SUMMER MATH PACKET ADVANCED ALGEBRA A COURSE 5
Updated May 0 MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of exciting math problems for your enjoyment over the summer. The purpose of the summer packet is to review the topics you have already mastered in math and to make sure that you are prepared for the class you are about to enter. The packet contains a brief summary and explanation of the topics so you don t need to worry if you don t have your math book. You will find many sample problems, which would be great practice for you before you try your own problems. The explanations are divided into sections to match the sample problems so you should be able to reference the examples easily. This packet will be due the second day of class. All of your hard work will receive credit. The answers are provided in the back of the packet; however, you must show an amount of work appropriate to each problem in order to receive credit. If you are unsure of how much work to show, let the sample problems be your guide. You will have an opportunity to show off your skills during the first week when your class takes a quiz on the material in the packet. This packet is to help you maximize your previous math courses and to make sure that everyone is starting off on an even playing field on the first day of school. If you feel that you need additional help on one or two topics, you may want to try math websites such as: www.mathforum.org or www.askjeeves.com. Math teachers will be available for assistance at the high school the week before school. Check the marquee or the school website for specific times, which are to be determined. Enjoy your summer and don t forget about the packet. August will be here before you know it! If you lose your packet, you will be able to access the packets on-line at the school website, www.oprfhs.org starting June 7 th. Extra copies will also be available in the counselor s office. See you in August! The OPRFHS Math Department I understand that the purpose of the summer packet is to review the topics my students has already mastered in math and to prepare for the next course
(Parent/Guardian Signature) (Date) SUMMER PACKET For Students Entering AAF Name Welcome to Algebra! This packet contains the topics that you have learned in your previous courses that are most important to this class. This packet is meant as a REVIEW. Please read the information, do the sample problems and be prepared to turn this in when school begins again. Enjoy your summer! I. Using your graphing calculator: (A TI-Nspire is required for this class.) Be able to perform basic operations on a graphing calculator including addition, subtraction, multiplication and division. Have an understanding of the order of operations and use parenthesis correctly. To enter ( ) = your keystrokes must include the parenthesis. = 9 where as = 9 Be able to graph lines on the graphing calculator.
. Use the graph button and enter the line f( x) = x. Your graph should look like the one below.. When graphing lines with slope equal to a fraction be sure to enter the equation, using parenthesis. Graph Y = x by entering f( x) = ( / ) x II. Solving equations and inequalities Be able to use the Addition Property of Equality: If a = b then a + c = b + c Be able to use the Multiplication Property of Equality: If a = b then a c= b c Solve linear equations and inequalities Problem: Solve the following equation using the Addition and Multiplication Property of Equality. x = x + = + Addition Property of Equality x = 7 x = 7 7 x = Simplify Multiplication property of Equality Problem: Solve the following inequality using the Addition and Multiplication Property of Equality. 6 7 y 0y 6 + 6 7y 6 + 0y Add 6 7 y 0y 0 0y 7y 0y+ 0y 0 Add -0y 7y 0 ( 7 ) y ( 0) Multiply by 7 7 divide by 7 7 and reverse the sign OR
0 y 7 Solve Absolute Value equations x = x if x is nonnegative, and x = x ( the inverse of x) if x is negative 5 = 5 8 = 8 Problem: Solve the following:. x = so x = or x =. 5x = 5x = or 5x = Separate into two equations using plus and minus. 5x = 5 or 5x = 7 Add to both sides 7 x = or x = 5 Multiply by 5 III. Functions Be familiar with function notation. Know that y = x can be written in function notation as f ( x) x =. Given a function, be able to find the values of f x = x, find each of the following: a) f 0 = 0 = Problem: Given b) f = = 9 = 5 c) f = = 8 = 5 f x. Be able to identify the domain and range from a set of ordered pairs or from a graph. The domain is the set of all first members in a relation and the range is the set of all second members in a relation. Problem: List the domain and range of the following relation: { 5,, 6,, 8,6 } Domain {5, 6,8} Range {,,6}
. Problem: List the domain and range of the following relation:. Domain: All real numbers Range: y Domain: {,, } Range: {,, } Be familiar with linear functions and inequalities and their graphs.. The graph of any linear equation is a straight line. y = mx + b is the slope intercept form of a linear equation. The y-intercept of a graph is the y-coordinate of the point where the graph intersects the y-axis and is represented by b. The slope of the line is represented by m.. Determine the equation of a line using the point slope equation. Error! Objects cannot be created from editing field codes. y y. To determine the slope of a line use the equation m = x x Problem: Find a linear function with a slope of and a y-intercept of 7. m = and b = 7 so the equation is y = x 7 Problem: Given the points ( 6, ) and (,5) find the equation of the line a) In slope intercept form: Step : Use equation above to determine slope 5 9 m = = = 6 9 Step : Use one of the two given points and the slope from above in the point-slope equation. 5
x = 6 ( ) y = y = x 6 m = Step : Simplify b) In standard form: y+ = x+ 6 y = x+ x+ y = Problem: Graph the following line using the slope and y-intercept. Using: m=, b= : 5 0,. ) Plot the y-intercept ) Use the slope and move down units and right 5 units to plot the point 5,8 OR move up units and left 5 units to plot the point ( 5,0 ). ) Your graph should look like the to the right. x+ 5y = 0 First put equation into slope-intercept form. y = x+ 5 Problem: Given the graph below, determine the equation of the line. 0 8 6-0 -8-6 - - 6 8 0 - - -6-8 -0 ) The y-intercept is, so b = ) Starting with the y-intercept of, move up units and right units to the next exact point on the graph. This would be the point (, ) ) The change in the y value is in the positive direction and the change in the x value is in the positive direction. Therefore the slope is m = ) Using the slope intercept equation and substituting the values of m and b into y = mx + b y = x 6
Graph absolute value functions using a table of values or by splitting the equation into it s two parts. x 0, f x = x x 0, f x = x Problem: Be able to graph the absolute value equation y x points AND by graphing the separate equations below. = + by making an x-y table and plotting Making a table of values and plotting points: y = x +, for all y 0 y = x, for all y 0 x y - - - - 0 0 Graph parabolic functions and absolute value functions and be able to translate (or shift) the f x = x graph of Problem: Given the parent function f ( x) = x translate the given functions accordingly. f x = x ** f x = x ** Where the graph shifted units to the right. ** f ( x) = x+ ** f ( x) = x + Where the graph shifted units down. 7
IV. Systems of Equations Solve systems of equations by graphing. y x= Problem: Solve graphically: y+ x= Solving both equations for y and graphing yields: Solving for y y = x+ y = x+ Graph (see graph on right) The solution to the system is (, ), which is the point where the two lines intersect. Solve systems of equations by substitution. In the substitution method you must get one variable of either equation by itself. Then substitute its value into the second equation. Problem: Solve using substitution. x+ y = 6 x+ y = Solve the first equation for y. y = 6 x Since y and 6 x are equivalent, substitute 6 x for y into the second equation. x+ 6 x = Use the distributive property. x+ 8x= Solve for x. x = Substitute for x in either equation and solve for y. x+ y = 6 + y = 6 8
The solution is the ordered pair (, ) y = Solve systems of equations by linear combination, which is a combination of linear equations that will eliminate a variable. When using this method, first put the equation in the form of Ax + By = C. Make sure the coefficients of either variable are opposites of each other. Add both equations together. Problem: Solve using linear combination: y = x y = x Put in Ax + By = C form x y = + x+ y = 0 Add the x ' s and add the y ' s y = Solving for y yields: Substitute Solve for : y =. y = into either of the two original equations: x = The solution is the ordered pair, x + = 0 V. Properties of Exponents Be able to multiply same bases using properties of exponents, divide same bases and raise a power to a power. Know the following properties of exponents: For any real number a and integers m and n m m n m+ n a m n a a = a = a, a 0 n a m n mn a = a m m n p mp np a a ab = a b n = b b 0 a = a ** a = m ** a = ** = a m m a a Simplify exponents using the order of operations. Problems: p m mp np 9
** d) VI. Factoring a) x 5x = 5x 7 6xy b) = xy 9 8xy 8 c) xy = 8xy 9 x x x xy = ( ) = = y y y 9 Be able to factor without using your calculator. Factor terms with a greatest common factor (GCF). Problem: Factor 5x 0x = 5x x 5x 5x is the GCF 5x x Factor trinomials of the form Problem: Factor x x bx c + + x 0 One way to do this is to: Look for pairs of integers whose product is -0 and whose sum is -. Pairs of factors whose product is -0 Sum of factors whose sum is - -,5, -5-0, - 9-0, -9 The desired integers are and -5. 0 = + 5 x x x x Factor trinomials of the form ax + bx + c x + 5x+ Problem: Factor Look for pairs of numbers who product is and then for pairs of numbers whose product is. Trial and error yields: x + 5x+ = ( x+ )( x+ ) Factor the difference of two squares. Use: a b = ( a+ b)( a b) Problem: Factor x 6 = ( x+ )( x ) Factor x 9= ( x+ )( x ) Factor perfect trinomial squares Use: a + ab + b = a + b a ab+ b = a b 0
x 0x+ 5 = x 5 Problem: Factor + x+ 9 = x+ 7 Factor VII. Solving quadratic equations Solve quadratic equations by factoring (you need to be able to do this without using your calculator) and be able to use the quadratic formula. Problem: Solve the following quadratic by factoring. x 5x = 0 x 7 x + = 0 Factor x 7= 0 or x + = 0 Set equal to zero x = 7 or x = Solve Problem: Solve using the quadratic formula: x + 5x+ = 0 a=, b= 5, c= Using the quadratic formula and simplifying: ± x = The solutions are: 5 5 5 + 5 x = and x = 6 6 b ± b ac x=, a 0 a VIII. Simplifying radicals Simplify radicals. Know how to use the following theorems: a b = a b a a b = b Problem: a) 50 = 5 = 5 = 5 b a a a ) 7 = 7 = 7 Add, subtract, multiply and divide radicals
Add and subtract like terms. Problems: a) 6 = 6 = 8 = ) 8 8 5 6 b x y x = x x y = x y = x xy 80 80 c) = = 6 = 5 5 d) a b a = b e) 6 + = 8 f ) 6 = 8 a Rationalize the denominator. It is standard procedure to write a radical expression without radicals in the denominator. This process is called rationalizing the denominator. Problem: = Multiply by in form of = = 6 6 Simplify
Sample Problems Complete the problems below, showing work where necessary. Feel free to do your work on separate sheets of paper which you should attach. Remember you will be required to turn this in. An answer key is provided for you, but in math class, the work is equally as important than the answer! II. Solve the following equations and inequalities. Graph the solution to the inequalities on the number line.. 9y 7y 7 = 9 5y =. 5 + = [5 + ].. ( x ) ( x ) x + = 8 5. x 5 x x + = 5 6 6 6. 9x+ x
7. ( y ) 9 ( y+ 5) 8. 5x + = 7 9. 7 z + = 6 0. 5 x = 5. x. x <
. x < 5. a + 8 III. Functions 5. Determine whether the following are functions. If the relation is a function, state the domain and range: a) {(, ),( 7,9 ),(, ),(,6 )} b) {(,9 ),(, ),( 6, 9 ),( 7, )} 6. State whether the following are functions. If they are the graphs of a function, determine the domain and range. a) b) 5
Using your knowledge of linear functions, answer the following questions. 7. Graph the following using the slope and y-intercept. a) y = x+ 5 b) x+ y = 6 8. Find the slope of the line containing the following points. ( 8, 7) and (, ) 9. Find a linear function whose graph has the given slope and y-intercept. Slope of, and a y-intercept of ( 0,9 ) 0. Given the points (,) and (, 7) find the following: a) the equation of the line in slope-intercept form. b) the equation of the line in standard form 6
. What is the equation in slope-intercept form of the line graphed below. Using your knowledge of functions, answer the questions below. - graph on the axis provided. f ( x) = x. f ( x) = x. f x = x+ + 7
f x = x. 5. Evaluate the following for a) f b) f ( 0) c) f ( ) IV. Systems of equations Solve the following systems of equations by graphing. 6. x y = x+ y = 5 7. y = x x y = 8 8
Solve the following by substitution. 8. y = 5 x x y = 9 x y = 9. x y = 6 Solve the following using linear combination. 0. x+ y = 7 x+ y = 7. 5x 7 y = 6 x+ 8y = 6 9
V. Properties of Exponents. Multiply the following: 5 a) x x b) a 7a 9 5 6 5 7 c) ( a )( 7a ) d) ( mn)( mn p ). Divide the following: a) a a 9 b) t t 7 c) m n mn 9 6 d) 8xyz 8 6 7 xyz. Simplify the following: a) ( x ) 5 b) ( x y ) 5 c) ( 9m n p ) d) ( a bc ) 5 VI. Factoring Factor the following: 5. x x 6. 6xy xy + 0xy 5 0
7. x + x 6 8. x + 8x+ 9. x 6x + 0. x x 7x. x 6x. 6x x 5. x 9. x 6x+ 6
VII. Solve the following quadratic equations. Solve by factoring: 5. x + 8x+ 5 = 0 6. x 8x= 0 7. x 8x 0 + = 8. x x= 5 Solve using the quadratic formula: 9. x + 6x = 0 50. x 5x=
VIII. Simplifying radicals Simplify the following: 5. 0 5. 7 5. 5. 7+ 7 55. 8 6 + 5 56. 8 7 57. 9 50 Rationalize the denominator 58. 59. 6
Answers:. y =. y = 7. x =. x = 6 6. x 5. x = -5 - - - - 0 5 6 7. y 5 6 8. 9 x = or x = 5 0 5 6 7 8 9 0 9. z = ±. x or x 0. x = or x =. < x < -5 - - - - 0 5 6-5 - - - - 0 5 6. x < 8 and x > also < x < 8. 0 a and a 0 5 6 7 8 9 0-5 - - - - 0 5 6 5. a) Relation is NOT a function b) Relation IS a function 6. a) Function D : x ( all real numbers) R: y b) NOT a function 7. a) 7. b) 8. m = 9. y = x+ 9
0. a). y = x+ 5 b) x y = 5. y = x.. 5. a) 5 b) c) 5 6. (, ) 7. (, ) 8. (, ) 9. (, 5) 0. (, ). (, ) 8. a) x. a) b) c) d) 6 a 5 t 8 mn 6x yz 6 6 5 b) c) d). a) b) c) d) 8a 9 a m n p 0 x 6 9x y 0 9 5 9 79m n p 0 5 0 a bc
5. ( ) x x 6. x y ( x 6xy + 0y ) x( x+ )( x ) 7. ( x+ 9)( x 7) 8. ( x+ 6)( x+ ) 9. ( x x+ ) 8 6 ( x ) x( x 9)( x+ 8). ( x+ )( x 6). ( x+ )( x 5). ( x+ 7)( x 7). 0. ( x 7) x x x 8 5. x =, 5 6. x = 0, 8. x = 9, 5 7. x =, 9. x = ± 0 50. x = 5. 5 5. 5. 5. 5 7 55. 7 56. 57. 59. 58. 5 ± 57 6