[Type the document subtitle] Math 0310

Transcription

1 [Typethe document subtitle] Math 010

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3 [Typethe document subtitle] Cartesian Coordinate System, Domain and Range, Function Notation, Lines, Linear Inequalities

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5 Notes-- Cartesian Coordinate System Page 5 We use a rectangular coordinate system to help us map out relations. The coordinate grid has a horizontal axis and a vertical axis. Where these two axes intersect is called the origin. The grid is also divided into 4 quadrants. Traditionally, we use Roman numerals to label these 4 quadrants. Let s begin by plotting some ordered pairs. A=(1, 4) C=(-,5) E=(-,-4) G=(,-1) B=(0,) D=(-4,0) F=(0,-) Find the ordered pair associated with the given points.

6 Page 6 We also use the coordinate system to graph solutions of equations in two variables. One of the most common equations we graph is linear equations. LINES There are several methods to graph lines: plot points by creating a table of values, plot the intercepts, use the slope and y-intercept. How do you know if the graph of an equation is a line? y x 5 y 4x 11 y x 1 x 9 0 y 5 y x 5 Make a Table of Values to graph the following: x y 6 5xy 0 What happens when we do NOT have a linear equation? What happens when x is squared and y is not?

7 Make a Table of Values to graph the following: yx y x Page 7 What happens when y is squared and x is not? Make a Table of Values to graph the following: x y x1 y

8 Page 8 What happens when x is under a square root sign? Make a Table of Values to graph the following: y x y x

9 Page 9 Cartesian Coordinate System 1. Find the ordered pairs associated with the given points: Make a table of values and graph the following equations on graph paper provided. (next page). y x. xy y 9 x x 1 y 4. 1 x y 5. xy 0 9. y x y x 6. yx 5

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11 . y x. xy x y Page xy 0 6. y x 5 7. y 9 x 8. x 1 y 9. y x y x

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13 Notes-- Relations and Functions Page 1 A relation is a set of ordered pairs. The domain is the set of all first coordinates and the range is the set of all second coordinates. Bob Carly Juan Bow Tie Flower Scarf {(Bob, tie), (Carly, bow),(juan, flower),(bob, scarf)} The domain would be {Bob, Carly, Juan} The range would be {tie, bow, flower, scarf} A function is a relation such that no two ordered pairs have the same first coordinate. An example of a function is: {(Sally, pink), (Carly, blue), (Adam, green), (Bob, blue)} The domain would be {Sally, Carly, Adam, Bob} The range would be {pink, blue, green} Another example of a function is: {(, ) : 5,,0,} A x y y x x B ( x, y): y x 10, x 6, 1,6 Find the domain and range.

14 Page 14 What if we have an infinite number of ordered pairs? We cannot make a list, but we can draw a picture of the relation. What is the domain? What is the range? How can we determine if a graph is a function? Remember definition: no two ordered pairs have the same first coordinate. This leads to the vertical line test. Do the following graphs represent a function? Find the domain and range.

15 Page 15 Relations and Functions Do all work on notebook paper. All work should be neat and organized. Write the following sets as sets of ordered pairs and identify the domain and range. 1. V ( x, y): yx7, x5,0,,1,. Z {( x, y) : y x, x,0,}. 1 X ( x, y): y x1, x1,,5, Determine the domain and range of each relation whose graph is given. Determine which of the following are graphs of functions

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19 Notes-- Function Notation Page 19 In algebra, we use function notation: Non-function notation: y x Re-written with function notation: fx ( ) x We read this as f of x f() () f() 9 f () f ( ) f ( ) Let fx ( ) x 1. Find f ( ), f (0), fa, ( ) fa ( h, ) fa ( h) fa ( ) h Let gx ( ) x. Find g ( ), g (0), ga, ( ) ga ( h, ) ga ( h) ga ( ) h

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21 Page 1 Function Notation and Operations of Functions Do all work on notebook paper. All steps should be shown. All work should be neat and organized. For the given functions, find the following values: f( ), f( 1), f( 0), f( 1), f( a), f ( ah ) 1. f ( x) x. f ( x) x f ( x) 5 x f ( x) x. f x ( ) x f Find ( f g)( x),( f g)( x),( f g)( x), ( x). g 6. f ( x) x 5x 1, gx ( ) x 8. f ( x) x, gx ( ) x f x ( ) x 5, gx ( ) x 9 9. f ( x) x x 5, gx ( ) x1 Let f ( x) x 1, Find the following: g( x) x 5x 1, mx ( ) x 4, and ( ) 1 px x. 10. f ( ) 11. m( ) 1. 5 f( ) 4 m( ) 1. f ( ) 14. g () 15. m( ) g() 16. ( mp)( 1) 17. ( g f)(0) 18. gx ( ) mx ( ) 19. mx ( ) p( x ) 0. g( x ) 1. p( x). f ( a). f ( a h) ga ( ) f ( ah) f( a) h 6. ga ( h) 7. ga ( h) ga ( ) h

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23 Notes-- Linear Functions-Part 1 Page Part1: Solve for y. Identify the slope and y-intercept. The slope-intercept formula of a line is: ymx b Where m is the slope of the line and b is the y-intercept. Solve for y. Identify the slope and y-intercept. Ex #1. x4y 10 Ex #. x 1 y 5 Ex #. 5y 15 0

24 Page 4 Notes-- Linear Functions Part Part : Find the x-intercept and y-intercept. The x-intercept is where the graph crosses the x-axis and the y-intercept is where the graph crosses the y-axis. Find the x-intercept and y-intercept. Ex. #1: 4xy 8 Ex. #: x5y 0 Ex. #: y7x 1 Ex. #4: x y 1 4 5

25 Notes-- Linear Functions Part Page 5 Part : Find the slopes of the lines passing through the following points. y y1 Formula for slope: m x x 1 m rise run Find the slopes of the lines passing through the following points. Ex #1: (7,0) and (0,4) Ex #: (, 5) and (1,9) Ex #: (, 5) and ( 1, 5) Ex #4: (7, ) and (7,5)

26 Page 6 Notes-- Linear Functions Part 4 Part 4: Graphing lines using the slope and y-intercept. 1. Solve the equation for y.. Identify m and b.. Plot b on the y-axis. rise 4. From b, use the slope to get more points. run Ex. #1: 4x y 1 Ex. #: xy 9 Ex. #: 7 yx Ex. #4: y Ex. #5: x 1 0

27 Page 7 Linear Functions Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Solve for y. Identify the slope and y-intercept. 1. x y 6. x4y 8. xy 0 Find the x-intercept and y-intercept. 4. x y y y x x4y x5y xy y x y 1 x 5 Find the slopes of the lines passing through the following points. 1. (6,0) and (0, ) 1. ( 4,1) and (, 5) 14. (, 7) and (6,) 15. (5,) and (9,) 16. (,1) and (,10)

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29 Graph the following lines. Find the slope and y-intercept. 17. x y m= 18. xy 4 m= 19. 4yx8 Page 9 m= b= b= b= 0. x5y 0 m= b= 1. 5 yx m= b=. y 4 m= b=. x m= 4. y 6 0 m= 5. 5x 15 0 m= b= b= b=

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31 Notes-- Equations of Lines-Part 1 Page 1 1. Find the equation of a line given the slope and y-intercept.. Find the equation of a line given the slope and a point. Find the equation of the line with the given information. Write answers in slopeintercept form, if possible. You will need to know formulas: Ex. #1: 1. Slope-intercept formula: ymx b. Point-Slope formula. yy m( x x ) 1 1 m ; y-intercept = 5 Ex. #: m 0; y-intercept = 1 5 Find an equation of a line given a slope and a point: Use the Point-Slope Formula: yy1 m( x x1) m slope Point ( x1, y 1) Ex. #: m 5; through (,1) Ex. #4: m ; through ( 4, ) 5

32 Page Extra Practice: m ; through (4, 1) Horizontal Equation: y number m 0 only has a y-intercept Vertical Equation: x number m is undefined only has an x-intercept Ex. #5: m 0; through ( 5,) Ex. #6: m is undefined; through (, 7)

33 Equations of Lines-Part Page Find the equation of the line passing through the given points. y y1 1. Find the slope first. m x x 1. Pick one point and now use the Point-Slope formula. y y1m( x x1). Write answers in slope-intercept form, if possible. Ex. #1: Passing through the points ( 1,) and (4,7) Ex. #: Passing through the points (, 4) and ( 5, 1) Ex. #: Passing through the points (5, 6) and (, 6) Ex. #4: Passing through the points ( 7, 4) and ( 7,8)

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35 Page 5 Equations of Lines Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Find the equations of the following lines. Write answers in slope-intercept form. 1. m 5 ; y-intercept 1. m 1; y-intercept 9 1. m ; y-intercept 4. m 0; y-intercept m 0; y-intercept 5 Find the equations of the following lines. Write answers in slope-intercept form when possible. m ; through ( 1, ) m ; through (,) 8 m ; through (,) 5 m ; through ( 4, 1) m is undefined ; through (, 7) Find the equations of the lines passing through the given points. Write answers in slope-intercept form when possible. 11. (,4) and ( 5,7) 1. ( 8,6) and (4, ) 1. (0,0) and (,) 14. (8, 4) and (, 4) 15. (5,) and (5, 6)

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37 Notes-- Parallel Lines Page 7 Parallel Lines have the same slope. Find the equations of the lines passing through the given points parallel to the given line. Write answers in slope-intercept form when possible. Ex #1: Through (,5); parallel to x7y Find the slope of the given line by solving for y.. Use the slope and the given point to write equation of line.. Write answers in slopeintercept form when possible. Ex #: Through ( 4, 9) ; parallel to y Ex #: Through (7, ) ; parallel to x 8

38 Page 8 Notes-- Perpendicular Lines Perpendicular Lines slopes are opposite reciprocals. (flip and change the sign) Find the equations of the lines passing through the given points perpendicular to the given line. Write answers in slope-intercept form when possible. Ex #1: Through (,5); perpendicular to y4x 5 Ex #: Through ( 7,) ; perpendicular to x5y Find the slope of the given line by solving for y.. Find the opposite reciprocal of the slope. We label this m. Use m and the given point to write equation of line. 4. Write answers in slope-intercept form when possible. Ex #: Through ( 4, 9) ; perpendicular to y 8 Ex #4: Through (7, ) ; perpendicular to x

39 Page 9 Parallel and Perpendicular Lines Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Find the equations of the lines passing through the given points parallel to the given line. Write answers in slope-intercept form when possible. 1. Through (1, ) ; parallel to yx 4. Through (,11) ; parallel to 5x 4y 8. Through (,5) ; parallel to x 7y 1 4. Through (, 7) ; parallel to y 1 5. Through ( 8, 9) ; parallel to x 5 Find the equations of the lines passing through the given points perpendicular to the given line. Write answers in slope-intercept form when possible. 6. Through ( 1,) ; perpendicular to y x 4 7. Through (,11) ; perpendicular to x 5y Through (,5) ; perpendicular to 4x y 1 9. Through (, 7) ; perpendicular to y Through ( 8, 9) ; perpendicular to x 7

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41 Notes-- Linear Inequalities Page 41 Graph the solution set of the linear inequalities: Steps: 1. Solve for y. Identify the slope and y-intercept.. Graph the line by plotting the y-intercept first (on the y-axis) and then use the slope to rise get other points, run.. Use a solid line if you have or. Use a dashed or dotted line if you have or.. Look at the y-intercept. Shade below the y-intercept if less than. Shade above the y-intercept if greater than. Ex. #1: yx Ex. #: 94x y Ex. #: 4x1y Ex. #4: x Ex. #5: y 4

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43 Page 4 Linear Inequalities in Two Variables Solve for y, if possible. Identify the slope and y-intercept. Graph. 1. yx 6. xy 1 y y m b m b. 1 y x 1 4. y 4x 5 y y m m b b 5. 4x y x y y y m m b b

44 Page x y 9 8. y 7x 8 y y m b m b 9. x y 10. x4y 0 y y m m b b 11. y 1. x 5 y y m m b b

45 Page 45 [Typethe document subtitle] Substitution Method, Elimination Method, Applications

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47 Notes-- Linear Systems in Two Variables Page 47 SUBSTITUTION METHOD: 1. Choose one of the equations and solve for one of the variables. Try to pick a variable with a coefficient of 1 or 1. This will eliminate having a lot of fractions.. Substitute this expression into the OTHER equation.. Solve the resulting equation. (It should now have only one variable) 4. Take this solution and substitute it into the expression obtained in step one. 5. Write your answer as an ordered pair. Ex. #1: Solve by the substitution method: 4x y4 xy Ex. #: Solve by the substitution method: xy9 4xy14

48 Page 48 ELIMINATION METHOD: 1. If necessary, rewrite both equations in the form of Ax + By = C.. Multiply either equation or both equations by appropriate numbers so that the coefficients of x or y will be opposites with a sum of 0.. Add the equations. 4. Solve this equation. 5. Substitute this solution back into one of the ORIGINAL equations. 6. Write your answer as an ordered pair. x7y1 Ex. #: Solve by the elimination method: x y Ex. #4: Solve by the elimination method: 4xy5 x8y10 SPECIAL SOLUTIONS: If both variables are eliminated when you are solving a system then 1. There is NO SOLUTION when the resulting statement if false.. There are INFINITELY MANY SOLUTIONS when the resulting statement is true. (They were the same line) Ex. #5: Solve by the substitution xy5 method: x4y7 Ex. #6: Solve by the elimination x6y14 method: 5x15y5

49 Page 49 Linear Systems in Two Variables Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Use the substitution method to determine the solution(s) of the following linear systems of equations. x y 7 1. y x 5 x y xy8.. x y 11 x y6 x y 8 x y9 x 6 y xy11 6 x y x y7 Use the elimination method to determine the solution(s) of the following linear systems of equations. x y 1 7. xy1 x y xy x y 5x4y6 x y 7 6x4y1 9. 5x y 4xy7 5 x y x5y6 4 x 5 y 10. 8xy6

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51 Notes-- Applications of Systems Part 1 Page 51 Solve word problems using a system of equations. Ex. #1: The sum of two numbers is 6. The larger number is 5 more than twice the smaller number. What are the numbers? 1. Define the variables.. Write the equations.. Solve. 4. Answer the problem. Ex. #: The difference of two numbers is 1. Twice the larger number plus the smaller number is 18. What are the numbers? Money problems: If I have quarters, I have 75. (value of coin) = $money If I have 9 nickels, I have 45. 9(value of coin) = $money (How many of the item)(value of the item)= $ Total money Ex. #: Jan has $5.90 in dimes and quarters. She has a total of coins. How many dimes and how many quarters does Jan have? Ex. #4: Elaine spent $5.96 on 1 and 6 stamps. She bought a total of 1 stamps. How many 1 stamps and how many 6 did she buy?

52 Page 5 Applications of Systems Part Solve word problems using a system of equations. Ex. #1: Alex sold tickets to the high school football game. Adult tickets cost $4.00 and student tickets cost $1.50. Alex collected $10 from the sale of the tickets. How many adult tickets and how many student tickets did Alex sell? 1. Define the variables.. Write the equations.. Solve. 4. Answer the problem. Ex. #: Yolanda bought pens and 1 notebook for $1.70. Four days later she bought pens and notebooks for $.6. What is the cost of each pen and each notebook?

53 Page 5 Applications-System of Equations Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Define the variable. Write a system of equations. Solve. 1. The sum of two numbers is 5. The larger number is 1 more than 5 times the smaller number. What are the numbers?. The sum of two numbers is 6, and their difference is 10. What are the numbers?. The difference of two numbers is 10. Twice the larger number minus the smaller number is 14. What are the numbers? 4. Juanita has $.80 in nickels and dimes. She has a total of 40 coins. How many nickels and how many dimes does Juanita have? 5. Shana spent $5.8 on 0 and stamps. She bought a total of 1 stamps. How many 0 stamps and how many stamps did she buy? 6. Devon sold 7 tickets to the choir show. Adult tickets cost $.50 and children s tickets cost $1.5. He collected $7.5 from the sale of the tickets. How many adult tickets and how many children s tickets did he sell? 7. Maritza bought 4 pencils and 1 eraser for 6. The next week she bought pencils and 7 erasers for 84. What is the cost of each pencil and eraser? 8. Coach Reeves buys basketballs and footballs for $60. Coach Johnson buys basketballs and 5 footballs for $7. What is the cost of each basketball and football?

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57 Notes-- Absolute Value Equations Page 57 Absolute value of a real number is the from zero on a number line. Distance is always positive. x x x 0 Process: 1. Isolate the absolute value expression. expression number. Determine the type of number the absolute value expression is equal to a. If it is equal to a NEGATIVE NUMBER the answer is NO SOLUTION. b. If it is equal to a POSITIVE NUMBER you will split into equations (without absolute value bars) ( solutions) expression number or expression ( number) c. If it is equal to ZERO rewrite the equation without absolute value bars and solve for the variable. (one solution) Example 1: x 5 1

58 Page 58 Example : x 4 11 Example : x5511 Example 4: x 14 4 Example 5: x599 Example 6: 4 x 17 Example 7: 5 x 7

59 Page 59 Absolute Value Equations Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Solve the equations. 1. x 8. x 1. x x x x x x x x x x 7

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61 Notes-- Absolute Value Inequalities Page 61 Recall that absolute value is the distance away from zero. The distance is ALWAYS positive. x 4 x Less Than: ( ), 1. Isolate the absolute value: expression number. RE-write without the absolute value sign. Use a "sandwich" inequality: number expression number. Solve. 4. Graph answer on a number line. 5. Write answer in interval notation. Greater Than: ( ), 1. Isolate the absolute value: expression number. RE-write without the absolute value sign. You must separate into inequalities: expression number or expression number. Solve. 4. Graph answers on a number line. 5. Write answer in interval notation. Example 1: x 4 8 Example : x 9

62 Page 6 Example : x 1 9 Example 4: x 511 Example 5: 5x Example 6: x 1 51

63 Page 6 Absolute Value Inequalities Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Solve the inequalities. Graph solutions on a number line. Write the answers in interval notation. 1. x. x. x 6 4. x x x x x 5 9. x x x 7 1. x x x x

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65 Page 65 [Typethe document subtitle] GCF, Grouping, Binomials, Trinomials, Solving Factorable Quadratic Equations

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67 Notes-- Greatest Common Factor Page 67 GCF: The largest number that will divide into all the terms AND the variable(s) that occur in all the terms (raised to the smallest power). Find the GCF of the following polynomials: x 1x 8x. 16a4b 8a Factor the following:. 8x 18xyx 4. 7xy5xy 14xy y(x9) 14 z(x9) xy ( ) 5 z( y)

68 Page 68 Notes-- Factor by Grouping GROUPING: This factoring technique is used when you have 4 or more terms. Try grouping the first two terms and the last terms. Factor out the GCF of each group. If the GCF matches exactly, then take out that GCF from both groups. Factor further, if necessary. Factor the following: 1. 10hn km hm 15kn. 4xz 6xz xz 14xz xy xy y x y

69 Page 69 GCF and Grouping Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Factor completely x x xy x 6x 18x 10x x( x y) 6 y( x y) 5. xx ( ) ( x ) 6. 8 yz(15 x) 6 yz(1 5 x) x( y) 5 z( y) 8. ax bx ay by 9. 6cf cg 6df dg x xy x y x y5x y 1xy 0xy y 7y 18y x x x x 54y 6x 7x y 1xy 4

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71 Notes-- Factoring: Difference of Squares Page 71 ALWAYS look for a GCF first. Difference of Squares: 1. a b ( ab)( ab) 5x 81y. 4 16x 1. x ( x ) 9 5. (x y) 49z 6. x (5x4) 9(5x4)

72 Page 7 Notes-- Factoring: Cubes Cubes: 1, 8, 7, 64, 15, 16, 1000 Sum of cubes: a b ( ab)( a abb ) Difference of cubes: Sum of cubes: a b ( ab)( a abb ) x 8 Difference of cubes: x 15 ALWAYS look for a GCF first x 7y. 40x 15y. 4 y y y xy75xy1xy 75xy

73 Page 7 Factoring Binomials Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Factor completely. 1. x x 18y. 6x 49y 1. 75x. 7x y 1xy 1. x (x) (x ) 4. x x ( x6) 8( x 6) 5. 81x 16y x x 45x ( x ) x 4y 4x y xy 7. (x y) 9z x 5x x 5 8. x x x x x x x x x 7y 0. 7x 7x 8x y 8xy 5

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75 Notes-- Factoring Trinomials Page 75 TRINOMIAL: A polynomial with terms. ALWAYS look for a GCF first. It is a good idea to make leading coefficient positive. Guess and check method is one of many methods, but not the only one. Factor the following: 1. x 14x 45. x 6x 10. 8x 16x x 7x 0 5. x 5x x x 7. 8x 6xy5y 8. x 4x 5 4

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77 Page 77 Factoring Trinomials Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Factor completely. 1. x 9x x 0x 5. x 9x x 9x. x x x 1x x 6x x 60x 5 5. x 4x x x x 6. x 16x x 16. 9x 16xy 4y 7. x 6xy 5y 17. x x x 1x x 7x 9. 1x 5x x 19x x 15x x 19x 6

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79 Notes-- Solve Quadratic Equations by Factoring Page 79 A quadratic equation has the form of ax bx c 0 Steps to solve by factoring: 1. Write in standard form: ax bx c 0. Factor.. Set each factor that has a variable equal to zero. 4. Solve each resulting linear equation x 1x 0. 1x 6x x 8x 1 x( x 8) 4. 7x x4 ( x4)( x 4)

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81 Page 81 Quadratic Equations Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Solve by factoring x 6x8 0 x 1x 0 6x x 0 4x 9 0 x 7x4 0 6x 41x7 0 1x 1x9 0 x 5x4 10 8x 1x95x x 5x0x x 4 9x 7x7 x( x 4) x 8x1 (x)( x 7) 1. (x5)( x4) ( x7)( x 4)

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83 Page 8 [Typethe document subtitle] Integer Exponents, Reducing Expressions, Performing Operations on Rational Expressions, Complex Fractions, Long Division, Synthetic Division, Solving Equations

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85 Notes-- Evaluating Integer Exponents Page 85 5 Negative Exponents: Take the reciprocal of base and change the sign of the exponent. x y = x y = Zero Exponent: 0 5 = 0 x = 4 0 (10 xy ) = ANYTHING (except zero) raised to the power of zero equals 1. Evaluate each of the following ( ) 4 4. ( 5) compared to

86 Page 86 Notes-- Simplifying Integer Exponents Review Negative Exponents: Take the reciprocal of base and change the sign of the exponent. x y = x y = Zero Exponent: 0 x = 4 0 (10 xy ) = Simplify each of the following. Your answers should have no NEGATIVE exponents. 1. Multiply the exponents to get rid of parentheses.. Make all exponents positive.. Clean up x y 5xy. 5 4 xy x y. 5xy 8xy xy 9 x y 0 5 6

87 Page 87 Integer Exponents Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Evaluate each of the following. Your answers should have NO exponents ( ) ( 7) ( 7) (4 ) ( 8) ( ) ( ) Simplify each of the following. Your answers should have no NEGATIVE exponents. 1. 8x y 4x y x y x y ( x y ) (4 xy ) (8 xy ) 8. 4 xy 0 7 5xy 4 4. x 5 4 y x y 4 x y xy x y 0. 4 x y 4 8xy x y 10x y 6 5

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89 Notes-- Reducing Rational Expressions Page 89 Review Reducing Fractions: x x Factor.. Cancel common FACTORS. 1. 8xy z 5 4 xyz. xy14 xy xyxy. x 6x16 64 x 4. 8x 7 x x9

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91 Page 91 Reducing Rational Expressions Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Reduce the following expressions to lowest terms. 1. 8x y 6xy x 7x 5 x x. 9ab c 4ab c 9. x x10 4 x. x x y xy xy 10. 4x 1x9 x 11x1 4. x 6 y x 11. x 9x0 6x 0x 5. 4x y 4x y xy x y 1. x 5x1 9 4x 6. x 5x6 x 4x1 1. x 8 x x x 7x 4x x 1 6x 4x

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93 Notes-- Page 9 Multiplying and Dividing Rational Expressions Directions: Perform the indicated operations and reduce to lowest terms. Even though the "indicated operations" are multiplication and division, what we need to do is FACTOR and REDUCE mp 5 4xy 18xy 9xy mp. 4x 5x0 6x9 xzz 1. Factor. Reduce. x x15 x x10 x 9x10 1x7x

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95 Page 95 Multiplying and Dividing Rational Expressions Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Perform the indicated operations and reduce to lowest terms xy 14 x y x y ab ab ab x y 14 ab a b 8 8a b 9xy 1xy x x 5x 15 xy y 10xy 5xy 6x 8 4x 1x4 10x 17x 5x 41x8 x 7x6 x x6 x 1 x x8 x x4 5x5x 6x4x x x x x x x x 10x 8 ( x 5) x x0 x x x x x x x x x x x x x 11x1 8x 1x 4x

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97 Notes-- Adding and Subtracting Rational Expressions Part 1 When adding or subtracting fractions that have the same denominator: 1. Keep the denominator and collect like terms in the numerators.. Try to reduce the expression by factoring. Directions: Perform the indicated operations and reduce to lowest terms. Page x 1 4 x 6 x x 5x 5x. x x 6 x x x x OPPOSITE DENOMINATORS x 5x 5. y y 6. x 7 x x1 x4 4x

98 Page 98 Adding and Subtracting Rational Expressions Part When adding or subtracting fractions that have different denominators, one must find the least common denominator (LCD) before adding or subtracting the fractions. Process to find the LCD: 1. Factor each denominator.. Write down one of every kind of factor.. Raise each factor to its highest power. Find the LCD: 5 7 abc abcd Notes xy 4xy Perform the indicated operations and reduce to lowest terms: x 8x. 6 x1 x4 1. Find the LCD. Re-write each fraction with the LCD. Collect like terms of numerators 4. Reduce, if possible.. x 1 x 4 x x4 x7 x1 x x4 x x0 4. x x6 x 9x10 x 6x8 5.

99 Page 99 Adding and Subtracting Rational Expressions Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Perform the indicated operations and reduce to lowest terms x x x x. n 9n xy xy 1. 5 x5 x. x x 5 4x7 4x7 1. x x x x6 5x1 x9 x x11 x x x1 x x5 x x x 1 x1 x 1 7x x y y x 4x5 x x15 4x x 5x x 1x x 8 x x1 x5 5x 5 x x x1 1x 8 5x y xy x y 17. x x x 1 x 6x5 18. x5 x1 x x1 x 5x6 x1 x 19. 6x 7x x 7x 0. x1 4x5 4x 4x15 8x 10x

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101 Notes-- Synthetic Division Page 101 Synthetic Division is a condensed method of long division. It is quick and easy. Unfortunately, it can only be used when the divisor is in the form of ( x a) Review long division: 9x 5x1 x 1 Synthetic division: 9x 5x1 x 1 x 5x x 1 4 x 5x 10x x x 15 x 5 Reminders: 1. Write both polynomials in standard form.. Fill in all missing terms with a place holder of zero.. Write your answer as a polynomial that is one degree less than the dividend (numerator).

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103 Page 10 Division of Polynomials Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Perform the indicated divisions using LONG division method. 1. 4x 6x6 4x x 7 x. 40x 44x17 5x 6. 4x x7 x. x 1x 10x19 x 7. 6x 0x 19 x x 1x 1x 4x x x x 9x1 x x Perform the indicated divisions using SYNTHETIC division. 9. x x5 x 1. 4 x x x x 5 5 x 10. x 7x x6 x x x x 7x5 x x 7x4 x 11 5 x x x x 4 holder for the constant) (Hint: need a place x x x x 1 x 6

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105 Notes-- Complex Fractions Page 105 A complex fraction is a fraction that has fractions in the numerator and/or the denominator. Single Fractions: Directions: 1. 1x 8y 7x 16y 4 Simplify the following fraction.. x 5x6 10x 5 4 x 6x Change the division to multiplication. Reduce x y 6 5xy 1 1 x 5 5 7x 14xy Multiple Fractions: 1. Find the LCD of all the fractions.. Multiply every term by the LCD.. Reduce

106 5. Page x x x x Multiple Fractions: 1. Find the LCD of all the fractions.. Multiply every term by the LCD.. Reduce 6. 1 x y x 7. 5 h 5 h h

107 Page 107 Complex Fractions Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Simplify the following fractions x x 15 9 x x x y 4 5x 4y x x y y 8x 1x y y 4 8xy 4y 6xy x y xy 9xy 1 6 x 7 1 x x y 4 x y 1 1 x y 1 1 x y 9. x y 1 y x x 1 x 1 x x 1 1 x y x y 5 4 h h 4 x 1 5 x 1

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109 Notes-- Rational Equations Page Find the L.C.D.. Multiply EVERY term by the LCD to get rid of all the fractions. (OR Cross-multiply, if you can**). Solve the resulting equation. 4. Check for extraneous solutions. (Substitute answers into the denominator to see if this would cause division by zero. You must throw out any solutions that cause division by zero because it is undefined.) x 8x x 1x x1 x x x 4. 6 x x x x

110 Page 110 x 5 5. x1 x x ** x 5x x1 ** 7. x 4 x 1 x x

111 Page 111 Rational Equations Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Solve the following equations. Do not forget to check solutions x x x 1 x1 x5 4 5 x1 x4 x x x 11. x 6 x1 x4 x 7x x x 1 x1 x x x 1. x 6 1 x 5 x 5. 1 x1 x x 14. x x1 9x x1 x1 4x x 4x1 8x x 8. x1 x1 10 x x x x 6x x x 10x0 x 4 4 x x1 1 5 x5 x x x5 5 x1 x

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113 Notes-- Applications Rational Equations Page 11 Consecutive integers: Remember that integers are only negative and positive whole numbers. They do not include any decimals or fractions. Two consecutive integers: x, x+1 Two consecutive odd integers: x, x+ Two consecutive even integers: x, x + Reciprocals: If the number is x, then its reciprocal would be 1 x. Define the variable. Write an equation. Solve. 1. The sum of a number and its reciprocal is 9. What is the number? 10. The difference of the reciprocals of two consecutive integers is 1. What are the integers?

114 Page 114. The sum of the reciprocals of two consecutive even integers is 9. What are the 40 integers? Work: time alone time alone time together 4. Paris can wash her car in 4½ hours. Her friend, Celia, can wash the same car in 7 hours. Working together, how long will it take them to wash the car? 5. Working together, Joseph and Dylan can write the computer program in 11 hours. Working alone, Joseph can write the computer program in 15 hours. How long does it take Dylan to write the program by himself?

115 Page 115 Applications-Rational Equations Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Find the solutions to the following problems. 1. The sum of a number and its reciprocal is 5. What is the number? 1. The sum of the reciprocals of two consecutive integers is 7. What are the 1 integers?. The sum of the reciprocals of two consecutive even integers is 7. What are the 4 integers? 4. Tameka can mow her yard in hours. Her brother Dante can mow the yard in hours. Working together, how long will it take them to mow the yard? 5. Brittany can prepare her report in 1 1 hours. Her co-worker, Firza, can prepare the report in 6 hours. Working together, how long will it take them to prepare the report? 6. Working together Debbie and Jim can clean the house in 4 hours. Working alone, Debbie can clean the house in 5 hours. How long does it take Jim to clean the house? 7. Working together, Elmer and his son can paint their house in days. Working alone, Elmer can spray paint the house in 4 days. How long will it take his son, using a brush, to paint the house by himself?

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117 Page 117 Rational Exponents, Radicals, Simplifying, Operations with Radicals, Equations, Complex Numbers

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119 Radicals Page 119 What does a radical sign look like? Here are some examples:,, 4, 5 Square root: Cube root: Fourth root: ( ) Fifth root: ( ) 5 Even root of a negative number is NOT real. Convert rational exponents to radicals: Odd root of a negative number is a negative number (5 x y) Convert radicals to exponents. Simplify where possible x 5 0 p 1 (5 xy)

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121 Page 11 Radicals Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Evaluate each of the following, if possible Convert the following expressions to radicals xy x Convert the following radicals to expressions with rational exponents. Simplify where possible x. 4 4 k. 7x y 6 4. a 15

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123 Notes-- Rational Exponents Page 1 Evaluate each of the following, if possible. 1. Make any negative exponents positive.. Change to radicals. Simplify ( 5) ( 5) 1 9. ( ) Simplify. All answers should have only POSITIVE exponents. 1. x 1 1 x 1. 5x 5 x x 10x 1 4 y y

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125 Page 15 Rational Exponents Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Evaluate each of the following, if possible ( 16) Simplify each of the following. Your answers should have no NEGATIVE exponents. 1. x x x 8x. x x x 8y 5x 9y x y 4x 1 y x x x 5x

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127 Notes-- Simplify Radical Expressions Part 1 Simplify the following: Page Prime factor the number.. For square root: Look for pairs. For cube root: Look for of a kind. For 4 th roots: Look for 4 of a kind. etc.. "Take out" the pairs, of kind, etc xy xy xyz x y z

128 Page 18 Notes-- Simplify Radical Expressions Part Rationalize the Denominator Rationalize the denominator means to eliminate any radicals in the denominator. A process to follow is: 1. Reduce the fraction, if possible.. Simplify the radicals. Rationalize by multiplying by "what you need". 4. Reduce again if necessary. Simplify the following: SQUARE ROOTS: y 4. 5x 0x x 6. 49x 9y CUBE ROOTS: 5 7. y y y 9x 9. 5

129 Page 19 Simplify Radical Expressions Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Simplify the following radicals x x 7 6. x y x x y x yz x yz 19. 7x x 0. y x 5y x 16. x y 5x x 5. 49x 9y

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131 Adding & Subtracting Radical Expressions Part 1 Review of collecting like terms: x 5x x 5 x Notes-- Page 11 Perform the indicated operations: Simplify all radicals. Add the coefficients of "like" radicals x 18y x x y 4y 1x xy 4. 50xy 54xy

132 Page 1 Notes-- Adding & Subtracting Radical Expressions Part 1. Perform the indicated operations: Simplify radicals.. Rationalize all denominators.. Find the LCD 4. Re-write all terms with LCD. 5. Combine like terms

133 Page 1 Adding and Subtracting Radical Expressions Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Perform the indicated operations x 7 15x 5 15x x 10y 8x 10y x 1x 48x 8. x 18x 5 x x 8x 9. 7x 1x 150y 4 4y x x x

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135 Notes-- Multiply Radicals Page 15 Multiplying a monomial by a monomial: Multiply the "outsides" Multiply the "insides" Simplify, if possible. ( )(4 15) ( y)(5 x) Multiply a square root by the SAME square root: ( )( ) ( 7 y)( 7 y ) ( 5 y)( 5 y) Perform the indicated operations and simplify your answers: 1. 5y 5y x 1. F.O.I.L.. Simplify. Combine like terms.. (4 )(5 8 6). x y 4. (5 x ) 5. ( 5x 1)

136 Page 16 Notes-- Divide Radicals Review: ( x5)( x 5) ( x 5)( x 5) ( 5)( 5) The "conjugate" of x 5 is x 5 The "conjugate" of 5 is 5 As you can see, the conjugate is found by changing the middle sign. When you multiply conjugates, you just need to square each term and then subtract. We use the conjugate to rationalize the binomial denominators. Rationalize the denominator of the following x y x 5y

137 Page 17 Operations with Radical Expressions Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Perform the indicated multiplications and simplify your answers x x x y4 x 5 y x 7 9. x x 6 Rationalize the denominator of the following radicals x 5 x x x y 19. x x y y

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139 Notes-- Radical Equations Page 19 Process: 1. Isolate the radical.. Get rid of the radical by raising both sides to the appropriate power. x x. Solve the resulting equation. 4. Check for extraneous solutions. x x 4 4 x x 1. 4x x 414. x x x4 5. x 4x x4x

140 Page 140 Extra Example: 0 x x Remember that a fractional exponent can be written in radical form. 5 5 x 5 x or x x x or x If you encounter an equation that has a variable raised to a fractional exponent, you solve it just like a radical equation. Get rid of the radical by raising both sides to the appropriate power. x x 5 5 x x 7. x x x 9

141 Page 141 Radical Equations Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Find the solutions(s) of the following radical equations. 1. x x5x. x x8x 4. x x x 10x x 5 7. x 40 x 1. x 51 x 14. x x 15. x4x x x x x x x x 4 11 x x x x 8x65 x 18. x x x 1 4

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143 Notes-- Complex Numbers Page 14 What happens when we want to solve the equation: x 4? In order to solve this equation, we must introduce 1 and the set of imaginary numbers. We will represent 1 with i. This leads to 1 i 1 1 i Therefore, 1 i Any square root of a negative number can be written in terms of i A complex number has a real part and an imaginary part. 5i We can add, subtract, multiply, and divide complex numbers. Perform the indicated operations: 1. ( 47 i) ( i). ( 5 i) (7 4 i). ( i)(4 5 i) 4. (4 i) Replace i with 1

144 Page 144 Rationalize the denominators. (Divide) Review: i i Review: 4 5 The "complex conjugate" of 5i is 5i Multiply: ( 5 i)( 5 i) 6. i 5i 7. i 54i

145 Page 145 Complex Numbers Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Perform the indicated operations. 1. ( i) (5 6 i). ( 5 i) ( i). 7( 9 i) 6. (5 i)( 4 i) 7. ( 5 i)( 5 i) 8. ( i)( i) 4. (4 i 7) i 5. (6 i)( 4 i) ( i) ( i) Rationalize the denominators i 16. i 1 i 1. 5 i i 17. 4i 5i 1. 4 i i i i i i i 15. i 0. 5i 1 4i

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147 Page 147 Extraction of Roots, Completing the Square, Quadratic Formula, Applications

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149 Notes-- Quadratic Formula Page 149 This is another method to solve quadratic equations. If the quadratic cannot be factored we have to have something else that will allow us to solve the equation. There are such methods completing the square and the quadratic formula. The quadratic formula is derived from completing the square on the general equation: ax bx c 0 b b 4ac You MUST memorize the formula: x a Process: 1. Write the equation in standard form: ax bx c 0. Identify ab,, and c.. Substitute numbers into formula. 4. Carefully do the arithmetic under the square root sign. 5. If possible, simplify the radical. 6. If possible, reduce the fraction. 1. x 4x1 0. 9x 18x xx ( ) 6x 11 x x x 5. (x)( x4) 7

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151 Page 151 Quadratic Formula Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Use the quadratic formula to find the solutions of the following quadratic equations x 9x 9 0 9x x 0 8x 10x 1 0 6x x 5 0 x 6x 8 0 8x x 0 4x 6x 9 0 x 8x 7 0 4x 4x x 4x x(x ) 7 1. x( x ) 1 1. ( x 5)(x 1) x x 14.

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153 Notes-- Page 15

154 Page 154 Notes-- Completing the Square This is another method to solve quadratic equations. If the quadratic cannot be factored we have to have something else that will allow us to solve the equation. There are such methods completing the square and the quadratic formula. Completing the Square is also used for other applications. Process: 1. Write the equation in standard form:. Move c to the left hand side of the equation. ax bx c 0 x bx c. If a is NOT = 1, divide all terms by a. Reduce any fractions. 4. Take 1 of the coefficient of x. 5. Square this and add to both sides of the equation. 6. Re-write left hand side as a squared binomial. 7. Solve the equation by the extraction of roots method. 1. x 8x11 0. x 6x18 0. x x x x x 5x7 0

155 Page 155 Completing the Square Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Use the Extraction of Roots method to find the solutions of the following quadratic equations x 5 x 80 x 11 x x x x 8 0 7x x 8 (x 7) 6 (4x 5) 9 ( x 1) 75 (x 5) 7 (5x 1) 50 Find the solutions of the following quadratic equations by completing the square x x x x x 4x 1 0 x 7 0 4x 9 0 x 7 0 5x 0 4x 4x 0 x 5x 4 0 4x 4x x 1x 7 0

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157 Notes-- Applications Quadratic Equations Page 157 Consecutive integers: Remember that integers are only negative and positive whole numbers. They do not include any decimals or fractions. Two consecutive integers: x, x+1 Two consecutive odd integers: x, x+ Two consecutive even integers: x, x + Reciprocals: If the number is x, then its reciprocal would be 1 x. Define the variable. Write an equation. Solve. 1. Find two consecutive positive integers. Find two consecutive odd integers whose product is 1. whose product is 5.. The sum of the squares of two even consecutive integers is 40. Find the integers. 4. The sum of a number and its reciprocal is 6. What is the number? 5. Three times the square of a number is 6 more than twice the number. What is the number?

158 Page 158 Area: Rectangle: Area = base x height or Area = length x width. Triangle: Area = 1 base x height or Area = base x height 6. The area of a rectangle is 5 sq. ft. The length is 1 ft. less than twice the width. What are the dimensions of the rectangle? 7. The height of a triangle is in. more than three times the base. Find the base and the height if the area of the triangle is 5 sq. in. Work: time alone time alone time together 8. One pipe can fill a reservoir hours faster than another pipe can. Together they fill the reservoir in 5 hours. How long does it take each pipe to fill the reservoir?

159 Page 159 Applications-Quadratics Equations Do all work on notebook paper. All steps should be shown. All work should be neat and organized. Find the solutions to the following problems. 1. Find two consecutive integers whose product is 7.. Find two consecutive odd integers whose product is 6.. The sum of a number and its reciprocal is 4. What is the number? 4. Find two consecutive even integers whose product is The sum of the squares of two consecutive integers is 5. Find the integers. 6. The sum of the squares of two consecutive integers is 61. Find the integers. 7. One number is 1 more than twice another number. The product of the two numbers is 1. What are the numbers? 8. The square of a number minus times the number is 14. What is the number? 9. The area of a rectangle is 4 sq. ft. The length is ft. more than 4 times the width. What are the dimensions of the rectangle? 10. The base of a triangle is 1 inch more than twice the height. Find the base and the height if the area of the triangle is 14 sq. in.? 11. The base of a triangle is 1 inch less than twice the height. Find the base and the height if the area of the triangle is 6 sq. in.?** 1. One pipe can fill a reservoir 1 hour faster than another pipe. Together they fill the reservoir in 4 hours. How long does it take for each pipe to fill the reservoir?** 1. Gary can process his job days faster than his co-worker, Bob. Working together, they can process the job in 6 days. Working alone, how long does it take each of the men to process the job?** **Determine a decimal approximation.

160

161 [Typethe document subtitle] Math 010

162 Cartesian Coordinate System-Answers Page 9 1. A(4,0) B(1,6) C(0,4) D(0,0) E(,) F( 6,0) G( 4, 5) H(0, ) I(7, 6). y x. xy x y 5. xy 0 6. y x 5 7. y 9 x 8. x 1 y 9. y x y x

163 Relations and Functions-Answers Page 15 5,17, 0,7,,, 1,5,,4 Domain= 5,0,,1, 1. V. Z,5, 0,,,5 Domain=,0, Range=,5 1 1,1,, 5, 5,,, 0 Domain= 1 1,, 5,. X 4. Domain=, Range=, Range= 17,7,,5,4 Range= 1, 5,, 0 ; Yes, it is a function 5. Domain= [,] Range= [0,] ; Yes, it is a function 6. Domain=, 7. Domain=, Range= [0, ) ; Yes, it is a function Range= (,]; Yes, it is a function 8. Domain= (,1] Range= (,0]; Yes, it is a function 9. Domain= [,] Range= [0,4] ; Not a function 10. Domain= [4, ) Range=, ; Not a function 11. Domain= [ 6,4] Range= [,4] ; Yes, it is a function 1. Domain=, 1. Domain= [] Range= [4]; Yes, it is a function Range=, ; Not a function 14. Domain= [, ) Range= (,5]; Yes, it is a function 15. Domain=, Range=, ; Yes, it is a function 16. Domain= [ 5,5] Range= [,] ; Not a function 17. Domain=, 18. Domain= (,1] Range= (,]; Yes, it is a function Range=, ; Not a function 19. Domain= [, ) Range= (, ] (0,4]; Yes, it is a function

164 Function Notation and Operations of Functions-Answers Page 1 1. f ( ) 7, f( 1) 5, f(0), f(1) 1, f( a) a, f( ah) ah. f ( ) 5, f( 1) 4, f(0), f(1), f( a) a, f( ah) a h f f f f f a a f a h a ah h ( ), ( 1) 1, (0), (1) 1, ( ), ( ) f ( ) 1, f( 1) 4, f(0) 5, f(1) 4, f( a) 5 a, f( ah) 5a ah h f ( ) 8, f( 1), f(0) 0, f(1), f( a) a, f( ah) a 4ahh f g x x x f g x x x ( )( ) 6,( )( ) 4 1 f x x f g x x x x x g x ( )( ) 11, ( ) f g x x f g x ( )( ) 4,( )( ) 14 4 f x ( f g)( x) x 4x 45, ( x) g x ( f g)( x) x8,( f g)( x) x14 8. f x ( f g)( x) x 19x, ( x) g x11 9. f g x x x f g x x x ( )( ) 5 4,( )( ) 6 4 f x x ( f g)( x) x x 6x 7x 5, ( x) g 5 x x 5x 5. a 5a x 4 x 1 1x 10x x 1. a 1. ah 1 6. a 6ahh 5a5h ah5

165 Linear Functions-Answers Page 7 1. y x 6; m ; y-intercept 6.. y x ; m ; y-intercept y x; 1 m ; y-intercept y x ; m ; y-intercept 8 5. y ; m 0; y-intercept 6. 9 y x 6 ; 4 9 m ; y-intercept x-intercept ; y-intercept 8. x-intercept 5 ; y-intercept 9. x-intercept 0 ; y-intercept x-intercept 5 ; y-intercept x-intercept 15 ; y-intercept m 6 m m 15. m m undefined

166 17. m ; y-intercept m ; y-intercept 19. m ; 4 y-intercept 0. m ; 5 y-intercept 0 1. m ; y-intercept 5. m 0; y-intercept 4. m undefined ; y-intercept NONE 4. m 0; y-intercept 5. m undefined ; y-intercept NONE

167 Equations of Lines-Answers Page y 5x. y x 9. 1 y x 4. y y 5 6. yx y x y x 9. y x 11. yx y x 4 y x 14. y x 5 Parallel and Perpendicular Lines-Answers Page 9 1. yx y x y x y 7 5. x y x 5 y x y x 4 9. x 10. y 9

168 Linear Inequalities in Two Variables-Answers Page 4 1. yx 6. xy 1 y x 6 m b 6 y x 6 m b 6. 1 y x 1 4. y 4x 5 1 y x 1 1 m y 4x 5 m 4 b 5 b x y x y y x m b y x 7 m b 7

169 7. 5x y 9 8. y 7x 8 5 y x 5 m b 7 y x 4 7 m b 4 9. x y 10. x4y 0 y x m b 0 y 1 4 x 1 m 4 b y 1. x 5 y m 0 b m undefined b NONE

170 Linear Systems in Two Variables-Answers Page ( 1, ) 8. Same line. Dependent.. ( 1,4). (,) 4. (0,4) 5. No solution. Parallel Lines. Inconsistent. 6. Same line. Dependent. Infinitely Many Solutions 7. ( 5, ) Infinitely Many Solutions. 9. (6, 1) 10., (, 4) 1. No solution. Parallel Lines. Inconsistent. 1. (,4) Applications-System of Equations-Answers Page , 1. 8,. 4, dimes and 4 nickels stamps and 9 stamps Adult tickets and 5 children s tickets pencils and 6 erasers 8. $9 football and $14 basketballs

171 Absolute Value Equations-Answers-Page , 8. 6, 6. 4, , , 7 4., , , (,) Absolute Value Inequalities-Answers-Page 6 9. ( 5,). (, ] [, ). [ 4,4] 4. (, 14) (14, ) (, ) 7. (4,6) 8. (,1) (4, ) 10. (, 4] [, ) , (, ) 14. (, 4) (6, ) 15. [0,]

172 1.. 5(x 7) (4 x x 9y 1) GCF and Grouping-Answers Page ( x y)(a b) 9. ( f g)( c d). x (x 9x 5) or x ( x 9x 5) 4. ( x y)(x y) 10. (x 4)(5 yx) or (4 x)( x 5 y) 11. xy(5 y)( x 4 y) 5. (x1)( x ) 1. (y7)( y 6) 6. yz(15 x)(4y z ) 1. 4( x x )( x 5) 7. 5( y)( x z) 14. ( y x )(9 y x) Factoring Binomials-Answers Page 7 1. ( x)( x ). (6x 7 y)(6x 7 y) (x 4 y)(9x 1xy 16 y ) (5x1)(5 x 5x 1). xy(x y)(x y) 1. (x)( x1)( x 1) 4. ( x 16) 14. ( x6)( x)( x x 4) 5. (9x 4 y )(x y)(x y) 15. (5x)( x)( x ) 6. ( x4)( x 10) 16. ( x y)( xy)(x 4 y) 7. (x y z)(xy z) 17. (x5)( x1)( x x 1) 8. ( x4)( x 4x 16) 18. (4x)( x)( x x 4) 9. (x1)(4x x 1) 19. (x1)(x1)( x)( x x 4) 10. (4x y)(16 x 1xy 9 y ) 0. 7 x( x y)( x y)( x1)( x 1)

173 Factoring Trinomials-Answers Page ( x7)( x ) 11. (x5)(x5) or (x 5). ( x8)( x 1). PRIME 1. (9x1)( x ) 1. ( x)(6x 5) 4. ( x)( x) or ( x ) 14. 5(x1)( x 7) 5. ( x7)( x 1) 6. xx ( 4)( x 4) 15. (5 x x1)( x ) 16. (9x y)( x y) 7. ( x 5 y)( x y) 17. ( x1)( x1)( x ) 8. (x1)(x 5) 9. (4x1)(x 7) 10. (x4)(x 1) 18. (x1)(x1)( x ) 19. (x5)(x ) 0. (5x)(x ) 1., 4. 8, ,, Quadratic Equations-Answers Page , , , , , 6. 1, , 1., 4

174 Integer Exponents-Answers Page or or x y y x y y 7x 4x 5y x y 8. 16x 5y 1 0 4y 9. 4 x 0. 64x 4 y 10 Reducing Rational Expressions-Answers Page 91 4x 1. y 9c. 4b x( x y). y ( x y) x y x x y x x 7 x 1 7. x 8. x x 5 ( x 5) 9. ( x) (x ) 10. ( x 4) 11. ( x ) x ( x 4) 1. ( x) x x4 x 4 9x x 1 ( x 1) Multiplying and Dividing Rational Expressions-Answers Page bx ay 5. 5y 6. ( x ) 5x 9. x x4 x. 5bx 4 ay 4. 5xy 5. x x 8 7. ( x ) x 1 8. x 10. x x 1 4x