ( 2, 0) & (,0) 13 Yes ; all real numbers; {y y {3,,6} ; {1,2,3} y=7x & 7 24

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