2015 SUMMER MATH PACKET

Transcription

1 Name: Date: 05 SUMMER MATH PACKET College Algebra Trig. - I understand that the purpose of the summer packet is for my child to review the topics they have already mastered in previous math classes and therefore will be prepared to take the class they are currently enrolled in. (Parent/Guardian Signature) Date

2 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: Problems that need to be solved for credit. 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 to do on a separate piece of paper before you try the problems that will be due. The explanations are divided into sections to match the sample problems so you should be able to reference the examples easily. How to receive full credit for the packet: Must be turned in the second day of class. The answers are provided in the back of the packet; however, always state why or how you formed your answer 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 when your class goes through this material during the first weeks of school. 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: Math teachers will be available for assistance at the high school the week before school. Check the school website for specific dates and times. ONLY NEED TO TURN IN THE FIRST 9 PAGES! (Keep the rest for reference during the year) Enjoy your summer and don t forget about the packet. August will be here before you know it! See you in August! The OPRFHS Math Department

3 Name: Date: Period CAT Summer Packet Practice Problem Summer For students entering College Algebra Trigonometry - (CAT) Welcome to CAT! This packet contains the topics that you have learned in your previous courses that are most important to CAT. Pg. #- 8 Problems that need to be solved and turned in for credit. Pg. # 9-4 Part I Calculator instructions for TI Nspire and TI 84.. Pg. #4-4 Part II-Part VIII Sample problems and explanations. Pg. #5-6 Answers to problems so that you can check to see if you solved them correctly. ONLY NEED TO TURN IN THE FIRST 9 PAGES! (Keep the rest for reference during the year) Enjoy your summer! Practice Problems Complete the problems below by stating why or how you formed your answer. Feel free to do your work on separate sheets of paper and remember you will be required to turn this in. An answer key is provided for you but in CAT the work is always more important than the answer! I. Perform the following operations using your graphing calculator.. Graph the following on your calculator and sketch the graph on the axis provided. 7 b) a) y x y x 7x 6

4 . Find the intersection of the following systems of equations using your graphing calculator. yx x5y 4 a) b) y x yx. Find the roots of the following using your graphing f x ( x ). calculator: 4. Find the roots of the following using your graphing f x 5x 5x. calculator: 5. Complete the following table for f x x 5x, using the ask function on your calculator. 6. Find the roots of the following by factoring: f x x x ( ) X Y 6 - II. Polynomials Multiply the following: 7. x 4x 6x a b ab b ab b 5x y x y

5 . x h 4x h 9 x 4x 9 Factor the following completely:. w 7w 0. x 6x y 0y 5 5. ab cb 6. x x x y 64z 8. 6y 96x x p q 5p q 4

6 III. Rational Expressions Simplify the following: x 4 x. x x. a a a a a a a a x x 4 x 8 x 4 4. a a a a a a 6 a a 5. a b a 5b a b a b y y y 7. a a b b b a 5

7 IV. Exponents Use the properties of exponents to simplify the following: 8. 4xy x 4 y 5 9. x x 0. x y z xy z Find x and x, when. a) x 5 bx ) 7 V. Radicals Write the following in simplest radical form: c d 4. x y xy y y 4 6

8 VI. Linear Functions 7. Write the equation of the line containing the points, 4 and 4, in slope intercept form. VII. Functions Using your knowledge of functions answer the following: 8. Determine whether the following are functions: {,, 7,9,,,,6 } a) b) {,9,,, 6, 9, 7, } 9. Determine the domain and range of the following graphs: a) b) Domain: Range: Domain: Range: f x 5x 4x for 40. Evaluate the following: a) f b) f c) f t d) f(a + h) f(a) 7

9 4. Following are the graphs of the functions.,,, f x x f x x f x x f x x Label each graph with the correct function. a) b) c) d) VIII. Quadratic Equations Solve the following quadratic equations: 4) x x 4 0 4) x x 44) 5 0 x 45) x x 5 0 8

10 I. Using your graphing calculator TI Nspire Instructions Be able to find the zeros, (also known as roots or x-intercepts) using the calculation menu. Problem: Find the zeros of ( ). Enter the function into the calculator as a graph. Press Menu, Analyze Graph, and Zero.. Highlight the upper and lower bounds and hit enter. 4. Repeat steps and to find the second zero. C. Know how to use the Table Set function.. First, use CTRL T to get the table visible on the screen. 9

11 . Access the Table Set function by pressing Menu, Table, Edit Table Settings. You can change the TABLE so that the x-value increases by different increments by changed the Table Step. You can also change the x-value to start with by changing Table Start. TI 84 Instructions (Keystrokes below are based on using a TI 8-Plus or TI 84 Calculator):.. Be able to do basic graphing b) Graph a) Graph y x 4 y x x 0

12 . Find the intersection of the lines yx yx using the intersection function on your calculator.. a) Solve the above Equations for y. y x y x b) Graph on calculator c) Press nd TRACE for the.calc menu. Your screen should look like the one below d) Press 5 for INTERSECT or cursor down and press ENTER. Your screen should now look like this: e) Press ENTER times, (once at each prompt) f) The x and y coordinates appear at the bottom of the screen. Your screen should look like the third one above. The solution to the system is, ) Find values using tables. a) Graph the line yx 7. Access the table by pressing nd GRAPH. The table screen should look like the one below:.

13 b). You can move up and down using the arrow keys while on the table while the cursor is on the x-values. 4) To see the graph of f x x, be able to set the window manually, using both your knowledge of functions and of the calculator a) In order to view the function, first press ZOOM 6 and then ENTER. Your graph will look like the one below. b) Now, set your window by pressing the WINDOW key and changing the values to look like the ones below. c) Your graph should look like the one below.

14 5) Be able to find the zeros, (also known as roots or x-intercepts) using the calculation menu. Problem: Find the zeros of f x x x 6. a) Enter the function into the calculator. Be sure to press ZOOM 6 in order to go back to the standard screen. b) Press CALC ( nd TRACE) to access the calculate menu c) Press (or cursor down) and hit ENTER for ZERO. Your screen will now look like this: d) Use the arrow keys to move the cursor close to where the function crosses the x-axis and press ENTER. Your screen should now look like this: e) Move cursor to the right of the zero and press ENTER again. Your screen should look like the one below. f) Press ENTER a third time and the x and y coordinates of that x-intercept will be on the bottom of the screen.

15 6) Know how to use the Table Set function. a) Access the Table Set (TBLSET) function by pressing nd WINDOW. You can change the TABLE so that the x-value increases by different increments. For example, cursor down to Tbl and change the number 0.5. Then the independent variable will increase by 0.5 b) Go to the TABLE ( nd GRAPH) and confirm this. 7) Store function To store a value in for a variable is used when evaluating a function or testing the solution to a problem. Enter the value you want to store, which can be an expression.. Press STO. The store symbol () is displayed.. Type the variable name, using Alpha if necessary 5x 4. Press enter. You entered value (ex. 5) will appear when you type in x and press enter. You can evaluate an expression for x = 5 by typing in x + 9 and when you hit enter the value of 9 will be seen on the screen. II. Polynomials: Basic Operations Expand Polynomials including multiplying two binomials, binomials by polynomials and cubing binomials. Multiply the following:. 7x y x y 5xy 4x y 5x y x x 4 x 5x. x 4 x x 6 x x 6x 8x x 4 x 5x 8x 4 4. a b a b a b a a b ab b x x 6x 8x 8 x x 6x x 8 Be a factoring expert! This is very important. You should be comfortable factoring out the greatest common factor from an expression, factoring by grouping, factoring quadratics when the leading coefficient is one and when the leading term is something other than one. For example, you should be able to factor the following:. x y 0x y 4x y 5x. x 9 x x 4

16 . x 7x x 4x 4. x 0x 8 x x 4 5. x 4 x 0 x 5x a b a ba ab b. Using: a b a ba ab b Factor: 5x y 5x y5x 5xy y Factor: x 7y x yx xy 9y III. Rational Expressions. Be able to reduce a rational expression. Problem: Simplify the rational expression below. 9x 6xy y x xy y x y x y x yx y x y x y 4 Factor the numerator and denominator x y x y x y x y 4 x y 4 x y Commutative Property Reduce fraction. Add, subtract, multiply and divide rational expressions. Problem: Multiply and simplify xx 4 x x x x x x x x 4 Multiply the numerator and denominator x x x x x x Factor and reduce x x x x 5

17 Simplify Problem: Divide and simplify Multiply by the reciprocal a b a ab b a b a ab b a b a ab b a b a ab b a ba ab b a ba b a ba ba ab b Factor and reduce a b Problem: Addition of rational expressions 5 x 5 x x x x x x 4 Factor the denominator. Find the least common denominator x 5 x LCD x x x x x x x Multiply by in the form x x6 5x x x x x Multiply fractions x 6 5 x x 4 Combine numerator and simplify x x 4 Problem: Subtraction of rational expressions. Changing the above problem to a subtraction problem: 5 x 5 x x x x x x 4 Following the steps above, find LCD and multiply by Remember to distribute the negative! x 6 5 x x 4 4x x 4 6

18 Problem: Solving rational equations. 4 x x4 4 x x4 Multiply both sides by the least common demoninator. x x 4 x x 4 4 x x4 Using the distributive law: x x 4 x x 4 x x 4 Simplify: 4 x 4 x x x 4 0 x 0 x 5 4x 56 x x x 8 0 x 5x 50 0 x 0 or 0 x 5 The solutions are: x 0, 5 Problem: Solve: 7x x 54 x x x 9 7x x 54 x x x 9 x x x x Multiply both sides by the LCD: 7x x 54 x x x 9 Distribute: x x x x x x Simplify: x 7x x x 54 x x x x Distribute the negative sign and continue simplifying: 7x 9x 6 x x 54 x 0x 9x 6 54 Subtract 54 and multiply by : x 0x 9x 90 0 x x 0 9 x 0 0 x 9 x0 0 Factor: Solve: x or x x or x 0 Since x cannot be equal to x 0., as the denominator in the original equation will be zero, the only solution is 7

19 Problem: Multiply by the LCD x x x x 4 4 x x x x x 4 x (Distribute) x x xx (Factor) x x (Reduce) Problem: Know the difference between solving and simplifying a rational expression. x x You would simplify: x x7 while you would solve: x x x x7 IV. Exponents Definitions and Rules for Exponents For any integers m and m and n (assuming 0 is not raised to a nonpositive power): 0 Zero as an exponent: a Negative exponents: n m n n n a b a b a n ; m n ; a b a b a Multiplying with like bases: m n m n a a a (Product Rule) Dividing with like bases: m a mn a ; a 0 (Quotient Rule) n a Raising a product to a power: n n n ab a b Raising a power to a power: a m n mn a (Power Rule) Raising a quotient to a power: n n a a ; 0 n b b b 8

20 Problem: Know the properties of exponents and be able to simplify all types of exponents, including negative and fractional exponents. For example: x x x 9 6 x x x x x Problem: Be able to find the th n root of real numbers. For example: Problem: Simplify using rational exponents. 4x x x 4x x x x 4 x x V. Radicals Simplify radicals. Express radicals in simplest radical form. x x 4 7y y 8x y y x 8x y 9 x x y

21 Multiplication of radicals x x x( x ) x x Addition of radicals. (Radicals must be the same.) x 4 x 6 x 5 4x 9x 5() x () x 0 x 6 x 4 x Note: Unlike radicals y y Do not combine. Subtraction of radicals. 8y 4 y ()() y 4 y 6 y 4 y 8 y 5y 5y y 5 y Division of radicals. 8x 8x 4x 4x 7x x 7 7y 7y 9y 9y y ( )

22 Finding Conjugates The conjugate of is The conjugate of 5 is 5 The conjugate of 6 is 6 The conjugate of 5 is 5 Rationalizing, Denominators or Numerators a a 5b 0ab 0ab 5b 5b 5b 5b 5 b Problem: Simplify radicals. Be able to express the following in simplest radical form: 5 x y z xy z xy 5 64 x y z ( xy ) x z Problem: Know the difference between exact versus approximate answers. For example, is an approximation. is exact, while VI. Linear Equations. Have knowledge of linear equations. Know how to use the following formulas: y y m, where x x x x y mx b y y m x x

23 When lines are parallel their slopes are the same. When lines are perpendicular mm their slopes are opposite reciprocals.. Solve systems of linear equations in two and three unknowns, using substitution, linear combination and matrices.. Be able to find the intersection of two lines on your calculator. yx Problem: Solve graphically: yx Solving both equations for y and graphing yields: The solution to the system is, Problem: Solve using substitution. x y 6 x4y 4 Solve the first equation for y. y6 x Since y and 6 x are equivalent, substitute 6 x for y into the second equation. x 4 6 x 4 Use the distributive property. x 4 8x 4 Solve for x. x 4

24 Substitute 4 for x in either equation and solve for y. xy 6 4 y 6 y The solution is the ordered pair 4, Problem: Solve using linear combination: x 4y x y 0 Add the x' s and add the y ' s y Solving for y yields: y. Substitute y into either of the two original equations: Solve for : x The solution is the ordered pair, x 0 VII. Functions Know how to use function notation and what it means. f () 5 (,5) Identify a function from a set of ordered pairs, a graph or an equation. Determine the domain and range of a function from a graph. Given the graph: Domain All Real Numbers Range y

25 4 Evaluate functions. f x 5x x Given f f f f h h h 5 f h h h h h h h h 5( ) [ ] 5 [ ] Recognize the graphs of the following functions: a) f x x b) f x x c) f x x d) f x x VIII. Quadratic Functions. Solve quadratic equations using factoring and the quadratic formula.. Solve equations in quadratic form a) x x 0 0 x x Factor Set each factor equal to zero x 5 0 Solve x i x or x 5 0 b) x 4 x 0 Substitute: a x a 4a 0 Solve for a: a or a Substitute: x a x or x x or x 9 Square both sides: 4

26 Answers: a. b. a., b.,. x.,. 4. x.4,.4 6, 7 6. x, x 5., , 6x 4x x x 0xy 4y a b 0a b ab 4ab 6b 5x 0x y 4y 6 4 x xh 4h h. w5w. x4x 7 4. y 5 5. b a c 6. ( x 6) x 7. y 8z y 8z a. ( x )( x x 9) 6( y 4 x )( y x)( y x) 0. ( p6q 5)( p q) x a. x a 5 ( x 4) a 4. x 4 a a b a ab b y ( y ) y x 6 6x b xy y6 y 8 8. a) b) no yes 7. 9a. 7 4xy 7z c d d y x 5 D R y 4

27 9b D x 4 R y a) f ( x) x b) f ( x) x c) f ( x) x a) b) c) 5t 4t 9 d) 5h 0ah 4h 4. x 4, d) f ( x) x x, 44. x 45. x 7, 5 5 6