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Brien McMahon High School SUMMER REVIEW ALGEBRA 2/ HONORS ALGEBRA 2 This packet is designed for you to review your Algebra 1 skills and make sure you are well prepared for the start of your Algebra 2 school year. Every topic includes an explanation, examples, and practice problems designed to ensure your readiness for next year. This packet is expected to be completed upon your arrival back to school and will be counted as a grade. *It is strongly recommended that calculators NOT be used to complete all sections of this packet since the objective of this packet is to verify your understanding of the concepts. Please show all of your work as this will be required of you throughout the entire school year. **It is strongly recommended to have a TI-83 or TI-84 graphing calculator for Algebra 2** Honors Algebra 2 – Complete sections I – XI Algebra 2 – Complete sections I - VIII Unit 10 I : Real Numbers and their Properties Number Systems: Natural Number (Counting Numbers) 1, 2, 3, … Whole Numbers (Introduce Zero) 0, 1, 2, 3, …. Integers (Introduce Negative Numbers) …, -3, -2, -1, 0, 1, 2, 3, ….. Note: The above number systems do not contain fractions or decimals. Rational Numbers (Numbers that can be written as the ratio of two integers.) When written as decimals, Rational Numbers terminate or repeat. 2 3 7 Examples: = 0.666…. = 0.75 7 3 4 1 Irrational Numbers - Number that when written as decimals neither terminate nor repeat. Examples: = 3.1415... 5 2.236067978 Real Numbers (All numbers on the number line.) This system combines the Natural #s, Whole #s, Integers, Rational #s, and Irrational #s. Properties of Real Numbers Let m,n, and p be positive real numbers. Property Addition Multiplication COMMUTATIVE m+n=n+m mn = nm ASSOCIATIVE (m + n) + p = m + (n + p) (mn)p = m(np) IDENTITY m + 0 = m, 0 + m = m m x (1) = m, 1 x (m) = m INVERSE m + (-m) = 0 1 m x =1, m 0 m DISTRIBUTIVE m(n + p) = mn + mp Additive inverse: the opposite of any m = - m Example: the additive inverse of 4 is -4 the additive inverse of -3 is 3 1 Multiplicative inverse: the reciprocal; the inverse of m is m 1 5 6 Examples: the multiplicative inverse of =3 the multiplicative inverse of 3 6 5 Unit 10 Practice Problems for Topic I: Complete the chart by placing a check in all the boxes that apply. Natural Whole Integers Rational Irrational Real Numbers Numbers Numbers Numbers Numbers -4 8 3 2 5 1 3 Identify the property being performed: 1. 5 4 4 5 2. 3(2 4) 4(2 3) 3. 4(1) 4 4. 10 10 0 1 5. 4(6 9) 4 6 4 9 6. 5 1 5 7. 0 36 0 8. (9 7)(5) 2(5) 9. (7 4) 1 7 (4 1) 10. 5(12 4) 5(12) 5(4) 11. (.5) (2) 1 12. (0.3)(0.3) 0 13. 36 jkm 36mjk 14. 29c 29c 0 Unit 10 II. Orders of Operations The rules for Order of Operations are as follows: FIRST: Perform operations inside grouping symbols. Grouping symbols include parentheses , brackets , braces , radical symbols , absolute value symbols and fraction bars. If an expression contains more than one set of grouping symbols, simplify the expression inside the innermost set first. Follow the order of operations within that set of grouping symbols and then work outward. SECOND: Simplify exponents. THIRD: Perform multiplication and division from left to right. (Remember that a fraction bar also indicates division.) FOURTH: Perform addition and subtraction from left to right. Hint: You can use the well-known phrase "Please Excuse My Dear Aunt Sally" to help you remember the Order of Operations. (Remember, however, that multiplication and division must be done in the order that they appear if they do not appear in parentheses. This is also true for addition and subtraction.) Please Excuse My Dear Aunt Sally Parentheses Exponents Multiplication Division Addition Subtraction Example 1 Simplify. - 4 2 24 3 2 (Note that there are no grouping symbols. Therefore the exponent only applies to the "4" and not the "-".) - 16 24 3 2 16 8 2 16 16 Example 2 Simplify. 4 (25 - (5 - 2) 2 ) 4(25 - (3) 2 ) 4(25 - 9) 4(16) 64 Example 3 Simplify. 3x – 4(8x + 2) + 5x distribute the -4 first 3x – 32x – 8 + 5x combine all like terms -24x – 8 Unit 10 Practice Problems for Topic II: Simplify each expression using the orders of operations. 3(4 11) 1.) 6 9 3 2.) 4( 2 3) 2 3.) 10 3 3 4.) 47 36 8 2 5.) 4( x 2) 3x 6.) 3(2 x 5) (5 3x) 3 2 3 1 14 2 7.) 8.) 2(x-1) – 5(x-3) 9.) 2(5 – 2)2 + 2(4 – 2)2 63 62 Evaluate each expression if a = 2, b = 3, c = -2, d = 4, f = -5 10.) cf 11.) c d 12.) ca 2 13.) (b cd )(ac) 14.) (cf ) (ab) 15.) dc a 16.) a2b2 17.) 5c + 3bc 18.) a – b – c – d Unit 10 Simplify each of the following. Hint – remember to use orders of operations and to combine like terms 19.) 3x – 4 + 7x – 8 – 10x – 2 20.) 4x + 2(x + 5) 21.) 3(x + 5) – 4(x – 6) 22.) 5x3 + 2x2 -7x – x3 + 5x2 – 18 23.) (5x2 + x – 4) – (9x2 – 4x – 11) 24.) 12x3(x4 – 5x2 + 2) 25.) (x + 2)(x2 – 5x + 1) 26.) 5xy – 20x + 6y – (10xy + 10x) 27.) 2x(3x2 – 4x + 2) – 5x(2x + 3) 28.) 5x2y2 + 3xy – 3x2y2 + 2x3y2 + 5xy Unit 10 III. Plotting Points on the Coordinate Plane Suppose you were told to locate "(5, 2)" where would you look? To understand the meaning of "(5, 2)", you have to know the following rule: The x-coordinate (the number for the x-axis) always comes first. The first number (the first coordinate) is always on the horizontal axis. So, for the point (5, 2), you would start at the "origin", the spot where the axes cross: ...then count over to "five" on the x-axis: ...then count up to "two", moving parallel to the y-axis: ...and then draw in the dot: *Remember: “- x” values moves left and “– y” values move down Unit 10 Practice Problems for Topic III: Plot the following points of the grid below and label them accordingly: 1.) A - ( 4, -5 ) 2.) B - ( 0, -4 ) 3.) C - ( -3, -1) 4.) D - ( -2, -2 ) 5.) E - ( 3, 0 ) 6.) F - ( 5,1 ) 7.) G -( -3, 7 ) 8.) H - ( 10, -2 ) 9.) I – ( -1,-4) 10.) J – (0,2) 11.) K – (-3,10) 12.) L – ( 4,0) 13.) M – (-2,1) 14.) N – (3,4) 15.) P – (0,0) Unit 10 IV: Solving Linear Equations Solving with a variable on only one side of an equations: Ex) -2x + 7 = 15 subtract 7 from both sides -2x = 8 divide -2 from both sides x = -4 -2(-4) + 7 = 15 check your solution using substitution 8 + 7 = 15 15 = 15 Solving with a variable on both sides of the equation: Ex) 8x – 10 = 2x + 14 Move the smaller x value to the other side 6x – 10 = 14 (subtracted 2x from both sides) 6x = 24 add 10 to both sides (your non x variable) x=4 divide both sides by 6 8(4) – 10 = 2(4) + 14 check by substitution 32 -10 = 8 + 14 22 = 22 Solve using the properties of real numbers: Ex) -3( 2x + 5 ) + 6x = 11x + 7 Use the distributive property -6x - 15 + 6x = 11x + 7 Combine like terms (on the same side of the equation) -15 = 11x + 7 subtract 7 from both sides -22= 11x divide both sides by 11 -2 = x -3( 2(-2) + 5) + 6(-2) = 11(-2) + 7 check by substitution -3(-4+5) - 12 = -22 + 7 -3(1) - 12 = -15 -15 = 15 Unit 10 Practice Problems for TOPIC IV: Solve the following equations and check your solutions. 1.) -3x + 4 = 11 2.) 1/2x - 8 = 3 3.) 8x – 24 = -6 + 18 4.) -6 + x – 4 = 7x + 2 2 5.) x + 6 = 18 6.) 6 – x – 5 = -4x – 3 – x 3 b 7.) -5 + =7 8.) 4.2m +4 = 25 4 9.) 4t + 7 + 6t = -33 10.) 6 = -z – 4 11.) 4m + 2.3 = 9.7 12.) 2a + 5 = 9a – 16 1 4 2 13.) + y= 14.) 6(y – 2) = 8 – 2y 3 6 3 m 15.) n + 3(n – 2) 10.4 16.) = 2.7 12 Unit 10 17.) -4.7 = 3x + 1.3 18.) 3w + 2 – w = -4 19.) Solve 2x -4 = -7. Justify each step. 20.) A taxicab company charges each person a flat fee of $1.85 plus an additional $.40 per quarter mile. A. Write a formula to find the cost for each fare. B. Use the formula to find the cost for 1 person to travel 8 mi. 21.) Find the dimensions of the rectangle given the area = 164 sq. ft. 3x+5 4 Unit 10 V: Using Slope and y-Intercept to Graph Linear Equations and to Write Equations of Lines There are 3 forms to write an equation of a line: Slope intercept form: y = mx + b m = slope, b = y intercept Point slope form: ( y-y1 )=m( x-x1 ) (x1, y1) is a point on the line Standard form: Ax + By = C y intercept: where x = 0 x intercept: where y = 0 *to find these values plug either x = 0 or y = 0 into the equation and solve for the other value I.) Graphing in slope intercept form, y = mx + b: Ex. 1 y = ( 2/3 )x – 4 Step 1: Identify “b” (0,-4) Step 2: Plot “b” on the grid Step 3: Use the slope to plot your next point (rise/run) ( 3, -2) , ( 6,0 ) Step 4: Continue step three until at least 3 points are plotted. Unit 10 Ex. 2 y = –2x + 3 1.) Plot “b” (0,3) 2.) Plot your slope (-2/1) Down 2 and right 1 3.) Repeat this for 3 points 4.) Connect your points Ex. 3 Ex. 4 X=3 This is a vertical line with an undefined slope y = 4 This is a horizontal line with a slope = 0 II.) Writing equations of lines in slope intercept form: Ex. 1 – Write and equation of a line that passes through the point (2,1) and has a slope of 3. y = mx + b plug 3 in for m, 2 in for x, and 1 in for y 1 = 3(2) + b simplify 1=6+b solve for b by subtracting 6 -5 = b y = 3x – 5 plug 3 in for m and -5 in for b Unit 10 Given two points (x1, y1) and (x2, y2), the formula for the slope of the straight line going through these two points is: Ex. 2 - Write an equation of a line that passes through the points (2, 4) and ( 5, 6 ). 6–4 start by finding slope using the formula above plug the slope and one of the points into the Point Slope Formula y – 4 = 2/3( x – 2 ) simplify by distributing the 2/3 y – 4 = 2/3 x – 4/3 add 4 to both sides to get y by itself y = 2/3 x + 8/3 Parallel Lines: two lines that have the same slope and never intersect Ex. y = 2x + 3 is parallel to y = 2x -10 Perpendicular Lines: two lines whose slopes are opposite reciprocals Ex. y = 2x + 3 is perpendicular to y = -1/2 x *2 and -1/2 are opposite reciprocals Practice Problems for Topic V. Sketch each of the lines on the graph below. 1. y = 2x 2. y = -3x Unit 10 3. 4. 5. y=6 6. x = -4 7. 8. Unit 10 9. y 3x 2 10. y = -4x+6 11. x=2 12. y = -5 13. 2x + 3y = 12 14. -3x + 27y = 8 Unit 10 15. Complete the table and graph the points on the graph below. 16. An hot air balloon is currently at an altitude of 10,000 feet. The pilot begins to descend the balloon at a rate of 50 feet per minute. 1. Write an equation for the altitude (a) of the balloon as a function of the time (t) 2. Find the altitude of the balloon after: a. 10 minutes b. 30 minutes c. 1 hour 3. How long will it take for the balloon to be: a. 7000 feet b. 2000 feet c, on the ground Unit 10 4. Complete the table and graph the altitude of the balloon as a function of the time in minutes that the balloon takes to begin its descent and land on the ground. 17. This summer you and a friend recorded your first CD. In order to produce your CD you need to hire a recording studio. One studio you called charges $250 for making a master CD and $3 to burn each CD after that. (a.) Write an equation to model the cost of burning the CDs. ________________________ (b.) How much will it cost to burn 75 CDs? ________________________ (c.) How many CD’s can be made for $3000? _______________________ More practice for Topic V. 1. y = 4x – 2 2. 6x – 3y = 15 slope = _______ slope = _______ y-intercept = _______ y-intercept = _______ x-intercept = _______ x-intercept = _______ Unit 10 3. y = 2x – 5 4. 4x + 3y = 12 slope = _______ slope = _______ y-intercept = _______ y-intercept = _______ x-intercept = _______ x-intercept = _______ 5. 3y = 4x – 9 6. 5x – y = 15 slope = _______ slope = _______ y-intercept = _______ y-intercept = _______ x-intercept = _______ x-intercept = _______ Find the slope and y-intercept for each line given below. Then write the equation in slope-intercept (y = mx+b) form. 7. A line that passes through the point (-5,6) with a slope of 4. 8. A line that passes through the points (4,1) and (7,-11) 9. A line is perpendicular to the line in #2 above and passes through the x-axis at 3. Unit 10 Find the slope and y-intercept and x-intercept of each line. Then write the equation of each line in y = mx + b form. 10. 11. Slope = _________ Slope = ________ y-intercept : ________ y-intercept: _________ equation: y = _________________ equation: y = ________________ x-intercept: _______ x-intercept:_______ 12. 13. Slope = _________ Slope = ________ y-intercept : ________ y-intercept: _________ equation: y = _________________ equation: y = ________________ x-intercept: _______ x-intercept:_______ Unit 10 14. 15. Slope = _________ Slope = ________ y-intercept : ________ y-intercept: _________ equation: y = _________________ equation: y = ________________ x-intercept: _______ x-intercept:_______ 16. 17. Slope = _________ Slope = ________ y-intercept : ________ y-intercept: _________ equation: y = _________________ equation: y = ________________ x-intercept: _______ x-intercept:_______ Unit 10 18. A tanker that is filled with oil pulls into an oil refinery. The tanker is going to offload the oil as shown in the graph below. 1. How many gallons of oil are in the tanker when it pulls into the refinery?_____________ 2. At what rate, in gallons per hour, is the tanker unloading the oil?___________________ 3. Write an equation for the amount of oil (l) after h number of hours: l = _____________ 4. Find the amount of oil after: a. 15 hours b. 25 hours 5. How long will it take for the tanker to have: a. 60,000 gallons left b. 25,000 gallons left 6. How long will it take? Unit 10 VI. Quadratics Multiplying Binomial Expressions: F.O.I.L. (First, Outer, Inner, Last) FOIL is the method by which two binomials can be multiplied using the distributive property. Example 1: Multiply the binomials using FOIL. x 3x 2Take the first term of the first binomial and distribute it through the second binomial. Then, take the second term in the first binomial and distribute it through the second binomial. x x x 2 3 x 3 2 ( x 2 ) (2 x) (3x) (6) ( x 2 5 x 6) Example 2: (x – 4)(x + 5) ( x x) ( x 5) (4 x) (4 5) ( x 2 ) (5 x) (4 x) (20) x 2 1x 20 Factoring Quadratic Trinomials: (x 2 bx c) : When factoring quadratic trinomials with no coefficient in front of the x 2 , ask yourself this question: What two numbers add to give "b" and multiply to give "c"? Example: Factor: x 2 7x 12 What two numbers add to give "7" and multiply to give "12"? Obviously, the answer to this question is 3 and 4. Therefore, the factors of x 2 7x 12 are x 3x 4. Example 2: Factor: x 2 - 10x 16 What two numbers add to give "-10" and multiply to give "16"? The answer to this question is -2 and -8. Therefore, the factors of x 2 - 10x 16 are x 2x 8. Graphing Quadratics Properties of quadratics: PARABOLA: graph of a quadratic function ROOTS: Value of x when y = 0 (where the graph crosses the x axis) VERTEX: maximum or minimum point on a parabola Unit 10 To graph a parabola: Fill in the t-chart (function table) by choosing values of x, then plug those values into your function to find y, then plot your ordered pair. Example: x x 2 4x 5 y -2 (-2)2-4(-2)-5 7 -1 (-1)2-4(-1)-5 0 0 (0)2-4(0)-5 -5 2 (2)2-4(2)-5 -9 5 (5)2-4(5)-5 0 The vertex is (-2,9), the lowest point of the graph. The roots are (-1,0) and (5,0), the two places where the parabola crosses the x axis. Practice Problems for Topic VI: Foil: 1.) (x – 2)(x + 6) 2.) (3x + 6)(x – 1) 3.) (x + 9)(-2x – 7) 4.) (5x + 1)( 3x – 2) Factor: 1.) x 2 x 56 2.) x 2 14 33 3.) x 2 7 x 10 4.) x 2 11x 12 5.) x 9 x 20 6.) x 2 10 x 16 7.) x 2 x 56 8.) x 2 19 x 90 2 9.) x 14 x 49 10.) x 2 x 72 11.) x 2 2 x 24 12.) 2 x 2 5 x 3 2 13.) x 4 x 4 14.) x 2 10 x 25 15.) x 2 9 16.) x 2 36 2 Unit 10 Complete the following chart and graph the function. Identify the vertex and the roots. 1.) x x 2 4x 4 y 2.) x x 2 6x 2 y 3.) x x 2 7x 6 y Unit 10 VII. Properties of Exponents Unit 10 VIII. Solving Systems of Equations Solving systems using: Linear combination Given the equation: 3x – 4 = 7x + 8 -3x -3x subtract 3x from both sides to get the x terms together -4 = 4x + 8 -8 -8 subtract 8 from both sides to get the term by itself -12 = 4x 4 4 Divide both sides by 4 (opposite operation) to solve for x. -3 = x Solution Substitution Given two linear equations: y = 5x -1 and -2x + 3y = 23 y = 5x -1 substitute (replace) y in the equation -2x + 3y = 23 with 5x -1 -2x + 3(5x -1) = 23 -2x + 15x -3 = 23 Distribute (multiply) 3 to both 5x and -1 13x – 3 = 23 Combine like terms (-2x and 15x) +3 +3 Add three to both sides 13x = 23 Divide by 13 to solve for x. x=2 Take 2 and substitute it into the equation y = 5x – 1, for x y = 5x – 1 y = 5(2) – 1 Solve for y. y = 10 – 1 so, y = 9 Your solution then is x = 2 and y = 9 written (x,y) as (2,9) Unit 10 GRAPHING: To solve the system by graphing, graph both linear equations on the same coordinate plane and their intersection point is their solution. Practice problems for Topic VIII. Solve the following systems by the indicated method. Linear Combination 1. 4x + 3 = 2x – 8 2. 9x – 5 = -6x + 25 3. -2x – 1 = -3x + 11 4. 10x – 40 = 6x Substitution 1. y = -x + 7 2. y = 3x – 5 3. x = y + 5 4. y = -7x 2x + 3y =14 -4x + y = -11 2x – 4y = 2 4x – 5y = -39 Graph 1. y = -3x – 2 2. y = -2x + 5 3. y = -x 4. y = 3x + 2 y = 2x + 8 y = 2x – 13 y=x+5 y = -3x – 4 Unit 10 IX. Solving Linear Inequalities EX 1: Solving linear inequalities is almost exactly like solving linear equations. Solve x + 3 < 0. Then the solution is: x < –3 -3 0 When the inequality sign is just greater than (>) or less than (<), we use an open circle to show the answer does not include this number. EX 2: Here is another example, along with the different answer formats: Solve x – 4 > 0. Then the solution is: x > 4 0 4 When the inequality sign is just “greater than or equal to” ( ), or “less than or equal to” ( ), we use a closed circle to show the answer includes this number. Solve –2x < 5. To solve "–2x < 5", I need to divide through by a negative ("–2"), so I will need to flip the inequality: * When solving inequalities, if you multiply or divide through by a negative, you must also flip the inequality sign. 2 x 5 5 x > 2 2 2 Practice Problems for Topic XI: 1. x – 7 < 1 2. x + 9 ≤ -6 3. -4y + 2 > 14 4. 16 – 3x ≥ 4 Unit 10 5. Wireless company A has a plan that charges $20 initial fee and $0.05 for every call. Wireless company B has a plan that has NO initial fee but charges $0.25. How much will it cost in each of the two different plans if you make 80 calls one month and then 110 calls the next month? Write two different equations, one for each plan and then graph both. Describe when you would use each of the different plans. Unit 10 X.Graphing Linear Inequalities Practice problems for Topic X: Unit 10 Unit 10 XI. Absolute Values and Their Graphs The absolute value of a number is its distance from zero. The graph of an absolute equation has a “V” shape because for any input value of “x” (either positive or negative) the output must be the corresponding positive “y” value. Think of the linear function y = x. It is a line of positive slope 1, going through the origin. You can calculate that if you input a positive “x” you’ll get a positive “y” out, a negative “x” input gives a negative “y” out. What happens with the function y = |x| ? Positive in yields positive out, but what about negative values for “x” ? They turn positive also, thus creating the “V” shaped graph. To produce an absolute value graph on your graphing calculator, go to your “y=” section (functions) , clear existing equations, press the MATH key, scroll over to the NUM pulldown, and choose “abs( “ . Enter “x”, and close parentheses. Press the ZOOM key and then press the number“5”. On screen you should have a 90degree “V” shape with its vertex at the origin. This is the most basic standard form of an absolute value equation. Any alteration will cause a transformation. Practice Problems for Topic XI: 1) y = abs(x) + 5 2) y = abs(x) – 5 3) y = abs(x – 5) 4) y = abs(x + 5) What has happened in each case? (how has the figure moved?) 1) ____________________________________________________________________________________________ 2) ____________________________________________________________________________________________ 3) ____________________________________________________________________________________________ 4) ____________________________________________________________________________________________ For number three and four, did the figure move in the direction you expected? _____________________________ Now try the following: 5) y = .5 abs(x) 6) y = .75 abs(x) 7) y = 2 abs(x) 8) y = 5 abs(x) What has happened in each case? (how has the figure changed?) 5) ____________________________________________________________________________________________ 6) ____________________________________________________________________________________________ 7) ____________________________________________________________________________________________ 8) ____________________________________________________________________________________________ What can you say about the effect of numbers between zero and one on the shape of the figure? ______________ ______________________________________________________________________________________________ What can you say about the effect of numbers greater than one on the shape of the figure? __________________ Unit 10 Write the absolute value equation for each graph shown. 1. y = _____________________ 2. y = ____________________ 3. y = ____________________ 4. y = _____________________ 5. y = ____________________ 6. y = ____________________ 7. y = _____________________ 8. y = ____________________ 9. y = ____________________ Unit 10 10. y = ____________________ 11. y =___________________ 12. y = ___________________ Unit 10