Algebra Final Exam Topics June 0 The following is a list of topics covered in our Algebra CPA class this year. During your review, you should look over all packets and worksheets you have received for all sections covered and spend time re-familiarizing yourself with the types of exercises we have encountered in each section. Suggested Practice is just that: suggested. The decision of how extensively you wish to review the topic list is solely yours. Luck is what happens when preparation meets opportunity. Seneca (mid- st century A.D. philosopher) Algebra classes will take the final exam on Monday, June 6 th, 0 from :05pm - :05pm Section Topic Suggested Practice Chapter. Expressions and Formulas Prerequisite Skills. Properties of Real Numbers Prerequisite Skills Names of Properties will not appear on final exam. Solving Equations pp. -7, #9-6, -6 odd, 67, 69, 77. Absolute Value Equations pp. 0-, #5-5, 7, 5, 5.5 Solving Inequalities pp. 7-8, #5-7 odd, -6 all, 5, 55.6 Solving Compound and Abs Val Inequalities pp. -5, #7- odd Chapter. Relations and Functions pp. 60-6, #7- all, - odd. Linear Equations pp. 66-67, #5-7 odd,, 5, 9, 6, 6. Slope pp. 7-7, #5-7 odd, -50 all. Writing Linear Equations pp. 78-79, #-9 odd, 5, 5.5 Scatter Plots Not on final exam.6 Special Functions pp. 9-9, #0- all, #.7 Graphing Inequalities pp. 98-99, #-5 odd, # Chapter 7 (portion ) 7.7 Compositions of Functions pp. 87-88, #7- odd, 5-5 odd 7.8 Inverses Functions pp. 9-9, #5-, 5-7 odd, 6 Chapter /. Solving Systems Graphically p., #-5 odd. Solving Systems Algebraically pp. 0-, #-7 odd. Solving Systems of Inequalities p. 6, -9 odd, #.5 Solving Systems in Variables pp. -, #-7.5 Determinants (x, x) p. 86, #5-9 odd.6 Solving Systems by Cramer s Rule p. 9, #, 5, 7 (x, x) Chapter 5
5. Monomials pp. 6-7, #-, 8-5. Polynomials pp. -, #-, 7-50 5. Dividing Polynomials pp. 6-7, #5- odd,, 5-9 odd 5. Factoring Polynomials pp. -, #5- odd, 6-5, 57, 58 5.5 Roots of Real Numbers pp. 8-9, #7- odd, 9-55 odd 5.6 Radical Expressions pp. 5-55, #5-, 5-5 odd, 57 5.7 Rational Exponents pp. 6-6, #8-9, -6 odd 5.8 Radical Equations & Inequalities pp. 66-67, #- odd 5.9 Complex Numbers pp. 7-75, #9- odd, 9-55 odd Chapter 9 9. Multiplying/Dividing Rational Expressions pp. 76-77, #5- odd 9. Adding/Subtracting Rational Expressions pp. 8-8, #5-9 odd 9.6 Solving Rational Equations and Inequalities p. 50, #- odd,, Chapter 6 6. Graphing a quadratic Function p. 9, #5-7 6. Solving by Graphing pp. 98-99, #-7 odd, -, 50 6. Solving by Factoring p. 0, #- all 6. Solving by Completing the Square pp. 0-, #-8 all, 5- odd 6.5 Solving by the Quadratic Formula pp. 8, #5-5 odd, 9-9 odd 6.6 Vertex Form of a Quadratic p. 6, #5- odd 6.7 Graphing Quadratic Inequalities Not on final exam Chapter 8 8. Midpoint and Distance pp. -5, #,, 7, 9, 5- odd 8. Circles pp. 9-0, #8-7, - odd 8. Ellipses pp. 8-9, #- odd, 7, 9, 8. Parabolas pp. -, #-7,, 6, 7,,, 6-9, 8, 9 8.5 Hyperbolas pp. 5-6, #-6, - odd 8.6 Conic Sections p. 5, #-9, -8 Chapter 7 7. Polynomial Functions pp. 50-5, #5- odd, - all,, 0-, 9- all 7. Solving Equations Using Quadratic Techniques p. 6, #7-0 all 7. The Remainder and Factor Theorems pp. 68-69, #- odd 7.5 Roots and Zeros pp. 75-77, #-9 odd, 5, 5 7.6 Rational Zero Theorem pp. 80-8, #5-9 odd, -9 odd Chapter. The Counting Principal pp. 65, #- all. Permutations and Combinations pp. 6, #- all Note: All topics may or may not appear on the final. Only two sections above have been explicitly crossed out, meaning they are definitely not on the final exam.
Algebra - Review for 0 Final Exam This is just a selection of problems we are reviewing in class. The final exam is on Tuesday, June 7, 0, :05 P.M. :05 P.M. Section 9. Multiply and divide Rational Expressions pg. 76-77. Simplify ab c c 8a bc 7a 9. Simplify w w 5. Simplify a a 5 a a ( a 5) ( a)( a) Section 9. Adding and Subtracting Rational Expressions p. 8-8: 7, 5,, 7 7. Find the LCM of 9 p q,6 pq,p 6pq Simplify each expression x x5y 5. 5 y y. a a a a a x 7. x 5x 6 x x x 6 ( x)( x)
Section 9.6 Solving Rational Equations and Inequalities p. 50:, 7 Solve each equation 6. s 5 s {-6, } 7. 9 t t t Section 5.5 Roots of Real Numbers p. 8:, 5,, 7, 5 Simplify 7. 8-5. 5 5. 6a 8 8a 7. 6x z 6x z 5. 7c 9 d c d
Section 5.6 Radical expressions p. 5: 7, 9,,, 5 Simplify 7. 7 8 08 7 9. ( 5 6)( 5 ) 5 5 5 6. ( ). 7 8 7 5.
Section 5.7 Rational Exponents p. 6:, 7, 55, 59 Simplify each expression b. ( ) 5 b 5 7. t t t 55. 7 7 6 7 7 59. xy z xy z z
Section 5.9 Complex Numbers p. 7: 9,, 7,, 5, 9,, 9, 5 Simplify each expression. 9. 8 9i. 00a b 0a bi. 6-5. i ( 5i ) 75i 7. i. ( 7 i) ( i ) 5i 5. ( i)( i ) 6 7i 9. 5 i 0 7 6 i 7. i i i 5 5 9. x 6 0 i 5. x 80 0 i 0
Section 6. Graphing Quadratic Functions p. 9. Find the y-intercept, equation of the axis of symmetry, the vertex and minimum for f ( x) x 9x 9 y-int: 9 equation of axis of symmetry: x =.5 Vertex: (.5, -.5) Minimum: -.5 Determine whether each function has a maximum or minimum value, then find the maximum or minimum. 5. f ( x) x 6x Min.; -. f ( x) 0x 5x 9 Min.; -
Section 6. Solving Quadratic Equations by Factoring p. 0 5. Solve x x8 0 (-, 7) 9. Solve x x (-, 7). Solve x 6x9 0 (-, ) 5. Find quadratic equation with roots (-, 7) x 5x 0 Section 6. Completing the Square p. 5. Given x 8x c, find value of c that completes the square, then write it as a perfect square 8, ( x 9). Solve x x0 0 (-, 0) 5. Solve x x 0 { }
Section 6.5 Quadratic Formula and Discriminant p. 8 Find (a) discriminant, (b) describe number and types of roots, (c) solve 5. x 6x 0 a) 0 b) irrational c) 8 5 9. x 5x 0 a) 9 b) rational c) {, }. Solve x 8 6 x 9/ 5. 5 6 x 0x 7 Section 6.6 Analyzing Graphs of Quadratic Functions p. 6 a) Write the equation in vertex form. Identify (b) the vertex, (c) the axis of symmetry and (d) direction of opening. 5.. y ( x ) a) y x x 8 a) y ( x ) b) (-, 0) c) x = - d) down y b) (-, -7) c) x = - d) up ( x ) 7
Section 7. Polynomial Functions p. 5 9. Find the degree and leading coefficient of 6x x x 8 Degree: Leading coeff: 6 For each graph, find (a) the end behavior, (b) whether odd or even degree, (c) number of real zeros. 9. a) As x then f ( x), as x then f ( x) b) odd c). a) As x then f ( x), as x then f ( x) b) even c) 0 6 Section 7. Solving Equations Using Quadratic Techniques p.. Write x 6x 0 in quadratic form ( x ) 6( x ) 0 7. Solve m 7m m 0 0, -, -. Solve d d, -,. Solve x 79 0-9, 9 9 i
These problems: see the similarity and the difference! Section 7.5 Roots and Zeros p. 75 7. Solve x 6 0and state the number and type of roots, i real, imaginary. For h( x) x 6x 8x 5, state the possible number of positive, negative and imaginary zeros. (Descartes chart problem) Answer Positive Negative Imaginary Total 0 0 0 5. Find all zeros for g( x) x 6x x 6 -, i 7. Find the polynomial of the least degree with roots of i,, -. f x x x ( ) 7 Section 7.6 Rational Zero Theorem p. 8 7. List all possible rational zeros of 6 h( x) 9x 5x 7,,,, 9, 7 9. Find all rational zeros of g( x) x x 5x 9x 0, 9. Find all zeros of 5 p( x) x x x x x 0 -, -, 5, i
Section 8. Midpoint and Distance Formula p. Find the midpoint of:. (-5, ) and (-, -7) (-, -). (5, 9) and (, 8) 7 7, Find the distance from: 5. (, -) to (-6, 0) 5 7. (9, -) to (, -) 7 Section 8. Parabolas p.. Write x y y 0 in vertex form. x ( y 7) 9 Find (a) the equation in standard form, (b) the vertex, (c) the focus, equations of (d) axis of symmetry and (e) directrix, and (f) the direction of opening. 7. y x a) x y b) (0, 0) c) (/, 0) d) y = 0 e) x f) right
. ( y 8) ( x ) a) ( 8) y x b) (, 8) c) (, 8) d) y = 8 e) x 5 f) left 9. 5 a) x ( y 8) b) (, -8) c) (, 8) x y y d) y = -8 e) x f) left 7. Write an equation in vertex form given vertex = (8, 6) and focus (, 6) ( 6) y 8 x Section 8. Circles p. 9 9. Write an equation in standard circle form given: center (-8, 7), radius = / ( x8) ( y7) 5. Find the center and radius of ( x ) ( y 7) 50 Center (, -7) Radius 5. Find the center and radius of x y x y 8 8 5 0 Center (9, 9) Radius 09
Section 8. Ellipses p. 8. Write a standard ellipse equation for the following graph x y 6 7 7. Write a standard ellipse equation given endpoints of the major axis (, ) and (, -), endpoints of the minor axis (, ) and (0, ). ( y) ( x) 6 Find the (a) center, (b) foci, (c) length of major axis, and (d) length of minor axis for: 9. ( x8) ( y) a) (-8, ) 8 b) ( 8 7,) c) d) 8. x 9y 7 a) (0, 0) b) ( 6,0) c) 6 d)
Section 8.5 Hyperbolas p. 5 Write a standard hyperbola equation for the following graph.. x y 5. Write a standard hyperbola equation given vertices of (-5, 0) and (5, 0) and conjugate axis =. x y 5 6 Find the a) center, b) vertices, c) foci, and d) slopes of asymptotes of : 9. ( y) ( x) a) (-, ) 6 9 b) (-, 0), (-, 8) c) (-, -), (-, 9) d) Section 8.6 Conic Sections p. 5 Given the following equations, a) what kind of conic section is it and b) determine the standard equation. 5. y x 0 a) hyperbola x y b). x 8y y 0 a) circle b) x ( y) 5
Chapter. Problems p.65 Solve each problem 7. The menu for a banquet has a choice of types of salad, 5 main courses and desserts. How many ways can one of each be selected to form a meal? 0. Abby is registering at a Web site. She must select a password containing 6 numerals. How many passwords are allowed if no digit may be used more than once? 5,00 Chapter. Problems p.6 Determine whether the problem is a permutation or combination. Then, find the number of possibilities.. The winner, and the first, second and third runners-up in a contest with 0 finalists. Permutation 500. Selection of 8 employees to attend a business seminar Combination 8 7. An arrangement of the letters in the word parallel Permutation 60