These are the skills you should be proficient in performing before you get to Pre-AP Calculus.

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1 Fort Zumwalt School District PRE-AP CALCULUS SUMMER REVIEW PACKET Name: 1. This packet is to be handed in to your Pre AP Calculus teacher on the first day of the school year. 2. All work must be shown in the packet OR on separate paper attached to the packet. 3. Completion of this packet will be counted toward your first quarter grade. 4. An assessment of these skills is likely to occur the first week of school. These are the skills you should be proficient in performing before you get to Pre-AP Calculus. Linear Equations graphing lines, defining slope, writing linear equations in point-slope and slopeintercept forms Factoring factor a variety of polynomial expressions, factor out GCF, difference of squares, perfect square trinomials, general trinomials, factoring by grouping Solve Quadratic Equations by Factoring- example: 2x x + 24 = 0 Solving Quadratic Equations by Completing the Square Solving Quadratic Equations using the Quadratic Formula Graphing: parabolas, square root function, absolute value, exponential functions Properties of Logarithms Solving Systems of Linear Equations Properties of Exponents Right Triangle Trig and Special Right Triangles Equations of Circles and Graphing Circles Rational Expressions Kahnacademy.org is a great resource for tutorials and extra practice. The highlighted skill is not in this packet, but you should be familiar with the shapes of those parent graphs before school starts. 1

2 Geometry: Pythagorean Theorem (right triangles): a 2 b 2 c 2 Find the value of x x x 5 x 8 x A square has perimeter 12 cm. Find the length of the diagonal. Solve for x and y x y 4 x y 45 2

3 x y y x The lengths of the legs of a right triangle are given. Find the hypotenuse. 9. a = 3, b = a = 9, b = 12 Find the distance between the points P1 and P P1 = (3, 6); P2 = (-4, -2) Solve the problem. 12. Find the length of each side of the triangle determined by the three points P1, P2, and P3. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. P1 = (-5, -4), P2 = (-3, 4), P3 = (0, -1) 13. Find the midpoint of the line segment joining the points P1 and P2. P1 = (5, -8); P2 = (-1, -6) Solve the problem. 14. If (-2, -1) is the endpoint of a line segment, and (-1, 2) is its midpoint, find the other endpoint. 2

4 15. Find the area A of a triangle with height 8 in and base 7 in. 16. Find the area A and circumference C of a circle of radius 6 in. Express the answer in terms of π. 17. Find the area of the shaded region. Express the answer in terms of π. The sides of the square measure 9 cm and a circle is inscribed. 18. A diagonal of a rectangle measures 12 cm. The length of the rectangle is 7 cm. Find the area and perimeter of the rectangle. Round to two decimal places, if necessary. Solve. If necessary, round to the nearest tenth. 19. If a tree 22.5 feet tall casts a shadow that is 9 feet long, find the height of a tree casting a shadow that is 17 feet long. 20. Solve the right triangle using the information given. Round answers to two decimal places, if necessary. 3

5 Simplify each of the following (2x 8 3 ) ( 5 6)( 5 + 2) Rationalize

6 Complex Numbers: For of x num r - r a s r s r nd s r r A w ys s subs ons nd -1 o s fy for rfor ng ny o r on p P i 5 M subs on S s w fy Subs Simplify. ny o r r w n +, -,, or ( w ys s fy ) p 2i(3 i) 2(3i) 2i(i) D s r 6i 2i 2 S fy 6i 2( 1) M subs on 2 6i S fy nd r wr n x for S, no nsw r n n n or RA I NA I (2 8i) 3(5 7i) 32. (3 4i) (6 4i)(6 4i) Rationalize i 5i 5

7 Equations of Lines: 35. State the slope and y-intercept of the linear equation: 5x 4y = Find the x-intercept and y-intercept of the equation: 2x y = Write the equation in standard form: y = 7x 5 Write the equation of the line in slope-intercept form with the following conditions: 38. slope = -5 and passes through the point (-3, -8) 39. passes through the points (4, 3) and (7, -2) 40. x-intercept = 3 and y-intercept = 2 6

8 Graphing: Graph each function, inequality, and/or system x 4 y x y 4 x y y 4x y + 2 = x y > 2 x y + 4 = (x 1) 2 Vertex: x-intercept(s): y-intercept(s): 7

9 Systems of Equations: Subst tut on: S 1 on for 1 r. R rr. substitue o 2 on. m nat on F nd oppos ff s for 1 r. M y on(s) by ons (s). Add ons r ( os 1 r ). S for r r. S for r. n nsw r k o n or on o s for 2 r. y 6 3x s 1 on for y 2 2(6-3 ) 4 8x 16 x 2 s s r solve on fy Substitute x 2 k o or 2 6x 2 y 12 y 1 on by ff s of y r oppos 8x 16 dd x 2 s fy Solve each system of equations. Use any method. 2x y x 2 y 1 2x y x y 14 2w 5z w 3z 10 8

10 Exponennts Express each of the following in simplest form. Answers should not have any negative exponents a o 51. 3c 52. 2ef (n3 p 1 2 ) c 1 e 1 (np) (3m) 2 (5m 7 ) 55. (2a 3 ) ( b 3 c 4 ) m(3a 2 ) 9

11 Polynomials Simplify x 3 9 7x 2 x m 6 (2m 5) 60. 3(b 4a 2 ) 2( 3a 2 7b) Multiply Polynomials- Use the distributive property ( x+3) ( x + 4) =x x + 4x +3x +12 = x 2 +7x +12 Multiply. 61. (3a + 1)(a 2) 62. (2x + 3)(5x 6) 63. (2c 5) (5x + 7y)(5x 7y) 10

12 4 Terms -- Factor by Grouping a.) Pair up first two terms and last two terms b.) Factor out GCF of each pair of numbers. c.) Factor out front the parentheses that the terms have in common. d.) Put leftover terms in parentheses. Ex: x 3 3x 2 9x 27 = Factor completely. Visit kahnacademy.org to watch tutorials on how to factor ax 2 + bx + c 65. z 2 4z x x k 2 2k 60 11

13 68. 10b 4 15b c 2 30c n z mn 2mt 2sn 2st x 2 52x x x x 2 18xy + 9y x 2 56x (2x + 1) 2 + 3(2x + 1) (x + 5) 2 (2x - 1) 2 + 4(x + 5) 3 (2x - 1) s quad at quat ons, t y to f t f st and s t h f t qua to o. S f you ab. If th quad at d s N t, us quad at f mu a. EX: x 4 x 21 x 4 x 21 0 (x 3)(x 7) 0 x 3 0 x 7 0 x x 7 Set equal to zero FIRST. Now factor. Set each factor equal to zero. Solve each for x. Solve each equation. 79. x 2 4x x x 81. 3x 2 + 7x - 20 = x(x - 10) + 24 = 0 12

14 QUADRATIC FORMULA allows you to solve any quadratic for all its real and imaginary roots. EX: Solve the equation: x 2 2x 3 0 a= 1 b = 2 c= 3 x = b ± b2 4ac 2a x = 2± 22 4(1)(3) 2(1) x = 2± 8 2 x = 2±2i 2 2 x = -1 ± i 2 Solve the equation by the Square Root Method. 83. (x - 5) 2 = (x + 6) 2 = 10 Solve the equation by completing the square. 85. x 2 + 4x = x 2 + 5x - 5 = 0 Find the real solutions, if any, of the equation. Use the quadratic formula. 87. x 2 + 4x - 9 = x 2 - x + 4 = 0 Find the real solutions of the equation. 89. x 4-10x = (x + 1) 2 + 7(x + 1) + 3 = 0 13

15 Find the quotient and the remainder using long division x 8-10x 3 divided by 2x 92. x 2 + 3x - 45 divided by x + 9 Use synthetic division to find the quotient and the remainder. 93. x 2 + 6x + 4 is divided by x x 3 - x is divided by x x 3-16x x + 30 is divided by x x 5 + 8x x x x + 13 is divided by x + 5 Use synthetic division to determine whether x - c is a factor of the given polynomial x 3-11x 2-35x + 98; x + 2 Use synthetic division to find the quotient and the remainder x 5-5x 4 + x - 4 is divided by x + ½ 14

16 Use synthetic division to determine whether x - c is a factor of the given polynomial. 99. x 3-5x 2-29x + 105; x x 3-20x 2-21x + 98; x 7 Algebra Review- Functions Find the value for the function Find f(4) when f(x) = x 2 + 5x Find f(-9) when f(x) = x Find -f(x) when f(x) = -3x 2 + 5x + 1. Solve the problem If f(x) = 8x 3 + 7x 2 - x + C and f(-2) = 1, what is the value of C? 105. Find f(-x) when f(x) = -2x2 + 5x - 4. Find the domain of the function. x 106. f(x) = x g(x) = x 3 x 3 49x 108. h(x) = x x 4 For the given functions f and g, find the requested composite function f(x) = 6x + 7, g(x) = 2x - 1; find (f o g)(x) f(x) =, g(x) = 4 ; find (f o g)(x) x 5 3x For the given functions f and g, find the requested composite function value f(x) = x + 3, g(x) = 3x; find (fog) (4) f(x) = 9x 2-6x, g(x) = 9x - 9; find (f o g)(2). 15

17 Find the inverse, f 1 (x), if possible f (x) 5x f(x) = 1 2 x f(x) = 8x+6 9x 6 ALGEBRA REVIEW (RATIONAL EXPRESSIONS) Reduce the rational expression to lowest terms x2 36 x x+2 6x 2 +16x x2 60x+32 x (x 2 +2) 8 (8x+9) 3x (x 2 +2) 3 16

18 Perform the indicated operations and simplify the result. Leave the answer in factored form x 2 x 2x2 9x x 2 +7x+12 x x 2 +4x x 2 +11x (x 4)(x 5) (x 5)(x 2) x 2 3x x

19 x x x x + 9 x 9 x 2 9 Solving Rational Equations = 14 x+5 x 5 (x+5)(x 5) 135. x 5 = x x

20 Change the following to logarithmic form = = k 5 = y 139. ( 3) x = 81 Simplify 140. log log log log 1000 ALGEBRA REVIEW (Circles) The standard form of an equation of a circle is (x h) 2 + (y h) 2 = r 2 where the center of the circle is (h, k) and the radius is r Write the equation of the circle if r=8 and the center is ( 5,9) Write the equation of the circle with radius = 6 and center is ( -3,4) Name the center and the radius of the circle with the given equation. a. (x + 2) 2 + (y 1) 2 = 4 b. x 2 + (y + 8) 2 = Sketch the graph of the circle x 2 + (y 2) 2 = 2 19