Basic ALGEBRA 2 SUMMER PACKET
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1 Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout Algebra II. There are a few eample problems at the beginning of each section, followed by practice problems for you to complete. You are epected to know how to do these problems and to have this packet done before returning to school in the fall. You will have a quiz on the summer packet when you get back to school. You will have approimately one week to check your answers, which are included in the back of the packet, and ask questions before taking several mini-quizzes on the material. These quizzes will be used to assess your understanding of all the concepts in this summer packet. Have a great summer!!
2 Table of Contents Glossary of Terminology and Formulas Page 3-4 Factoring Polynomials Page 5 7 Solving Quadratic Equations Page 8-9 Solving Equations for a Variable Page 10 Solving Systems of Equations in Two Page 10-1 Variables Eponents Page 13 Radicals Page 14 Multiplying Polynomials Page 15 Writing the Equation of a Line Given Two Page 16 Points Graphing Linear Equations Page 17 Answer Key Page.
3 Glossary of Terminology, Formulas and Rules Sum Quotient Product Difference Absolute Value Simplify Term Definition/Eample The answer to an addition problem. The answer to a division problem. The answer to a multiplication problem. The answer to a subtraction problem. Make negative numbers positive and keep positive numbers positive. To make less comple; is simplified Solve for value Determine the value of the unknown; ( )( + 3) = 0 = and = -3 are the solutions Solve for a variable Isolate the desired variable, usually or y (see Solving Equations on page 9) Evaluate Determine the value when there is no unknown; 4(3 + ) = 0 Reciprocal 0 is the evaluated answer Interchange the numerator and denominator. Given 5 4 reciprocal is 3. Given 3 (remember 3 ), the reciprocal is 1 Slope Rise over run or the change in y over the change in Distribute 3( ) = 3 6 Coefficient The number in front of a variable Variable A symbol that represents a number, usually a letter Constant A number without a variable Eponent The power to which a base is raised Base The epression be raised to a power Binomial The sum or difference of two terms Trinomial The sum or difference of three terms Polynomial The sum or difference of two or more terms Like Terms Terms that have the same variables and powers Greatest Common Factor (GCF) 9 3 y 6y, 3y is the GCF Inequality Epressions that are not equal; 3 11 Ordered Pair The and y value of a point on a graph Parabola The graph of a quadratic function Proportion Statement of equal fractions Domain The set of all possible -values Range The set of all possible y-values Radical An epression that has a square root, cube root, etc. Scatterplot A graph that contains a series of and y coordinates System of Equations Multiple equations containing multiple variables 4 5, the 1 3
4 Solution The answer to an equation Undefined Slope A vertical line has an undefined slope Slope of Zero A horizontal line has a slope of zero Perpendicular Two lines intersecting at a 90 angle Parallel Two lines with the same slope Formula Quadratic Formula b b 4ac Given a b c 0; a Slope y y1 Given two coordinate pairs; m Distance 1 Given two coordinate pairs; d y y 1 1 Midpoint 1 1 Given two coordinate pairs;, y y Pythagorean Theorem With legs a & b and hypotenuse c; a b c Slope/Intercept Form of a y = m + b with m = slope, y-intercept at (0, b) and (, y) Linear Equation represents any point on the line 4.
5 Factoring Polynomials GCF stands for greatest common factor. This is the largest number (including variables) that divides evenly into a group of terms. Eamples of finding the GCF of the following groups of terms. 3 6, 4, , 4 & 30 all have a 6 in common., 3, all have an in common So the GCF is 6 1. First look to see whether they have a number factor in common.. Net, determine which variables each term has in common. Take out the smallest eponent. a , 50, y, 4, 40y b. c. 5 y 3y 7 3 3, 18, 6 y y y In Algebra 1, you learn to multiply using the distributive property. Eample: Distribute 4( 1) = 8 4 FACTORING is the opposite of multiplying. In other words, when you factor a polynomial, you write it as a multiplication problem. If a polynomial has a GCF, one way to factor it is to use the reverse distributive property. Eample: Factor 3 7. What is the GCF of all of the terms? Write it on the line below, then work backwards with the distributive property to fill in the terms in the parenthesis. ( - 9) GCF 5.
6 Practice factoring the following polynomials: TRINOMIALS When factoring a trinomial, you should always look for a GCF first. For now, we will focus on how to factor the trinomials. Trinomials are usually of the form a b c. When a=1, the trinomial is very easy to factor Eample: 9 0 ( - 4) ( - 5) (start with times to get ) Note: Diamond problems may help here! What factors of 0 add up to -9? Try When a is a number greater than 1, the process requires more of a guess and check method. Eample: 1 ( ) ( ) (Start with factors of ( 3) ( 3 7) (Find factors of the last number and write them in. It s okay to guess. You have to try something to get started. To see if you have the correct combination, test the inside and outside products. Can you get them to add up to the middle term? If not, switch your numbers around and try again.) 14.) ( 7) ( 3) These are not going to add up to -, so try again. 7 6 This combination works! The 7 would need to be negative, so the answer is ( - 7) ( + 3). 6.
7 Try the following problems: a 16a r 11r 3 MIXED FACTORING As stated earlier, you should always look for a GCF first before factoring the trinomial. Eample: What is the GCF of ? Factor this out first, then factor the trinomial. *3 is the GCF 3(5 7) 3(5-7)( + 1) Try the following Factor the following polynomials completely: 13. 3n 8n
8 SOLVING QUADRATIC EQUATIONS The key component to a quadratic equation is an equals sign. If it does not have an equals sign, it is not an equation and CANNOT be solved! At this point, we have learned two ways to solve quadratic equations: ZERO PRODUCT PROPERTY QUADRATIC FORMULA Must be factored & set equal to zero. Can be used to solve any quadratic equation. ZERO PRODUCT PROPERTY: 1. Set equation equal to zero.. Factor completely. 3. Set each factor equal to zero and solve. Eample: 4 = = 0. ( 7)( +3) = = 0 and + 3 = 0 = 7 = -3 Eamples for you to try: y y 40y
9 QUADRATIC FORMULA: 1. Set equation equal to zero. QUADRATIC FORMULA:. Apply quadratic formula. a +b + c = 0 = b ± b 4ac a EXAMPLE: = 0 = 4 ± (4) (4)(3)( 7) (3) 4 ± = 6 4 ± 100 = 6 4 ± 10 = 6 = 1, 7 Eamples for you to try
10 SOLVING EQUATIONS FOR A VARIABLE SOLVING FOR Y: We often need to manipulate an equation so that it is solved for a different variable. For eample, when graphing, it is often more helpful to have an equation in y= form. Solve for y does not necessarily mean that you will find a value for y. It means that the equation should say y= when you are finished. EXAMPLES: Solve the following equations for y. 7( y 8) 3( ) 8 7 7y = 8 7 (Combine ) 7y = 8 7 (Subtract 3 from both sides.) 7y 50 = 5 7 (Add 50 to each side.) 7y = (Divide by 7 on both side.) y = Try these eamples: y y 1 5 6( )y Solving Systems of Linear Equations in Two Variables Solving by substitution Consider this system: 10y y 4 Look for the equation that is easiest to solve for or y. In this case, we chose to solve the second equation for. 4y 4 4 4y y Now replace the in the other equation with (- - y). 10y 3( y) 14 10y 6 6y 14 16y y 8 y
11 Find by substituting 0.5 for y in either original equation. (.5) 3 The solution is the coordinate pair (-3, 0.5). NOTE: When both variables drop out of the equation, you either have no solution (false statement), or infinitely many solutions (true statement). Solving by Elimination First, rewrite the equation so that the s and y s are lined up vertically. Net, decide what to multiply by to make the coefficients of either the s or the y s the same numbers with opposite signs. Consider the same system as above: 10y y 4 In this case we chose to make the coefficients opposites: multiply the top equation by two and the bottom equation by three to get: 0y 6 8 1y 6 1 If we add the two equations together, we eliminate the : 3y 16 y 0.5 Finally, go back and substitute 0.5 for y in either original equation: 10(0.5) Again, the solution must be written as a coordinate pair (-3, 0.5). NOTE: When both variables drop out of the equation, you either have no solution (false statement), or infinitely many solutions (true statement). Eamples Use either substitution or elimination. Determine which one would be easiest before beginning. 30. y = y = 7 + y = 15 9y =
12 Solve each of the following equations for y (3y ) 7 6y y Solve the following systems of equations. Choose the best method for each problem y = y = y = 13 y = = 4y y = 10 4y = 5 4y = 38. y = y = y = y = 10 1.
13 EXPONENTS RULES: EXAMPLES: 1) m n = m + n b 5 b =(bbbbb) (bb)=b 7 ; 9 4 = 13 ) m n m n a 8 6 a a ; 5 6m 3m 4 m 3) m n m n ( ) ( ) ; 7 14 (4b ) 16b Eample Problem: EXAMPLES for you to try: ab 3 7 8ab 11 5 (3 z ) 0 6 z 3 z 0 6 z 10 9 z 0 6 z 9 z z y y y v y v 45. y 3 y 46. y y 48. y y
14 RADICALS PERFECT SQUARES TO MEMORIZE: 1 = 1 = 4 3 = 9 4 = 16 5 = 5 6 = 36 7 = 49 8 = 64 9 = = = 11 1 = 144 SIMPLIFYING RADICALS In order to simplify a square root, you must rewrite it as the multiplication of the factors! Be on the lookout for perfect squares. For eample, simplify the following square root: What are factors of 45? is the perfect square! E 1: 7 E : = = 6 Simplify the following SIMPLIFYING RADICALS Rewrite the following radical epressions in simplest form
15 Multiplying Polynomials You might want to use a generic rectangle. Eamples: a) (3 )(4 5) b) ( 1)( 3) Multiply and simplify. 57. ( 3) 58. (4 )(3 5) 59. ( 3)( 7) 60. ( 5)( 3) 15.
16 Writing the Equation of a Line Given Two Points Find the equation of the line through the given points. Eample ( 6, 5 ) and ( 9, ) Find the slope between the two lines using the slope formula. y y 1 1 m m Plug this value into the general equation of a line, y = m +b Also, pick either of the coordinate points to use and plug in the values of and y into the equation. You can now solve the equation for b. 1(9) b 9 b 11 b Therefore the equation of the line is y 1 11 OR y 11 Find the equation of the line through the given points. 61. ( 0, 4 ) and ( -1, -5 ) 6. ( 4, ) and ( 8, -1 ) 16.
17 GRAPHING LINEAR EQUATIONS When an equation is in slope intercept form, it is easy to graph! Slope-Intercept Form: WARM UP: Graph the line y = ½ + 3 Remember y = m +b -Start with b, the y-int: (0, 3) -Use the slope m = ½ to graph the line (rise over run, from the y-intercept go up one, then right two) Graph each of the following equations on separate sets of aes. 63) y 7 64) 3 y
18 Solve the following for y and then graph. 65) 4 y 6 66) 4 y 1 67) + 3y = 7 68) 3 + ¼ y = 18.
19 Answers to the practice problems throughout the packet. 1.) 3( + 5).) ( + 6) 3.) 3(3 1) 4.) ( )( 3) 5.) ( 9)( + 3) 6.) ( 3)( + 7) 7.) (5 )( 4) 8.) (3a + )(a 6) 9.) (4r + 1)(r 3) 10.) (3 4)( + 1) 11.) ( + 3)( 1) 1.) 5( + 1)(3 5) 13.) (3n + 5)(n + 1) 14.) (3 )(3 + ) 15.) ( 1)( + ) 16.) ( + 6) 17.) ( 3)( + 3) 18.) 3( + 6) 19.) = -8, 0.) 1.) y = -0, -, 0.) 4 6 4, 3 3.) ) =, 5 5.) ,1.8 6.) , ) 8.) y = , ) y = ) 3 1 y 4 30.) (6, 9) 31.) (4, -3) 3.) 10 y 9 34.) (-7, 3) 35.) (3, 8) 36.) Infinitely many solutions 37.) (, ) 38.) (, -5) 39.) No Solution 41.) 7 18y 4.) 45.) 6 5 y 3 46.) y ) 49.) 6 50.) 40.) 3a b y 43.) 7 44.) y 5 v y 48.) 53.) ) 55.) 56.) 1 57.) ) ) ) ) y = ) y = ) ) y
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