Chapter Algebra Review Performing well in calculus is impossible without a solid algebra foundation. Many calculus problems that you encounter involve a calculus concept but then require many, many steps of algebraic simplification. Having a strong algebra background will allow you to focus on the calculus concepts and not get lost in the mechanical manipulation that s required to solve the problem. The Problems You ll Work On In this chapter, you see a variety of algebra problems: Simplifying eponents and radicals Finding the inverse of a function Understanding and transforming graphs of common functions Finding the domain and range of a function using a graph Combining and simplifying polynomial epressions What to Watch Out For Don t let common mistakes trip you up. Some of the following suggestions may be helpful: COPYRIGHTED MATERIAL Be careful when using properties of eponents. For eample, when multiplying like bases, you add the eponents, and when dividing like bases, you subtract the eponents. Factor thoroughly in order to simplify epressions. Check your solutions for equations and inequalities if you re unsure of your answer. Some solutions may be etraneous! It s easy to forget some algebra techniques, so don t worry if you don t remember everything! Review, review, review.
8 Part I: The Questions Simplifying Fractions Simplify the given fractions by adding, subtracting, multiplying, and/or dividing. 5 9. y z 8 y z. 5 6 0. y 5 y 5 6 7. 5 7 0. y z 8 y z 00 5 8 0.. y z yz yz. 5 6 5. y y z z y 0 5 5. 5 7 0 y y Simplifying Radicals 8 Simplify the given radicals. Assume all variables are positive. 6.. 50 7. y y 5. 8 0 50 8. 5 6 6y 0 0 y 6. 0 y z 5y z 6 7
Chapter : Algebra Review 9 8 5 7. y z y z 8. 7 0 8 y 6 5 0 y 7 y Writing Eponents Using Radical Notation 9 0 Convert between eponential and radical notation. 8 5 9. Convert y z to radical notation. ( Note: The final answer can have more than one radical sign.). Use the horizontal line test to determine which of the following functions is a oneto-one function and therefore has an inverse. (A) y = (B) y =, 0 (C) y =, 8 (D) y =, 6 (E) y =, 5. 0.. Use the horizontal line test to determine which of the following functions is a oneto-one function and therefore has an inverse. (A) y = + 7 (B) y (C) (D) (E) y = cos y = sin y = tan 0. Convert y 5 z to eponential notation. The Horizontal Line Test Use the horizontal line test to identify one-toone functions.. Use the horizontal line test to determine which of the following functions is a oneto-one function and therefore has an inverse. (A) y = + + 6 (B) y (C) y (D) y = + 8 (E) y 5 Find Inverses Algebraically 9 Find the inverse of the one-to-one function algebraically.. f ( ) = 5 5. f ( ) =, 6. f( ) 8 5
0 Part I: The Questions 7. f ( ) = 5 + 7 Linear Equations 7 Solve the given linear equation. 8. f ( ). + 7 = 9. f ( ). ( + ) = ( + ) The Domain and Range of a Function and Its Inverse 0 Solve the given question related to a function and its inverse. 0. The set of points {(0, ), (, ), (5, 6)} is on the graph of f ( ), which is a one-to-one function. Which points belong to the graph of f ( )? 5. ( + ) = 7 + ( 8) 6. 5 5 0 7. 5 0. f ( ) is a one-to-one function with domain [, ) and range (, ). What are the domain and range of f ( )? Quadratic Equations 8 Solve the quadratic equation. 8. Solve = 0.. Suppose that f ( ) is a one-to-one function. What is an epression for the inverse of g ( ) = f ( + c )? 9. Solve + 8 7 = 0 by completing the square.
Chapter : Algebra Review 0. Solve + = 0 by completing the square. Absolute Value Equations 8 5 Solve the given absolute value equation. 8. 5 7. Solve 6 + 5 = 0. 9. 5 8. Solve + = 0. 50. 6 7. Solve 0 + 7 5 + 0 = 0. Solving Polynomial Equations by Factoring 7 Solve the polynomial equation by factoring.. + 5 = 0 5. 5 5 5 5 Solving Rational Equations 5 55 Solve the given rational equation. 5. 0 5. 8 + + 5 = 0 5. 6. + = 0 5. 5 0 7. 8 = 0 55. 5 6
Part I: The Questions Polynomial and Rational Inequalities 56 59 Solve the given polynomial or rational inequality. 56. < 0 Graphing Common Functions 6 77 Solve the given question related to graphing common functions. 6. What is the slope of the line that goes through the points (, ) and (5, 9)? 57. + 6. What is the equation of the line that has a slope of and goes through the point (0, 5)? 58. 0 65. What is the equation of the line that goes through the points (, ) and (, 8)? 59. Absolute Value Inequalities 60 6 Solve the absolute value inequality. 60. 66. Find the equation of the line that goes through the point (, 5) and is parallel to the line y 8. 67. Find the equation of the line that goes through the point (, ) and is perpendicular to the line that goes through the points (, ) and ( 6, ). 6. 5 7 6. 5 68. What is the equation of the graph of y after you stretch it vertically by a factor of, shift the graph units to the right, and then shift it units upward?
Chapter : Algebra Review 69. Find the verte form of the parabola that passes through the point (0, ) and has a verte at (, ). 75. Find the equation of the fourth-degree polynomial that goes through the point (, ) and has the roots,, and, where is a repeated root. 70. Find the verte form of the parabola that passes through the point (, ) and has a verte at (, 6). 76. A parabola crosses the -ais at the points (, 0) and (6, 0). If the point (0, 8) is on the parabola, what is the equation of the parabola? 7. A parabola has the verte form y = ( + ) +. What is the verte form of this parabola if it s shifted 6 units to the right and units down? 77. A parabola crosses the -ais at the points ( 8, 0) and (, 0), and the point (, ) is on the parabola. What is the equation of the parabola? 7. What is the equation of the graph of y = e after you compress the graph horizontally by a factor of, reflect it across the y -ais, and shift it down 5 units? 7. What is the equation of the graph of y after you stretch the graph horizontally by a factor of 5, reflect it across the -ais, and shift it up units? Domain and Range from a Graph 78 80 Find the domain and range of the function with the given graph. 78. 7. Find the equation of the third-degree polynomial that goes through the points (, 0), (, 0), (0, ), and (, 0).
Part I: The Questions 79. Adding Polynomials 8 87 Add the given polynomials. 8. (5 + 6) + ( + 6) 8. ( + 7) + ( + 9) 80. 85. ( 5 + 6) + ( + + 8) 86. ( + + ) + ( + 6) 87. ( 6 + ) + (5 + ) End Behavior of Polynomials 8 8 Find the end behavior of the given polynomial. That is, find lim f ( ) and lim f( ). 8. f () = 6 0 5 + Subtracting Polynomials 88 9 Subtract the given polynomials. 88. (5 ) ( + ) 89. ( + ) ( 5 + ) 8. f () = 7 9 + 8 5 7 + 9 90. (8 + 5 + ) ( + 5 )
Chapter : Algebra Review 5 9. ( + ) ( + ) ( 5 + 6) 9. (0 6 + + 6) ( + 0 + 8 ) Long Division of Polynomials 98 0 Use polynomial long division to divide. 6 98. Multiplying Polynomials 9 97 Multiply the given polynomials. 99. 8 9. 5 ( ) 6 00. 9. ( + )( 5) 0. 5 5 95. ( y + 6)( y ) 6 5 0. 96. ( )( + ) 97. ( + + )( + )
6 Part I: The Questions