Pre-Calculus Summer Packet

Transcription

1 Pre-Calculus Summer Packet Name ALLEN PARK HIGH SCHOOL Summer Assessment Pre-Calculus Summer Packet For Students Entering Pre-Calculus Summer 05 This summer packet is intended to be completed by the FIRST DAY of school. This packet will be graded and count as the first grade of the marking period. You should be working on the packet for the class you are taking NEXT YEAR. Feel free to me if you have any questions regarding your packet by ing me at: We encourage you to check your answers and re-work any problems that were incorrect. We epect you to spend at least hour each week on your summer math packet. This packet is not designed for one intense 0 hour session the day before school starts, so begin now! This summer math packet will be worth 50 points and will be the first recorded grade of the marking period. Bring your questions and concerns regarding any problems you may have had difficulty with to your class on the first day of school, September 4 th. Start off the year with a great start by completing the packet to the best of your ability. APHS/MATH Page

2 Pre-Calculus Summer Packet Summer 05 Show all work for all problems, regardless of their level of difficulty. Answers will be in the form of positive, negative, whole numbers, fractions and decimals. Leave answers in fraction form unless the question contains decimals. Solving Absolute Value Equations Absolute value equations usually have two different cases when you are trying to solve them. This is illustrated in the following eample. If 4, what numbers can you substitute into that will give you a result of 4? There are two different answers; 4 4 and 4 4, therefore 4 and 4. Here s another eample with an equation: Solve: 5 Isolate the absolute value first, by dividing both sides by to get 5 Case : Drop the absolute value on the left Leave the right side as is 5 subtract from each side divide by on each side 4 Notice, there are two answers (although occasionally, there is only one answer). Case : Drop the absolute value on the left Change the right side to negative 5 subtract from each side 8 divide by on each side Solve each equation k t5 t 7. a a Solving Absolute Value and Compound Inequalities Combine the rules for solve absolute value equations and solving inequalities. You will have cases like absolute value equations. In case two you must flip the inequality symbol and change the sign of the side of the inequality without the absolute value symbol. To graph the solution to an absolute value inequality on a number line, use the following rules: For < and, your solution is where the two inequalities will overlap one another (the intersection). For > and, your solution must include both parts of your solution, (the union). Don t forget about open and closed circles on your graphs. Eample: Solve and graph 9 Case : Case : APHS/MATH Page

3 Pre-Calculus Summer Packet Summer 05 Eample: Solve and graph 9 Case : 9 Case : Solve each inequality. Graph the solution set on a number line. 9. 8a or 7. 5 c or 5 7 Completing the Square Eample : Solve by completing the square factor the left side of the equation = Solve by completing the square. 7. v 4v n 4n 9 Equation of a Circle The equation of a circle: h y k r where, hk is the center and r is the radius. Use the information provided to write the standard form equation of each circle. 9. center: 6, radius: 0. center: 5,4 radius: 6 APHS/MATH Page

4 Pre-Calculus Summer Packet Summer 05 Simplifying Comple Numbers Remember: i Eample : Simplify. 8ii 4 9i Eample : Simplify. multiply 6i i distribute 6 i 64 44i i multiply by the conjugate of the denom. 4 5i i 4 5i foil the top and bottom 4 5i 4 5i 8 0i i 5i simplify 6 0i 0i 5i 8 i 5 simplify 6 5 i 4 Simplify.. i i 8 i. 8 i. 5i 4i 4. 6 i 7i Identify the verte and ais of symmetry of each. Then sketch the graph. Ais of Symmetry: b a b b Verte:, f a a Eample : Find the verte and ais of symmetry of f ( ) b 8 8 Ais of symmetry: a 4, b 8, c 7 a 4 8 Verte: Use the value from the ais of symmetry and substitute it into the given equation. f ( ) The verte is at, Find the verte and ais of symmetry of each equation. Then sketch the graph. APHS/MATH Page 4

5 Pre-Calculus Summer Packet Summer f ( ) 8 7. y Divide using synthetic division. Eample : Divide using synthetic division. Eample : Divide using synthetic division With a remainder of 0. With a remainder of 8. 4 is a factor of is NOT a factor of 4. Divide n 0n 08n 86 n 0 Solve each equation with the quadratic formula. b b 4ac a 0. 4p 4 9 p. 7 6 Solving Eponential Equations Eample : Solve. APHS/MATH Page 5

6 Pre-Calculus Summer Packet Summer make the bases the same distribute the eponent 44 9 since the bases are the same, set the eponents equal to each other solve for Solve each equation.. v Simplifying Monomials Eample : Simplify. Your answer should contain only positive eponents. 4 ab reduce the fraction 5 6 6a b 4 ab 5 6 a b ab a b 4 5 a b 4 7 a b 6 7 ab 6 a 7 b 4 5 subtract eponents of common bases since b has a negative eponent move it to the denominator b now has a positive eponent. APHS/MATH Page 6

7 Pre-Calculus Summer Packet Summer 05 Eample : Simplify. Your answer should contain only positive eponents. 4 a b a b 4ab a b 4 4ab a b 4 4ab a b 4 4 ab 4 ( ) 4 a b ab a b 5 5 a b 5 b 5 a 4 4 ( ) numberator : multiply the coefficients and add eponents of common bases reduce the coefficients subtract eponents of common bases move a 5 to the denominator Simplify. Your answer should contain only positive eponents. 4. a b ba ab 4 5. y y m n m 6. 4 n Factoring Quadratics Eample : Factor completely. Factor out the GCF multiply = What are the factors of (from ) that add up to + = substitute for Pull out the GCF for... ( ) and for...( ) Notice the ( ) is the same for both parts... this becomes ONE factor and the #s outside the ( ) give you ( ) APHS/MATH Page 7

8 Pre-Calculus Summer Packet Summer 05 Eample : Factor completely. multiply 8= 4 Factor Completely. 8 8 What are the factors of 4 (from 8) that add up to the middle term? = = substitute 6 4 for 4 Pull out the GCF for 6... ( ) and for ( ) 4 Notice the ( ) is the same for both parts... this becomes ONE factor and the #s outside give you ( 4) n 6n 6n n 6n 4. a 4 difference of squares 4. k k difference of squares 44. 6n 9 difference of squares sum of cubes 46. a 8 gcf/difference of cubes 47. a 7 sum of cubes sum of cubes Sketch the graph of each line or inequality. 49. y y 5. 4y 5 5. y 5 5. y y APHS/MATH Page 8

9 Pre-Calculus Summer Packet Summer 05 Sketch the graph of each function Solve each system of equations or inequalities by graphing. 6. y y 9 6. y 4 y 6. y 9 y APHS/MATH Page 9

10 Pre-Calculus Summer Packet Summer y y5 66. y 5 y y 6 y 8 Logarithms Epanding and Condensing Logarithms Eample : Epand the logarithm log y z. mult add in epansion y log log log y log z log log y log z z from mult. from div. zz div. by z subt. in epansion Eample : Write as one logarithm (condense) log 4a log 4b log 4c ab log4 a log4 b log4 c log4 a log4 b log4 c log4 a b log4 c log4 c move coefficients to eponents add becomes mult subt. becomes div. Epand or condense each logarithm. 67. log log z u log 6 v loga logd 7. log log y 7. log 5log y log z Solving Logarithmic Equations log log log 0. Eample : Solve the equation log log log 0 log log ( ) APHS/MATH Page 0

11 Pre-Calculus Summer Packet Summer 05 Solve each equation. 7. log 6 log 7 log 74. log log log log log 6 6 (not solved the same as 7 & 74) Simplify each rational epression. 76. n n n n9 n m m m4 49 m n n n n 7 4 Solve each equation. Remember to check for etraneous solutions. 80. m 4 r 8. 5m 5m 4r r r Find the discriminant b 4ac of each quadratic then state the number and type of solutions. 8. r 6r Radical Epressions Simplify m Write each epression in radical form n 94. Simplify. Your answer should contain only positive eponents with no fractional eponents in the denominator y u v u v u v Find all rational zeros. 97. f ( ) f ( ) APHS/MATH Page

12 Pre-Calculus Summer Packet Summer 05 Approimate the real zeros of each function to the nearest tenth. 99. f ( ) 00. f 4 ( ) 4 5 Evaluate each function for the given value f ( n) n 5n n 9n 0 find f (). 0. f ( ) 5 find f( ) f ( ) find f ( 5). f ( ) 5 find f( a ). Solve the system of equations algebraically using elimination or substitution. 05. y 4 y y0 0 4y y 7 y y 9 y 09. y 0. y APHS/MATH Page