SUMMER MATH PACKET ALGEBRA TWO COURSE 229

Transcription

1 SUMMER MATH PACKET ALGEBRA TWO COURSE 9

2 MATH SUMMER PACKET INSTRUCTIONS 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: or Math teachers will be available for assistance at the high school the week before school. Check the school website for specific dates and times. 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, starting May 17th. See you in August! The OPRFHS Math Department

3 Introduction to Algebra Two The following 1st year Algebra topics are essential prerequisites for Algebra Two (Course #9: Algebra Two): Operations & Properties of Real Numbers Solving Equations (One step, Two Step & Quadratic) Problem Solving Graphs Linear Functions: Slope, Graphs, and Model System of Equations in Two Variables Solving by Substitution or Elimination Polynomials: Addition & Multiplication Factoring: Common Factors, Factor by Grouping, Factoring Trinomials & Difference of Two Square

4 FUNCTIONS Be familiar with function notation. Know that y x can be written in function notation as f ( x ) x. Example: Given f ( x ) x, find each of the following: a. f (0) b. f () c. f ( ) 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. The range is the set of all second members in a relation. Example: List the domain and range of the relation (5,),(6,4),(8,6) Domain = 5,6,8 the set of all first members in a relation Range =,4,6 the set of all second members in a relation Example: Find the domain and range from the graphs below. Domain: all real numbers Domain: 0 x 7 Range: all real numbers Range: y 4

5 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 the following equations Example 1 x 4 1 x Addition Property of Equality x x 17 Multiplication Property of Equality 17 x Example 16 7y 10y 4 LINEAR FUNCTIONS AND GRAPHS Be familiar with linear functions and inequalities and their graphs. Slope-Intercept form of a linear equation is y mx b. The y-intercept of a graph is the y-coordinate of the point where the graph intersects the x-axis and is represented by b. The slope of the line is represented by m. Remember that the graph of any linear equation is always a straight line. Determine the equation of a line using the point-slope equation. y y 1 To determine the slope of a line m, use the equation m x x 1 Example: Find a linear function with a slope of and a y-intercept of -7. If the slope =, then m. If the y-intercept = -7, then b = -7. Therefore the equation is y x 7.

6 Example: Given the points (6,-4) and (-,5), find the equation of the line in slope-intercept form We first need to find the slope of the line. Using the two ordered pairs and the slope equation, calculate the value of the slope. y y 1 m x x 1 5 ( 4) m Simplify 6 9 m 9 m 1 label the ordered pairs ( x, y ) and ( x, y ) 1 1 We now need to find the y-intercept of the line. Using the slope, one of the ordered pairs and slope-intercept form, calculate the value of the y-intercept. y mx b Substitutem 1and (6,-4) into the equation. 4 1(6) b 4 6 b Subtract 6 from both sides of the equation. 10 b Now substitute the values for m and b into slope-intercept form to obtain the equation: y 1x 10 or y x 10 Example: Graph 4x 5y 0 using the slope and the y-intercept. First, write the equation in slope-intercept form. 4 4 y x 4 where m and b Plot the point (0,4)

7 Use the slope to move down 4 units and to the right 5 units from the point (0,4) to plot the next point. Continue plotting points using the slope. Draw a straight line through all of the points to complete the graph. Example: Given the graph below, write the equation of the line in slope-intercept form. Look at the graph and find the y-intercept. Remember that the y-intercept is the point where the graph crosses the y-axis. Therefore b = 5

8 Next find another point on the line by counting a rise and run (slope). Doing this allows us to say that the slope is down and to the right. Therefore m. So the equation is y x 5 SYSTEMS OF EQUATIONS Solve by graphing. y x 1 y x Graph the equation y x 1, then graph the equation y x. Look for the point of intersection of the two graphs. The point of intersection is the solution to the system of equations. The solution is (-1,). Example: Solving systems by substitution x y 6 Solve by substitution. x 4y 4 Solve the first equation for y y x 6 Substitute the first equation x 4( x 6) 4 into the second equation for y. Distribute x 8x 4 4 Combine like terms 5x 4 4 Isolate the variable 5x 0 x 4 Now take your value for x and substitute into either equation ( 4) y 6 to find y.

9 Solve for y. 8 y 6 y 14 The solution is (-4,14) Example: Solving systems by linear combinations/elimination. This method is a combination of linear equations that will eliminate a variable. When using this method, first put the equation in the formax By C. You then want to make one of the variable terms opposites of each other. Then add the equations together to eliminate a variable. Solve by elimination Put the equations in Ax By C form 4y x 1 y x x 4y 1 x y 0 Add the equations together. y 1 Solve for y. Substitute into either equation to find x. The solution is 1 y 1 x 1 x 1 1, EXPONENTS Be able to multiply same bases using properties of exponents, divide same bases and raise a power to a power Know the properties of exponents. For any real number a and integers m and n. a a m a m n a, a 0 n a m n m n a m n a m n a 0 a 1 m n p mp np a b a b m a n n p a b mp np 1 a a a m 1 a m

10 Examples: 4 7 ( x )(5 x ) 4 (5 a )( a ) ( 4 x y ) ab c 5 a bc FACTORING Factor terms with a greatest common factor (GCF) Factor Factor each expression Factor out the GCF 4 5x 0x 5x x 5 4x 5 x ( x 4) Factor trinomials of the form x bx c Factor Look for factors of the first term, then factors of the last term. x x x 10 x x and 10 5 Write the expression as a product ( x )( x ) of two binomials. Use the 5 and to combine to make ( x 5)( x ) the middle term.

11 Factor Look for factors of the first term, then factors of the last term. x x 16 x x and Write the expression as a product ( x )( x ) of two binomials. Use the 4 and 4 to combine to make ( x 4)( x 4) the middle term, which is 0. Factor trinomials of the form ax bx c Factor x 5x Multiply the first and last number 6 Find factors of 6 that can combine to 6 make the middle coefficient Replace the middle term x x x Factor out the GCF for the first two terms x(x ) 1(x ) and the second two terms. Write the expression as a product ( x 1)(x ) of two binomials. SOLVING QUADRATIC EQUATIONS Solving Quadratic Equations by Factoring When solving a quadratic equation by factoring, remember that the equation must equal 0 and be written in standard form ax bx c 0 Solve by factoring x 5x 14 0 Factor ( x 7)( x ) 0 Set each factor = 0 x 7 0 or x 0 Solve for x x 7 or x

12 Solving Quadratic Equations by Quadratic Formula When solving a quadratic equation using quadratic formula, remember that the equation must equal 0 and be written in standard form ax bx c 0 Use the quadratic formula x a b b 4ac Solve by using quadratic formula x 5x 1 0 Find the values for a, b and c a =, b = 5, c = 1 Substitute the values into the formula Simplify x x Note: If you can simplify the radical, go ahead and simplify the radical and find two solutions.

13 ALGEBRA TWO SUMMER PACKET REVIEW PROBLEMS NAME Complete the problems below. Show all of your work. Feel free to do your work on a separate sheets of paper which you should attach. Remember you will be required to turn this in. 1. Determine whether the following are functions. If the relation is a function, state the domain and range. (1,19),(,11),(6, 9),(7,11) (, ),(7,9),( 11,1),(,6). State whether the following are functions. If they are the graphs of a function, determine the domain and range.

14 Solve the following equations.. 9y 7y (5y ) 5 ( x ) 5 4( x ) x x 5x 1x x x (y ) 9(y 5) 10. (5x 1) z x 4 5 ====================================================================================

15 Graph the following using the slope and y-intercept 1. y 4x 14. x y Find the slope of the line containing the points ( 8,7 ) and (,-1 ). 16. Find a linear function whose graph has the given slope and y-intercept. Slope of and a y-intercept of ( 0, 9 ) 4

16 17. 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. 18. What is the equation in slope intercept form of the line graphed below.

17 Using your knowledge of functions, answer the questions below 19. f ( x) x 0. f ( x) x 1. f x ( ) ( x 1) 4

18 . Evaluate the following for f ( x) x a. f() b. f(0) c. f(-) Solve the following systems of equations by graphing. x y. x y y x 1 4x y 18

19 Solve by the following by using substitution. 5. y 5 4x x y x y x y 6 Solve the following by using linear combination/elimination. 7. x y 7 x 4y x 7y 16 x 8y 6

20 Simplify the following expressions. 9. x x a 7a ( a )(7 a ) ( m n )( m n p ). a a t 4t 7 5. m n mn x y z x y z 7. 5 ( x ) 8. ( xy ) 9. 5 (9 m n p ) 40. ( a bc ) 4 5 Factor the following expressions. 41. x 1 x 4. x 8x 1 4. x x x 16x 45. x x 7x 46. x 16x x x x x 16x 64

21 Solve by factoring. Remember that the quadratic equation must equal 0 before factoring. 50. x 8x x 8x 0 5. x 8x x 4x 45 Solve using the quadratic formula x before solving. b b 4ac. Remember that the quadratic equation must equal 0 a 54. x 6x x 5x x 8x 4 0