Bishop Kelley High School Summer Math Program Course: Algebra 2 A
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1 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 16 pages of this packet provide eamples as to how to work some of the problems in the packet. You will keep pages 1 16 for yourself for reference and you will turn in everything starting with p. 17. Please make sure and read the course requirement regarding calculators which is at the top of the first page 17. No matter when you have math, this packet is due on the day of orientation This material will be graded, and points awarded at the discretion of each teacher. A test on this material will be administered during the first week of the class. An additional resource for help with this packet is It provides videos of about 10 minutes in length on hundreds of different math topics. Math Teachers will be available in C 1 the following dates/times if you need help. Date Monday, July 7 th Thursday, July 0 th Monday, August rd Time 9 10:0am 9 10:0am 9 10:0am 1 P age
2 ORDER OF OPERATIONS: 1. Do operations that occur within grouping symbols (parentheses, brackets, absolute value bars, radicals).. Evaluate powers if there are any.. Then do multiplications and divisions as you see them, in order, from left to right.. Finally, do additions and subtractions as you see them, in order, from left to right. EXAMPLES: 16 Since there are no grouping symbols or eponents, do the division first 16 From left to right, do the addition net Since there are grouping symbols, do inside them first Net do the powers Now do inside the brackets Time to multiply 10 7 From left to right, do the subtraction 6 60 EVALUATING VARIABLE EXPRESSIONS: 1. Substitute values for the variables.. Simplify using the order of operations. EXAMPLES: 5 ; 1 ; 5, 15 y y ;, y rh r h 1 () ( ) P age
3 COMBINING LIKE TERMS: Like terms have the same variables and the same eponents. You can add and subtract like terms by combining the coefficients (the numbers in front) and leaving the variables and eponents the same. EXAMPLES: y y yy yy a b a b 6a1bab 9a10b y 7 10y 7 8y 79y 17 9y SOLVING EQUATIONS: *Eliminate any fraction by multiplying the entire equation by the LCD. *Distribute and combine like terms as needed. *Get the variable by itself by moving across the equal sign, again always looking for like terms to combine. *If a variable drops out, look at the remaining part of the equation. If you are seeing a true statement, then the solution is all real numbers. If you see a false statement, then the solution is the empty set. EXAMPLES: 7 a 6a8a 7a16a8a 7a0a8 a 8 a True--all real numbers False--empty set P age
4 LITERAL EQUATIONS: These equations use the same process as any equation, they simply contain more than one variable. EXAMPLES: 1 A bh for h A bh A h b L b q c for q L bcqc L bccq Lbc q c P rsst rt for s Prt rsst P rt s(r t) P rt s r t SOLVING AND GRAPHING LINEAR INEQUALITIES 1. Solve as you would an equation.. If multiplying or dividing by a negative, reverse the inequality symbol.. On the number line, an open circle is used for the > and < symbols and a closed dot is used for the and symbols. The > and will have the line pointing to the right and the < and will have the line pointing to the left. EXAMPLES: (Look at the intersection of the graphs) 1 or 57 or 1 1 or >6 (Solve each inequality separately) P age
5 GRAPHING LINEAR EQUATIONS: Methods for graphing equations: 1. Table of values. a. Rewrite equation in slope intercept form.. b. Using the chosen value, solve for y. c. Plot the points on the graph and connect.. Intercept method. a. Set equal to zero and solve for y to find the y-intercept. b. Set y equal to zero and solve for to find the -intercept. c. Plot the intercepts and connect.. Point slope. a. Given one point and the slope, plot that point on the grid. b. Start at the point and use rise over run to count the boes to the net point. c. Plot the point and connect.. Slope and y- intercept. a. Using ym b form, m is the slope and b is the y intercept. b. Plot the y intercept. c. Count rise over run to the net point. GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES *Put in slope intercept form. *Graph using slope intercept form. *Make a boundary line. The symbols > and < use a dotted line and the symbols and use a solid line. *Use a test point. (The origin works nicely.) Plug the coordinates of the test point into the inequality. *If the result is a true statement shade the side including the test point. *If the result is a false statement shade the other side. FORMULA FOR SLOPE: Given two points, and, y y, the formula for slope is: 1, 1 y y m P age
6 WRITING EQUATIONS OF LINES: POINT-SLOPE FORM: y y m 1 1 SLOPE-INTERCEPT: ym b STANDARD FORM: ABy C *Given the slope and y intercept: m, y intercept b 7 5 y 7 5 *Given the slope and a point: Method 1 Method through 8, 1 Use the slope and point and put in the slope intercept form to find b : through 8, 1 Use the slope and point and put in the point slope form: m m y mb 1 8 b 16b 5 b Equation: y 5 y y m 1 1 y1 8 y1 6 y 5 *Given two points: Method 1 Method *Given two points: Method 1 Method,1 and, 9,1 and, 9 Find slope: 1 ( 9) 10 ( ) 5 Use one of the given points (it does not matter which one) and the slope and put in the slope intercept form to find b : ymb 1b 1b b Equation: y 1 ( 9) 10 ( ) 5 Use one of the points (it does not matter which one) and the slope and put in the point-slope form: y y1 m 1 y1 y1 y 6 P age
7 PARALLEL LINES HAVE THE SAME SLOPES. PERPENDICULAR LINES HAVE OPPOSITE RECIPROCAL SLOPES. THE EQUATION OF A VERTICAL LINE THROUGH ab, : a THE EQUATION OF A HORIZONTAL LINE THROUGH ab, : y b *Write the equation of the line that is parallel to y 5,. The slope of the given line is and since parallel lines have the same slopes, use the and the point and either method in the previous eamples to write the equation of the line. and passes through *Write the equation of the line that is perpendicular to y 6 and passes through,. 5 The slope of the given line is. To write the equation of a line perpendicular to 5 5 this slope, you will need the opposite reciprocal slope, which is. Use this slope and the point and either method in the previous eamples to write the equation of the line. SYSTEMS OF EQUATIONS: SUBSTITUTION METHOD: *Solve one of the equations for one of the variables. (Get one of the variables by itself look for a single *Substitute into the other equation to find the remaining variable. EXAMPLES: or y y y 9 Take the second equation and solve for. 9 y Substitute this back into the top equation for. 9y y Solve for y 18 8yy 11y 18 11y y Substitute back into any equation and find. 9( ) 1 Solution: 1, 7 P age
8 y 7 5y 1 Take the top equation and solve for y. y 7 Substitute into the second equation for y Now find y by substituting into any equation. 1 8 y Solution: 1 8, ELIMINATION METHOD: *Eliminate one of the variables by turning them into opposites of each other. This may involve multiplying one or both equations by numbers to make this happen. *Add the system together and one of the variables will drop out. *Substitute back into either equation to find the remaining variable. EXAMPLES: y 59y 1 5y 59y 1 150y10 157y 7y 7 y Decide which variable to eliminate. Let s do the 's. To turn the 'sinto opposites, you will need to multiply one of the equations by a positive number and one by a negative to ensure they drop out, but at the same time they must be opposite values. Now add the system Substitute back in either equation to find. Solution:, 1 8 P age
9 y 1 56y Eliminate the y'sby multiplying the top by y 1 56y 96y 56y Add the two together and find Substitute back into any equation to find y 1 y 1 1 y y 1 11 y 8 Solution: 1 11, P age
10 RULES OF EXPONENTS: a b a b When you multiply with the same base, you add the eponents. Any coefficients in front will be multiplied. a b a b When raising a power to a power, you multiply the eponents. Any coefficients a b c ac bc in front will be raised to that power. y y a b ab When dividing, subtract the eponents and put your answer where the largest eponent was in the problem to avoid negative eponents. Reduce any coefficients as a fraction or a whole number. 0 1 EXAMPLES y y y 1 y abc abc abc abc yz y z 7 yz abc abc abc abc 8a b c y 18y Notice that I reduced y 5 10 as 18 5 and subtracted the eponents. The moved to the denominator as that is where the largest value was in the 5 problem and the y is in the numerator as that is where the largest y value was in the problem. Let s do an eample missing all the rules together: y y y y y y y 9 y y 9 y 9y P age
11 Sometimes you will still have negative eponents in the problem. These must be turned to positive eponents and it is an easy process: if the negative eponent is in the numerator, you move it to the denominator and it goes positive. If the negative eponent is in the denominator, move it to the numerator and turn it positive. y y 5 The 5 moved to the denominator and went positive and the top and also turned positive. 5 y moved to the 0 5 yz so all the top has for now is a. But notice the and the z have negative eponents, so must move up to the top. The 5 and the neither has a negative eponent with them. 7 y stay where they are; z 5y y y 15 y Use the rules of eponents to simplify this before tackling the 15y z z 5 y z 5 y z 5 y z 5 y 5y negative eponents. Then make any negative eponent into the positive by moving it SIMPLIFYING RADICALS: An epression containing radicals is in simplest form if the following is true: No perfect square factors other than 1 are under the radical. No fractions are under the radical. No radicals appear in the denominator of a fraction. Product Property The square root of a product equals the product of the square roots of the factors. ab a b when a and b are positive numbers. Quotient Property The square root of a quotient equals the quotient of the square roots of the numerator and denominator. a a when a and b are positive numbers. b b 11 P age
12 Make sure to find the largest perfect square that will divide into the number under the radical. Simplify the epression: Simplify the epression: Simplify the epression: Simplify the epression: P age
13 Add, Subtract, Multiply Radical Epressions The number under the radical is called the radicand. Two radical epressions are like radicals if they have the same radicand. EXAMPLES: Simplify: = ` 1 P age
14 (Think FOIL here): POLYNOMIALS: ADDING AND SUBTRACTING POLYNOMIALS: A polynomial is written in standard form when the terms are placed in descending order, from largest degree to smallest degree. The degree of each term of a polynomial is the eponent of the variable. The degree of a polynomial is the largest degree of its terms. When a polynomial is written in standard form, the coefficient of the first term is the leading coefficient. A monomial is as polynomial with only one term. A binomial is a polynomial with two terms. A trinomial is a polynomial with three terms. To add or subtract polynomials, combine the coefficients and keep the variables and eponents the same. If subtracting, run the negative sign through the polynomial. EXAMPLES: Add and write your answer in standard form: P age
15 Subtract and write your answer in standard form: Subtract 5 1 from 5 6 Set this up as a subtraction problem: MULTIPLYING POLYNOMIALS: You will use the distributive property and the idea of FOIL, along with the eponent rules to simplify these polynomials. EXAMPLES: ab ab8ab b ab abab 8ab ab b 6ab ab 6ab 5 y5y 5yy5yy 10 y15y6 y 10 11y6 y Think FOIL m p m pm p 9m 6mp6mpp 9m 1mpp 5 Use distributive property P age
16 SCIENTIFIC NOTATION: A number is written in scientific notation when it is epressed as or equal to 1 and less than 10, and n is an integer. a 10 n, where the absolute value of a is greater than To write numbers in decimal notation: If n is positive, move the decimal point to the right n places. If n is negative, move the decimal point to the left n places. EXAMPLES: , 000, 000 n 7, move the decimal 7 places to the right n 6, move the decimal 6 places to the left. To write in scientific notation, move the decimal to make the number between 1 and 10, and count the number of places you moved the decimal. A large number is a positive n value and a small number is a negative n value. 9 5, 10,000, move the decimal point 9 places, positive for the large number move the decimal point 8 places, negative for the small number 16 P age
17 Name Summer Math Packet Students entering Algebra A ****Reminder: The Bishop Kelley course catalog states that a graphing calculator TI 8/8 series (TI 8 preferred) is required for Algebra A. Please consider purchasing the calculator when they go on sale during the summer. The best sales prices are often at the end of July and stores sell out quickly. Directions for Summer Math Packet: 1. Use a pencil. Show all work in packet do not do work on loose leaf.. This will be turned in when you attend orientation. Write the mied number as an improper fraction. Write each fraction as a mied number Simplify each epression and epress answer in simplest form. This portion is to be done without calculator so show all work. After this page, you may use a calculator on everything Use order of operations to simplify each epression. If the answer is not a whole number, leave the answer as a reduced fraction NOT a decimal (16) P age
18 1. 7(6 ) 7 7(7 ) ( ) ( ) (8 ) ( )() Evaluate when =, y=1 and z = y z 1. 5z 8 Simplify. Use the distributive property when necessary. Show work d d (y 9 y) (5y y) 6. ( r 6) (8 r 5) 7. Subtract 5 from [ (y 5)] 0. 5( y) (y 1) 8. y 18 P age
19 Solve. Make sure you show work and not just an answer. Any answers that are not whole numbers should be left as reduced fractions not decimals. c a 1. y (5) 8. 6p p 9. 7( ) ( ) 5( 1) LITERAL EQUATIONS Solve the following for the designated variable.. P L W for L. a y 8 z for y 19 P age
20 Write an equation and solve. 5. (+16) () Graph and epress the inequality in interval notation Solve and graph on a number line. Epress solution in interval notation ( ) (6 9) ( ) Solve Find the slopes given the two points. Simplify. 5., 5and 7, 9 5. (-8,-) and (9,-1) 0 P age
21 55. Given the graph, find the following: slope: -int: y-int: 56. Use a table to graph y y y Graph without an -y chart. (Draw arrows on the ends and use a straight edge ) (Hint: use y=m+b) 57. y 58. y 8 y y P age
22 59. y y Write the equations of the following lines in slope intercept form. Leave fractions in your equations no decimals. 60. With slope of 1 and passes through 5, passes through, and 7, That is parallel to y 6and passes 6. That is perpendicular to 5y 10 through 1,. And passes through, 7.. SYSTEMS OF EQUATIONS Solve the following systems by the substitution method. Leave any non integer values as fractions. 6. y 5 y y 1 6y 8 P age
23 Solve the following systems by the elimination method. 66. y 5y y 11 6y Simplify. Remember to epress final answer with no negative eponents a a 69. ( a ) 70. ( y )( y ) 71. ( ) bc d 7. 1 a 7. ( a ) ( a ) 7a) 0 b) ( ) ( y ) 6y 76. y z 18 y z ( ab c d) 1abc d (epress answer as eponent) 80 (5 yz ) (5 a b c )(6 a b c ) P age
24 POLYNOMIALS Add or subtract to simplify. Write in descending order of eponents Multiply and simplify. Write in descending order of eponents. 7 y 5y y y 5y y 90. Factor out Greatest Common Factor ab 1ab ab Factor terms by grouping 9. ab a b 6 9. ac a c 8 Factoring trinomial when the leading coefficient (also known as a ) is 1. You should be able to work these mentally. You MUST be good at this PRIOR to entering the door of Algebra A so practice more online if you need more practice! (None of these are prime!) a 5a Page
25 101. b 8b Factoring trinomials when the leading coefficient (also known as a for trinomials) is NOT 1. These require trial and error. (and NONE of these are prime!) You MUST be good at this PRIOR to entering the door of Algebra A so practice more online if you need more practice! y 1y a a p p y y P age
26 Factoring Difference of squares a 5b Simplify the following radicals. ( no decimals here) a b y y a 0a P age
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