George Ranch High School Pre-Calculus PAP Su Summer Packet Problems, 2017

Transcription

1 Name Date Due 4 th day of class (Thursday, August 1st, 017) George Ranch High School Pre-Calculus PAP Su Summer Packet Problems, 017 Pre-Calculus is a very exciting class. Because we will need to hit the ground running next fall, it is necessary for you to refresh your memory on some math basics before the beginning of class in August. The problems outlined in this summer packet will assist you in practicing the skills you will need for Pre-Calculus PAP. It is recommended you either do a problem a day until you are done or do a section a week until you are done. The completed summer packet is due on the fourth day of class. Students who have the completed packet ready on the first day of class will receive a 10-point bonus! The first two days of class will be spent reviewing and discussing the packet. The first quizzes and tests will also be based on the problems in the summer packet. Work must be shown to support each answer and should be done neatly in the packet or on a separate sheet of paper. Please circle/box all final answers. Any problems and/or packet that is not accompanied by appropriate work will be considered incorrect and not receive credit. (In other words think of it like a science lab and report If your work cannot be followed and replicated to come to the same answer/conclusion it was a bad experiment. Thus you did the problem poorly/incorrectly.) If you get stuck and/or need help try the notes and web pages listed in each section. If you need further assistance please Mrs. Guidry (rguidry@lcisd.org) and/or Dr. Hren (mhren@lcisd.org). We are looking forward to seeing you in Pre-Calculus PAP in August. SECTION I: Solve the equation. 1. ( x ) 4 = ( x+ 5) 1 = x ( x 8) 1 Page

2 SECTION II: Solve the equation for y.. ( y + 1) 9x = xy + y = 4 SECTION III: Determine if the following relations are functions. Explain why or why not. Need help??? Try (Be sure to notice the different videos under the different titles on the left had side of the page) 5. y = ± x 6. {(, 6) (0, 8) (-, 6) (0, 9)} 7. Graph 8. y = 5 x Page

3 SECTION IV: Find the equation of the following lines: Need help??? Try (Be sure to notice the different videos under the different titles on the left had side of the page) 9. Contains (-4, 5) and has the slope of. 10. Passes through the points ( 9,6) and ( 9, 6). 11. Perpendicular to the line x + 4y = 7 and passes through (-, 5). Page

4 SECTION V: Graph the following equations using the most appropriate method. Be sure to include key points. 1. f ( x) = -x + 1. f ( x) = x 14. ff(xx) = (xx ) ff(xx) = (xx + 44) 4 Page

5 SECTION VI: Factor the following equations. Factoring The Difference of Two Squares This gives us a way to factor any expression that is the difference of two squares. x y = ( x + y)( x y) Difference of two squares factors as the product of the sum of two terms and the difference of the two terms. To do this, it is necessary that you can notice perfect squares quickly: EXAMPLE: List the perfect squares of 1 through 15: 1, 4,,, 5,,, We see that the formula to factor the difference of two squares requires us to notice what terms are being squared. 16a 81b = 4a 9 b ( ) ( ) 4 ( 11x ) ( 7 ) x 49y = y To factor the difference of two squares: ( ) ( ) 16a 81b = 4a 9 b 1.) Find the terms that are squared: 4 a & 9 b.) Take the sum of the two terms times the difference of the two terms 16a 81b = 4a 9b = 4a + 9b 4a 9b ( ) ( ) ( )( ) We can t factor the sum of two squares. It is called a prime polynomial. Factoring Perfect Square Trinomials: Notice the form that a perfect square trinomial takes on. ( x + y) = x + xy + y ( x y) = x xy + y 1.) The first and the last term are perfect squares..) The middle term is twice the product of the expressions being squared in the first and last terms. Factoring The Sum or Difference of Two Cubes: We also have nice formulas for the sum and difference of two cubes: a a b + b = = ( a b)( a + ab + b ) ( a + a ab + b ) To do this, it is necessary that you can notice perfect cubes quickly: EXAMPLE: List the perfect cubes of 1 through 7: 1, 8,,, 15,, Factor 15x 16 15x 16 = a = 5 x and b = 6 ( 5x) ( 6) = ( a b)( a + ab + b ) = ( 5x 6) ( 5x) + ( 5x)( 6) + ( 6) ) = ( 5x 6)( 5x + 0x + 6) 5 Page

6 Factor the following 16. x 11x x y x 19. x 19x x + 19x x 14 16a b. 1a + ab 5b. 15x y x p q s r s 6 Page

7 SECTION VII: Quadratics Need help??? Try Solve the following quadratic equations by using square root, factoring, or the quadratic formula. x 6. x + 10x = 1 7. x 1x 7 = = 9 Solve by completing the square 9. rr + r - = vv 1 = 10vv 1. 7xx = 14xx bb 9bb 8 = 9. pp 8pp + 1 = 6 4. aa + aa = 0 7 Page

8 5. Write the equation of the quadratic function in standard form y = a( x h) + k the vertex is (,1) and passes through the point (,) 5., given 6. Graph the following quadratic equation: f (x) = -x + 8x 1 a) Determine whether it opens up or down b) Find the vertex & determine if it is a maximum or minimum value c) Find x & y intercepts d) Domain & Range 8 Page

9 Section VIII: Functions Notation and Combinations Need help??? Try Given the following functions, ( x) = ( x ) 1 x f and g ( x) =, find: x + 5 a. f (g()) b. g(f()) c. gg () d. (g + f)() ff 8. If gx ( ) = x 5x+ 1 then find: a. g(-6) b. g(x-) 9. Given f ( x) = x 1 and g( x) = x + x 8, find ( g f )( x) 40. Given f ( x) = 4x + and g( x) = x, find ( g f )( x) 41. Given f( x) = 4x x+ find f( x h) 9 Page

10 Section IX: Rational Expressions Need help??? Try Simplify the following rational expressions. 4. x x x x x x 0 x + 4 x x x x x 4 x 47. Solve the equation: = x + 1 x Page

11 Section X: Exponents and Radicals Need help??? Try (problems 50 and 51) Use the laws of exponents to simplify the following expressions x x 5 100m n 5mn 50. 5x 6y ( ) 5 x y m n m 5m n 5. 6 y 0 Simplify the following expressions y 7m 8 1 x x 8w x + x y 8xy ( ) + ( 6)( ) 11 Page

12 Division: (utilize long or synthetic division) 61. xx xx xx+18 xx+8 6. xx + xx xx+1 xx+6 6. xx xx +19 xx+14 xx+18 1 Page

13 64. xx xx xx+14 xx +14 Section XI: Word Problems 65. A full mile is 4 ½ laps around the outside of the track. If you want to run of a mile, how many laps should you run? 66. The diameter of a tree varies directly as its age. A 0-year-old tree is 6 inches in diameter. How old will the tree be when it is 4 inches in diameter? a. What is the constant of variation? b. How old is the tree? 1 Page

14 67. Assume that the rate at which a cricket chirps varies linearly with the temperature. Crickets make 70 chirps per minute at 60 F and 100 chirps per minute at 80 F. a. State what the independent and dependent variables are. b. Write the particular equation that expresses this relationship. c. What is the temperature-intercept? What does this tell you about the real world? d. What is the slope? What does it represent in terms of this problem? 68. The population of students at Brien McMahon High School is modeled by the 1t + 90t equation: rt () = where t is the number of years since t + a). How many students do we have today (010)? b). Does the number of students ever stabilize? Why? c). What is the maximum number of students in BMHS according to this equation? Give the year (to two decimals), # of students, and how you got the answer. 14 Page

15 69. The Brick Oven Bakery sells more loaves of bread when it reduces its price, but then its profits change. The function y = -100(x ) + 00 models the bakery s profits, in dollars, where x is the price of a loaf of bread in dollars. The bakery wants to maximize its profits. a. What is the domain of the function? Can x ever be negative? Explain. b. Find the daily profit for selling the bread at $.00 per loaf; at $1.5 per loaf. c. What price should the bakery charge to maximize its profits? d. What is the maximum profit? 70. During a free fall, a skydiver falls 16 feet in the 1 st second, 48 feet in the nd second, and 80 feet in the rd second. If she continues to fall at this rate, how many feet will she fall in the 10 th second? Section XII: Logarithm Solve the following logarithmic equation for x. Remember to check for extraneous solutions. 71. log ( x ) = 7. log ( x + ) = 15 Page

16 7. log (7) = x 74. log x (15) = 75. log (x 1) = log ( x) + log x= Write a small paragraph describing what an asymptote is. Be sure to describe the various types of asymptotes and their effects on graphs. 16 Page

17 Section XIII: Right Triangles Use the Pythagorean theorem to find the missing side of each triangle. Leave your answers in simplest radical form Page