PreCalculus. American Heritage Upper School Summer Math Packet

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1 ! PreCalculus American Heritage Upper School Summer Math Packet All Upper School American Heritage math students are required to complete a summer math packet. This packet is intended for all students entering PreCalculus for the school year. This packet is designed to help students prepare for their net math class at American Heritage School. Although they should have been presented all of the concepts contained in this packet in their previous classes, this packet will give them an overview of the essentials before entering their new math class. This packet is not designed to be busy work, but designed to help make their transition to the net level an easier one by giving them the opportunity to refresh material that they may have forgotten from previous classes. A quiz will be given on selected topics from this packet on the third day of school. This packet is intended to take approimately two weeks to complete. Students should not cram all of the information the week before school begins. If more eamples are needed on any topics presented, we suggest a search of the Internet for the specific topic. There is an abundance of helpful and FREE math resources that one can find online. Students are epected to print out their own copy of the packet, complete the packet showing all work on separate paper, copy answers on the answer sheets provided AND copy answers on the packet itself. Since the answer sheets will be turned in, having answers on the packet itself will be useful in studying for the quiz. On the first day of school, students will turn in the answer sheets and etra pages showing work. The packet is worth 10% of the first quarter homework grade. Please make sure all work is neat and easy to follow. Come to class prepared to answer and ask questions. Students will review the material in class and then will be quizzed on it. Students should be diligent in their efforts and pay attention to detail to ensure they have a great start to new school year.! Knowledge, Integrity, Compassion

2 PreCalculus Summer Math Packet GEOMETRY REVIEW The lengths of the legs of a right triangle are given. Find the hypotenuse. 1) a = 3, b = 4 2) a = 9, b = 12 The lengths of the sides of a triangle are given. Determine if the triangle is a right triangle. 3) 1, 2, 3 4) 15, 20, 25 5) 15, 36, 39 Find the distance between the points P 1 and P 2. 6) P 1 = (3, 6); P 2 = (-4, -2) Solve the problem. 7) Find the length of each side of the triangle determined by the three points P 1, P 2, and P 3. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. P 1 = (-5, -4), P 2 = (-3, 4), P 3 = (0, -1) Find the midpoint of the line segment joining the points P 1 and P 2. 8) P 1 = (5, -8); P 2 = (-1, -6) Solve the problem. 9) If (-2, -1) is the endpoint of a line segment, and (-1, 2) is its midpoint, find the other endpoint. 10) Find the area A of a triangle with height 8 in and base 7 in. 11) Find the area A and circumference C of a circle of radius 6 in. Epress the answer in terms of π. 12) Find the area of the shaded region. Epress the answer in terms of π. The sides of the square measure 9 cm

3 13) Find the perimeter. Approimate the result to the nearest tenth using 3.14 for π. 6 cm 3 cm 14) 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. 15) 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. Solve the right triangle using the information given. Round answers to two decimal places, if necessary. 16) b = 2, A = 25 ; Find a, c, and B. 2

4 ALGEBRA REVIEW (EXPONENTS) Simplify the epression. Epress the answer so that all eponents are positive. Whenever an eponent is 0 or negative, assume that the base is not 0. 17) (3y) 3 18) (53) -2 Simplify the epression. 19) ) (2-1 ) -3 21) Simplify the epression. Epress the answer so that all eponents are positive. 22) -2y2 y4 23) (9 3 ) -2 24) y -1 25) (-6y5) -6 z3 Simplify the epression. 1 1/2 26) 64 27) 32 4/5 28) 49-3/2 29) 27-4/3 3

5 ALGEBRA REVIEW (LINES) Find the slope of the line containing the two points. 30) (8, -7); (-3, 8) 31) (-5, 7); (-3, 7) 32) (-9, 6); (-9, 5) Find the general form of the equation for the line with the given properties. Graph the line. 33) Slope = - 3 ; containing the point (0, 5) 5 10 y Find an equation for the line with the given properties. 34) Parallel to the line y = -2; containing the point (6, 4) 35) Perpendicular to the line y = 2-4; containing the point (1, -2) 4

6 ALGEBRA REVIEW (DIVISION) Find the quotient and the remainder using long division. 36) divided by 2 37) divided by + 9 Use synthetic division to find the quotient and the remainder. 38) is divided by ) is divided by ) is divided by ) is divided by + 5 Use synthetic division to determine whether - c is a factor of the given polynomial. 42) ; + 2 Use synthetic division to find the quotient and the remainder. 43) is divided by Use synthetic division to determine whether - c is a factor of the given polynomial. 44) ; ) ; - 7 5

7 ALGEBRA REVIEW (FACTORING) Factor completely. If the polynomial cannot be factored, say it is prime. 46) ) ) ) ) 15z z ) t + 9t 2 52) ) ) f() = ) f() = Factor completely. 56) (2 + 1) 2 + 3(2 + 1) ) 3( + 5) 2 (2-1) 2 + 4( + 5) 3 (2-1) 6

8 ALGEBRA REVIEW (RATIONAL EXPRESSIONS) Reduce the rational epression to lowest terms. 58) ) ) Reduce the epression to lowest terms. 61) ( 2 + 2) 8 - (8 + 9) 3 (2 + 2) 3 62) (2 + 5)4-2 2(2) (2 + 5) 2 Perform the indicated operations and simplify the result. Leave the answer in factored form. 63) ) ) )

9 67) ) ) ) 1 ( - 4)( - 5) - 3 ( - 5)( - 2) 71) ) ) ) )

10 ALGEBRA REVIEW (EQUATIONS) Solve the equation. 76) = Solve the equation. 77) 3( + 4) = 4( - 8) 78) = ) = 14 ( + 5)( - 5) Solve the equation by factoring. 80) = 0 81) = 0 82) ( - 10) + 24 = 0 83) - 4 = Solve the equation by the Square Root Method. 84) ( - 5) 2 = 4 85) ( + 6) 2 = 10 Solve the equation by completing the square. 86) = 3 87) = 0 Find the real solutions, if any, of the equation. Use the quadratic formula. 88) = 0 89) = 0 Find the real solutions of the equation. 90) = 0 9

11 91) 4( + 1) 2 + 7( + 1) + 3 = 0 Solve the equation. 92) = 6 93) 6-3 = 4 ALGEBRA REVIEW (INEQUALITIES) Solve the inequality. 94) ) ) ALGEBRA REVIEW (COMPLEX NUMBERS) Write the epression in the standard form a + bi. 97) (4 + 3i) - (-9 + i) 98) (-5 + 8i)(3 + i) 99) 6i(2-7i) 100) i Perform the indicated operations and simplify the result. Leave the answer in factored form ) ) 5-45i 5-4i Solve the equation in the comple number system. 103) = 0 104) = 0 105) Find f( + 1) when f() =

12 ALGEBRA REVIEW (FUNCTIONS) Find the value for the function. 106) Find f(4) when f() = ) Find f(-9) when f() = ) Find -f() when f() = Solve the problem. 109) If f() = C and f(-2) = 1, what is the value of C? Find the domain of the function. 110) f() = ) Find f(-) when f() = Find the domain of the function. 112) h() = ) - 4 For the given functions f and g, find the requested function and state its domain. 114) f() = 3-2; g() = Find f + g. Solve the problem. 115) Find (f + g)(1) when f() = - 5 and g() = + 6. The function f is one-to-one. Find its inverse. 116) f() = , 0 117) f() =

13 Evaluate the epression using the graphs of y = f() and y = g(). 118) Evaluate (fg)(-1). For the given functions f and g, find the requested composite function value. 119) f() = + 3, g() = 3; Find (f g)(4). 120) f() = 9 2-6, g() = 9-9; Find (f g)(2). For the given functions f and g, find the requested composite function. 121) f() = 6 + 7, g() = 2-1; Find (f g)(). 122) f() = 7-5, g() = 4 3 ; Find (f g)(). 12

14 ALGEBRA REVIEW (CONICS) Write the standard form of the equation of the circle with radius r and center (h, k). 123) r = 8; (h, k) = (5, 9) Write the standard form of the equation of the circle with radius r and center (h, k). 124) r = 10; (h, k) = (-2, 0) Write the standard form of the equation of the circle. 125) y Solve the problem. 126) Find the equation of a circle in standard form where C(6, -2) and D(-4, 4) are endpoints of a diameter. 127) Find the equation of a circle in standard form with center at the point (-3, 2) and tangent to the line y = 4. Find the center (h, k) and radius r of the circle with the given equation. 128) ( + 2) 2 + y 2 = 4 129) 2 + (y + 8) 2 = 81 Find the equation of the parabola described. 130) Verte at (6, 2); focus at (7, 2) 131) Verte at (1, 7); focus at (1, 5) Find the verte, focus, and directri of the parabola with the given equation. 132) (y + 1) 2 = 20( - 2) 133) ( + 4) 2 = 8(y - 2) 13

15 Find the verte, focus, and directri of the parabola. Graph the equation. 134) 2-12 = 12y - 96 y Write an equation for the graph. 135) y (1, -2) -5 Find the center, foci, and vertices of the ellipse. 136) 16( - 1) 2 + 9(y + 2) 2 = 144 Graph the equation. 137) 4( + 2) (y + 1) 2 = 64 y Find an equation for the hyperbola described. 138) center at (6, 1); focus at (0, 1); verte at (5, 1) 14

16 Find the center, transverse ais, vertices, foci, and asymptotes of the hyperbola. 139) ( + 4) (y - 2)2 36 = 1 Graph the hyperbola. 140) ( + 1) 2-9(y + 2) 2 = 9 y

17 !"#$%&'(&()*+(,,#"*!%'-#.*/0)1#")* * 2%,#3* ****!#"5673* * Geometry Eponents *

18 30. Lines ##$" Factoring " 46. Division !"

19 Rational Epressions Equations !"

20 Inequalities Functions " Comple Numbers !"

21 Conics " #$%&" #$'&" " #%(&" " See graph.!" "

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