Summer Work for students entering PreCalculus Name Directions: The following packet represent a review of topics you learned in Algebra 1, Geometry, and Algebra 2. Complete your summer packet on separate paper and remember to show all of your own work. Work is to be done NEATLY on separate paper with the answers clearly written on the blanks provided. You will need graph paper for some problems. You will staple your work to the back of the packet. Do not do the whole packet right away or you will forget some of the concepts before the fall. Do not leave the packet until the end of the summer or you will have forgotten some of the concepts. You have learned how to do everything in this packet at some point during your Algebra and Geometry courses, there is nothing new. Use your notes to help you with the packet. If you do not remember how to do some of these problems, watch a video on khanacademy.org (Algebra, Geometry, or Trigonometry section) or a You Tube video to help you remember the curriculum. The summer math packet is due on the first day of school. You will have a test during the first week of school to make sure you have done the packet and are ready for your new math class. We are looking forward to seeing you in August. Have a great summer!! Find the measure of each angle in the problem. 1) Supplementary angles with measures 2p + 3 and 3p + 2 degrees 1) 2) Complementary angles with measures 4x and 5x - 27 degrees 2) Simplify the expression. 3) 112 3) 4) 75 36 4) Rationalize the denominator. 8 5) 7 5) The lengths of the legs of a right triangle are given. Find the hypotenuse. 6) a = 10, b = 20 6) 1
The lengths of one leg, a, and the hypotenuse, c, of a right triangle are given. Find the length of the remaining leg. 7) a = 7, c = 25 7) Given the side lengths of a right triangle, find the indicated value. 8) opp = 4, adj = 33 Find sin θ. 8) Solve the problem. 9) Find the length of a leg in a 45-45 -90 triangle if the hypotenuse is 16 ft. If your answer is not an integer, leave it in simplest radical form. 9) 10) Find the length of the long leg and the hypotenuse in a 30-60 -90 triangle if the short leg is 6 in. If your answer is not an integer, leave it in simplest radical form. 10) 11) Find the length of the short leg and the hypotenuse in a 30-60 -90 triangle if the long leg is 17 m. If your answer is not an integer, leave it in simplest radical form. 11) 12) Find the length of the hypotenuse in a 45-45 -90 triangle if the legs are 12 2 ft. If your answer is not an integer, leave it in simplest radical form. 12) Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. 13) Find tan B when a = 9 and b = 7. 13) 14) Find cos A when a = 17 and c = 10. 14) 15) Find sin B when b = 3 and c = 4. 15) Find the approximate value of the expression rounded to two decimal places. 16) sin 70 16) 17) cos 57 17) 18) tan 32 18) Use a calculator to find the value of the acute angle θ to the nearest degree. 19) tan θ = 13.2894 19) 20) cos θ = 0.2286 20) 21) sin θ = 0.8659 21) 2
Find the remaining sides and angles of the triangle ABC, with right angle at C. Round to the nearest tenth if necessary. 22) B = 19, c = 15 22) Solve the right triangle ABC where C is the right angle. Round side lengths to one decimal place and express angles to the nearest tenth of a degree. 23) b = 150, c = 430 23) Solve the problem. 24) From a distance of 49 feet from the base of a building, the angle of elevation to the top of the building is 65. Estimate the height of the building to the nearest foot. 24) Find the distance between the pair of points. 25) (2, 2) and (-6, -7) 25) Find the midpoint of the line segment whose end points are given. 26) (1, 5) and (-7, 6) 26) Write the standard form of the equation of the circle with the given center and radius. 27) (-9, 8); 3 27) Write the algebraic expression without parentheses. 28) - (6z - 7w + 3y) 28) Simplify the exponential expression. 29) (-4x 4 y -5 )(2x -1 y) 29) Add or subtract terms whenever possible. 30) -5 3 + 9 27 30) Rationalize the denominator. 2 31) 4-7 31) Perform the indicated operations. Write the resulting polynomial in standard form. 32) (5x 2 + 4x + 7) + (2x 2 + 3x - 7) - (5x + 2) 32) Find the product. 33) (4xy - 3)(8xy + 7) 33) 34) (3x - 4) 3 34) Factor out the greatest common factor. 35) 21x 4-6x 3 + 15x 2 35) 3
Factor using the formula for the sum or difference of two cubes. 36) 125x 3-27 36) Factor by grouping. Assume any variable exponents represent whole numbers. 37) x 3 + 8x - 5x 2-40 37) Factor the trinomial, or state that the trinomial is prime. 38) 7x 2-23x - 20 38) 39) x 2 + 5x - 24 39) Factor completely, or state that the polynomial is prime. 40) 4x 3-484x 40) Factor the perfect square trinomial. 41) x 2-10x + 100 41) Factor the difference of two squares. 42) 16x 4-1 42) Find all numbers that must be excluded from the domain of the rational expression. x - 7 43) x 2 + 3x - 28 43) Multiply or divide as indicated. 44) x2 + 16x + 64 x 2 + 10x + 16 x 2 + 5x + 6 x 2 + 11x + 24 44) Add or subtract as indicated. 4 45) x + 3-2 x - 3 45) 46) x + 5 x 2 + 6x + 8 + 5x - 2 x 2 + 7x + 12 46) Simplify the complex rational expression. 47) 1-5 x 1 + 5 x 47) 4
Solve the linear equation. 48) -7x - 7-2(x + 1) = 2x - 7 48) 49) x + 9 3 = 37 12 - x - 2 4 49) Solve the equation by factoring. 50) 6x 2 + 19x + 15 = 0 50) Solve the absolute value equation or indicate that the equation has no solution. 51) 8-3 4 x = 6 51) Solve the quadratic equation by the square root property. 52) (2x + 2) 2 = 16 52) Solve the quadratic equation by completing the square. 53) x 2 + 16x = -43 53) Compute the discriminant. Then determine the number and type of solutions for the given equation. 54) 8x 2 = -5x - 2 54) Solve the quadratic equation using the quadratic formula. 55) 2x 2 = -4x - 1 55) Solve the radical equation, and check all proposed solutions. 56) 5x - 4 = 4 56) Solve the inequality. 57) 5x - 1 5 57) Use the given conditions to write an equation for the line in slope-intercept form. 58) Passing through (2, 3) and (5, 2) 58) Graph the line whose equation is given. Use graph paper. 59) y = 3x - 2 59) Graph the equation in the rectangular coordinate system. Use graph paper. 60) 4y = 8 60) 61) 4x - 8 = -20 61) 5
Use the given conditions to write an equation for the line in the indicated form. 62) Passing through (2, -4) and parallel to the line whose equation is y = -8x + 3; slope-intercept form 62) Solve the problem. 63) The function P(x) = 0.95x - 60 models the relationship between the number of pretzels x that a certain vendor sells and the profit the vendor makes. Find P(600), the profit the vendor makes from selling 600 pretzels. 63) Evaluate the function at the given value of the independent variable and simplify. 64) f(x) = -2x + 1; f(-3) 64) 65) f(x) = x 2-5; f(x - 4) 65) 66) f(x) = x + 20; f(-4) 66) 67) f(x) = x2 + 6 ; f(-3) 67) x 3 + 6x Given functions f and g, perform the indicated operations. 68) f(x) = 9x - 6, g(x) = 4x - 8 Find f - g. 68) 69) f(x) = 5x + 9, g(x) = 9x - 6 Find fg. 69) For the given functions f and g, find the indicated composition. 70) f(x) = x 2-2x + 1, g(x) = x 2-2x - 1 (f g)(3) 71) f(x) = -5x + 7, g(x) = 6x + 4 (g f)(x) 70) 71) Use the vertex and intercepts to sketch the graph of the quadratic function. Use graph paper. 72) f(x) = -x 2-8x - 7 72) Find the coordinates of the vertex for the parabola defined by the given quadratic function. 73) f(x) = 9 - (x + 2) 2 73) 74) f(x) = -x 2 + 4x - 8 74) Find the axis of symmetry of the parabola defined by the given quadratic function. 75) f(x) = -2x 2 + 4x + 4 75) 6
Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. 76) f(x) = x 2-2x - 5 76) Find the x-intercepts (if any) for the graph of the quadratic function. 77) f(x) = 2x 2 + 10x - 48 77) Add or subtract as indicated and write the result in standard form. 78) 9i + (-8 - i) 78) Find the product and write the result in standard form. 79) (-7-9i)(3 + i) 79) Divide and express the result in standard form. 80) 3-9i 5-4i 80) Solve the quadratic equation using the quadratic formula. Express the solution in standard form. 81) 6x 2-5x + 8 = 0 81) Find the domain of the given function. 82) f(x) = 9 - x 82) 83) f(x) = x x - 9 83) Find the range of the function. 84) f(x) = x 2 + 3 84) 85) f(x) = 3x - 15 85) Describe how to transform the graph of f into the graph of g. 86) f(x) = x 3 and g(x) = -x 3 86) Describe how the graph of y=x 2 can be transformed to the graph of the given equation. 87) y = (x + 9) 2-3 87) Use the Leading Coefficient Test to determine the end behavior of the polynomial function. 88) f(x) = x - 4x 2-5x 3 88) Find the zeros of the polynomial function. 89) f(x) = x 3 + 6x 2 - x - 6 89) 7
Divide using long division. 90) -6t4-4t 3-9t 2-5t - 15 2t 2-2t + 3 90) Divide using synthetic division. 91) x5 + x 3-5 x + 3 91) Solve the system of equations by the substitution method. 92) -5x + 4y = 6 x - 2y = 0 92) Solve the system by the elimination method. 93) 7x + 21y = 21 6x - 7y = -7 93) Graph the function by making a table of coordinates. Use graph paper. 1 x 94) f(x) = 5 94) Approximate the number using a calculator. Round your answer to three decimal places. 95) e -2.9 95) Write the equation in its equivalent exponential form. 96) log 2 16 = x 96) Write the equation in its equivalent logarithmic form. 97) 6 3 = 216 97) Evaluate the expression without using a calculator. 98) log 5 125 98) Solve the equation by expressing each side as a power of the same base and then equating exponents. 99) 16 x + 8 = 64 x - 2 99) Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 100) 5 x = 20 100) Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. 101) log 3 (x - 2) = 1 101) 8