Summer }Issignment for.jl<p Ca(cu(us <BC

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1 Summer }Issignment for.jl<p Ca(cu(us <BC You are receiving this packet because you are currently enrolled in AP Calculus BC for the school year with Mrs. Squires. This course covers a full year of college calculus; much will be expected of you. Because of the demands of the AP curriculum, there is not much time for repetition and drill. You may feel at points throughout the year that more time is needed on certain concepts; you will have to find time to. work on your own, or schedule meetings for extra help after school. I expect, as students enrolled in AP Calculus, that you have already have the basics of algebra mastered. Solving equations, working with algebraic expressions, and factoring, for example, should all be a part of your mathematical repertoire. The problems selected in this summer packet are to help refresh and sharpen your algebra skills. Once school starts, we will only spend a few days reviewing the topics covered in this summer assignment, and they may appear on any assessment throughout the year.!assignment:! The following packet consists of two parts...!part 11 covers basic topics with an example and/or explanation that will help you to complete the problems that follow (Note: The numbers skip from #45 to #74).!Part 111 covers more challenging and advanced problems with no examples or explanations. If you are struggling, I encourage you to look for help on line, including any of the following websites: modules/ index/ htm It is expected that this work will be completed and ready to hand in on the first day of school. No late submissions will be accepted. This assignment will be counted towards the homework portion of your grade for term 1. Selected questions within the packet will be checked for work shown and accuracy. Assessments throughout the year will also assess your mastery of these algebraic skills. You are welcome to collaborate with other students, but the work in this packet should be of your own effort. Have a great summer and I'm looking forward to working with you next year! Mrs. Squires Lsquires@braintreema.gov

2 !Part l:l Complete all problems, except those that are crossed out. Your work may be done in this packet or on a separate piece of paper. Complex Fractions When simplifying complex fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common denominator of all the denominators in the complex fraction. Example: _7 6_. _7 6_ x+l x+l x+l -7(x+l)-6-7x-13 = = = 5 5 x+l 5 5 x+l x+l -2 3x -2 3x x x-4 x x-4 x(x-4) -2(x-4) + 3x(x) -2x+8+3x 2 3x 2-2x+8 = 1 = = = 1 x(x-4) 5(x)(x-4)-l(x) 5x 2-20x-x 5x 2-21x x-4 x-4 Simplify each of the following a ~ 1. a 2. x x-3 S+a 5+_!Q_ _ x+2 2x-3 x ~ 4. x+l x 5. 3x-4 x x+- x+l x 3x-4

3 Functions To evaluate a function for a given value, simply plug the value into the function for x. Recall: (/ 0 g )(x) = f(g(x)) OR f[g(x)] read"/ofg ofx" means: plug the inside function (in this case g(x)) in for x in the outside function (in this case, f(x)). [Example: Given f(x) =2x 2 +1 and g(x) = x-4 find f(g(x)). f (g(x)) = f (x - 4) = 2(x-4) 2 +1 = 2(x 2-8x + 16) +I = 2x 2 - I6x f (g(x)) = 2x 2 - I6x + 33 Let f(x) = 2x +I and g(x) = 2x 2-1. Find each. 6. /(2) = g(-3) = 8. f(t+i)= f (x +h)-f (x) 9. f[g(-2)]= 10. g[f(m+2)]= 11. = h - Let f (x) =sin x Find each exactly (;)= ( 2 ;)=--- Let f (x) = x 2, g(x) = 2x + 5, and h(x) = x 2-1. Find each. 14. h[f(-2)]= 15. f[g(x-1)]=

4 Find f (x +h)- f(x) for the given function/. h ) 7. f(x) = 9x+3 I~. f (x) =s- 2x Intercepts and Points of Intersection To find the x-intercepts, let y = 0 in your equation and solve. To find the y-intercepts, let x = 0 in your equation and solve. Example: y = x 2-2x - 3 x-int. (Let y = 0) 0 = x 2-2x-3 0=(x-3)(x+1) x = -1 or x == 3 x - intercepts (-1, 0) and (3, 0) y-'int. (Let x = 0) y == 0 2-2(0)-3 y =-3 y-intercept (0,-3) Find the x and y intercepts for each. 19. y =2x y=x 2 +x y = x-/16-x y 2 =x 3-4x

5 Use substitution or elimination method to solve the system of equations. Example: x 2 + y2-l6x+39 =0 x 2 -y 2-9 = 0 Elimination Method 2x 2 -l6x+30=0 x 2-8x+15 =0 (x-3)(x-5)=0 x =3 and x =5 Plug x = 3 and x = 5 into one original 3 2 -/-9 = /-9 = 0 -y 2 = 0 16 =y 2 y=o y=±4 Points of Intersection (5, 4), (5, -4) and (3, 0) Substitution Method Solve one equation for one variable. y2 = -x 2 +16x-39 (1st equation solved for y) x 2 -(-x 2 +16x-39)-9 = 0 Plug what y 2 is equal to into second equation. 2x 2-16x+30 = 0 (The rest is the same as x 2-8x+15 = 0 previous example) (x-3)(x-5) = 0 x =3 or x-5 Find the point(s) of intersection of the graphs for the given equations. x+y=8 x 2 + y =6 2x-3y = x-y = 7 x+y = 4 5x-4y =6 Interval Notation 26. Complete the table with the appropriate notation or graph. Solution Interval Notation Graoh -2 < x ~ 4 [-1, 7).. J.. 8

6 Solve each equation. State your answer in BOTH interval notation and graphically. x < 2x-I~O -4:>:2x-3<4 Domain and Range Find the domain and range of each function. Write your answer in INTERVAL notation f (x) =.,...Jx f (x) =3 sin x 33. f(x)=x-l Inverses To find the inverse of a function, simply switch the x and the y and solve for the new "y" value. Example: f (x) = ~x +1 Rewrite f(x) as y y = ~x+ 1 x = {jy+1 Switch x and y Solve for your new y ( x ) 3 = ( {)y +1r Cube both sides x 3 = y + 1 y = x 3-1 Simplify Solve for y 1-1 (x) = x 3-1 Rewrite in inverse notation Find the inverse for each function. 34. f(x)=2x+l 35. x2 f(x) = 3 5

7 Also, recall that to PROVE one function is an inverse of another function, you need to show that: f (g(x)) =g(f(x)) = x Example: x-9 If: f(x) = -- and g(x) = 4x+9 show f(x) andg(x) are inverses of each other. 4 f (g(x)) = ( x-9) g(f(x)) = ' (4x +9)-9 =x-9+9 4x+9-9 =-- 4 =X 4x = 4 =X f (g(x)) =g(f(x)) = x therefore they are inverses of each other. Prove/and g are inverses of each other. x3 36. f(x) =- g(x)= x 37. f(x) =9-x2, x ~ 0 =~9-x 2

8 Equation of a line Slope intercept form: y =mx +b Vertical line: x = c (slope is undefined) Point-slope form: y- y 1 = m(x-x 1 ) Horizontal line: y = c (slope is 0) 38. Use slope-intercept form to find the equation ofthe line having a slope of 3 and a y-intercept of Determine the equation of a line passing through the point (5, -3) with an undefined slope. 40. Determine the equation of a line passing through the point (-4, 2) with a slope of Use point-slope form to find the equation of the line passing through the point (0, 5) with a slope of2/ Find the equation of a line passing through the point (2, 8) and parallel to the line y = ~x-i Find the equation of a line perpendicular to they- axis passing through the point ( 4, 7). 44. Find the equation of a line passing through the points (-3, 6) and (1, 2). 45. Find the equation of a line with an x-intercept (2, 0) and a y-intercept (0, 3). 7

9 Fill in the unit circle: (_... _) 0 _o - (_,_) 0-0 (_,_)

10 Reference Triangles (If you don't remember how to do this, google "reference triangles") 49. Sketch the angle in standard position. Draw the reference triangle and label the sides, ifpossible. 2 a. -tr b c. d Unit Circle You can determine the sine or cosine for basic angles by using the unit circle. The x-coordinate of the circle is the cosine and the y-coordinate is the sine of the angle. See Unit Circle worksheet for more practice. ExampIe: sm-=-.tr.fj 4tr 1 cos-= tr 50. a.) sin tr= b.) cos 4. 1ltr c.) sin(- tr)= d). sm-= 2 6 e.) cos2tr = f.) cos(-tr) = q

11 Inverse Trigonometric Functions: ecall: Inverse Trig Functions can be written in one ofways: arcsin(x) sin- 1 x >0 cos 1 x >() tan- 1 x >O sin- 1 x <0 tan- 1 x <0 Example: Express the value of"y" in radians. -1 y =arctan.j3 Draw a reference triangle. -1. n n This means the reference angle is 30 or -. So, y = - - so that it falls in the interval from 6 6 -n n 1r -<y<- Answer: y = For each of the following, express the value for "y" in radians y =arcsm y =arccos(-1) 78. y=arctan(-1) 2 10

12 Example: Find the value without a calculator. Draw the reference triangle in the correct quadrant first. Find the missing side using Pythagorean Thm. 6 Find the ratio of the cosine of the reference triangle. 6 cose = {7;" v61 For each of the following give the value without a calculator.

13 IPart ll:j Complete all problems. Your work should be on a separate piece of paper Are the following statements true? If not, explain in words why not? 2k k = 2. --=-+ 3. x+ y = x + y 2x + h x + h p + q p q a= 3a 5. 3~= 3a _ a+b=3a+b 6 3 b 3b b b c c Factor each of the following completely a -b 2 8. a 3 -b x 3 -r y x 2-21x x 2 ~ x x 2 + 6x 3-9x 13. (x+1) 3 (4x-9)-(16x-i-9)(x~1/ 14. (x -1_) 3 (2x - 3)-(2x -12)(x -1/ 15. (2x-1) 2 (x-3)4-(x -r 1)(2x-1) Factor x - a in such a way that.[x-fa is a factor Simplify: xl ~x / 2 x(5x-+ 1)- 3(x h ~1) + (x + h) 17. xi 19. ~ h x-2 14 (x-1) (x -7-1) 3 (4x-9)-(16x ~ 9)(x -i- 1) 2 3x(x + 1)- 2 (2x ~ 1) (x - 6)(x-1) 3 (x-1). 2. )3 2x(x-1) -3(x...:...1 (x - N(2x - 3) - { 4 x - 1) (x - N 23 ' x x 2, 18x (x-1) (2x-1) Solve the equation x 2-21x-18=0 26. x + 3x 2-5x-15 = x - 9x =O )( == 0?x: +Sx -~ 29. Write as a single fraction with denominator in factored form: =O x +1 x Graph the equation y = x - x and answer the following questions. a) Is the point (3, 2) on the graph? b) Is the point (2, 6) on the graph c) Is the function even, odd, or neither? d) What is the y intercept? e) Find the x intercepts.

14 31. Find all intercepts of the graph of y = x x Show work to determine if the relation is even, odd, or neither. x2 32. f(x) =2x f{x) = -4x 3-2x 34. f (x) = 4x 2-4x f(x)=- 2 X Find the equation of the straight line that passes through the point (2, 4) and is parallel to the line 2x + 3y - 8 = Find the equation of the line that is perpendicular to the line 2x + 3y - 8 =0 at the point (1,2) 38. The line with the slope 5 that passes through the point (-1,3) intersects the x axis ata point. What are the coordinates of this point? 39. What are the coordinates of the point at which the line passing through the points (1, -3) and {-2,4) intersects they axis? 40. Given f (x) = Ix find f(1) - f(5). 41. Find all points of intersection of the graphs of x 2 + 3x - y = 3 and x..._ y = If the point (-1, 1 ) lies on the graph of the equation kx 2 - xy + y 2 =5, find the value of k. 43. Write the equation of a graph that is a function. 44. Write the equation of a graph that is not a function Find the domain for each of the following functions ~ h(x) = k(x} =.Jx - 5x x 2-21x - 18 Vx p(x) =-;:===== 48. y =ln(2x-12) -Jx 2 -x For the function y =5 -..J9- x, a) find the domain, b) find the range, and c) determine whether the function is odd, even, or neither. -0.5x x < Let f(x) =.~ a) draw the graph of f(x). b) find the domain, c) find the range. { vx + 2 X2: Find f(x + ~x) for f(x)= x - 2x-3. \J Sketch the graph of each function 2x (-:r,-1) (1 x < f(x) = ~ f(x) J =..2x 2 [-1,2) l-1 x > 0 1 -x +3 (2,x)

15 54. State the domain, range and intercepts of the function y = 2-x Given f (x) = x - 3 and g( x) = JX complete the following 55. f(g(x)) = 56. g(f(x)) = 57. f(f(x)) = Given f(x) = - - and g(x) = x - 5 complete the following x f (g(7)) = 59. g(f(v)) = 60. g(g(x)) = Let f(x) = 2x - 2. Complete the following: 61. Sketch the graph of f(x). 62. Determine whether fhas an inverse function. 63. Sketch the graph of r- 1 (x) 64. Give the equation.for r- 1 (x) Simplify using only positive exponents. Do not rationalize the denominator. _ ) 4x ( x - 4 ) If f(x) = x 2-1, describe in words what the following would do to the graph of f(x). 67. f(x) f(x-4) 69. -f(x + 2) 70. 5f(x) f (2x) 12. jt(x)i 73. (calculator) The dollar value of a product in 1998 is $78. The value of the product is expected to decrease $5. 75 per year for the next 5 years. Write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t =8 represent 1998). 74. (calculator) A business had annual retail sales of $124,000 in 1993 and $211,000 in Assuming that the annual increase in sales follows a linear pattern, predict the retail for (calculator) In order for a company to realize a profit in the manufacture and sale of a certain item, the revenue, R, for selling x items must be greater than the cost, C, of producing x items. If R =69.99x and C =59x + 850, for what values of x will this product return a profit? 76. (calculator) Suppose that in any given year. the population of a certain endangered species is reduced by 25%. If the population is now 7500, in how many years will the population be 4000? 77. A piece of wire 5 inches long is to be cut into two pieces. One piece is x inches long and is to be bent into the shape of a square. The other piece is to be bent into the shape of a circle. Find an expression for the total area made up by the square and the circle as a function of x Let f(x) = ~x -t- 2 and g(x) = x - 2. Which of the following are true? I. g(x) =r- 1 (x) for all real values of x. II. (t c g)( x) == 1 for all real values of x. Ill. The function f is one to one

16 80. Let f(x) = -J3- x. Find an expression for r- 1 (x). (Be sure to state any necessary domain restrictions.) 83. Let y = 3sin (2x - :r) + 2. Determine the period, domain, and range of the function Evaluate: Answers for 90 & 91 must be in radians. 84. coso 85. sino 89. sin n 90. arccos.j3 91. arctan1 2 1[ 1t 86. tan 87. cos ![ Sin Find the solution of the equation for 0 ~ x < 21r 2sin 2 e=1-sine 95. Which of the following expressions are identical? 2 2 a) cos x b) (cosx) 2 c) cosx 96. Which of the following expressions are identical? 1 1 a) (sin xt 1 b) arcsinx c) sin x- d) sinx Solve for x. 97. lne 3 = x 98. lnex = lnx ~ lnx = ern 5 = x 101. ln1-fne =x 102. ln6 ~ 1nx-ln2 = ln(x - 5) = ln(x - 1) - ln(x - 1)