Incoming Magnet recalculus / Functions Summer Review ssignment Students, This assignment should serve as a review of the lgebra and Geometry skills necessary for success in recalculus. These skills were taught in previous math courses. Our hope is that this review will keep your mind mathematically active during the summer, identify weaknesses in lgebra and Geometry, if they exist, and prepare you for the fun and challenging year ahead. ecause of the diverse backgrounds of the students coming into the magnet program some of the problems may be more challenging than others. We expect that you will do your best with this material and make an attempt of all the problems. irections: - nswer all questions on a separate sheet of paper. - Show all work. - arefully and neatly label your problems and solutions, including the original problem. - If your answer involves radicals or π, give an exact answer and a decimal approximation using a calculator This assignment will be collected on the first day of school. Enjoy your summer. See you in ugust ready to learn!!! I. onvert from one kind of units to another: 1) 159 cm = mm ). m = km ) 18 inches = feet 4) feet = 4 miles 5).6 yards = feet II. Find the perimeter and area of each of the following figures. 95 1) ) 47 5 m 9 4m 1 m Triangle III. For each of the following circles: 1. If the radius is 5. cm, find the area and the circumference.. If the circumference = 6π m, find the radius and the area.. If the area = 14 π cm, find the circumference and the diameter. IV. Simplify. 1) 8 ) 4 7 ) 4) 16 a b 5) 8 + 18-6) 1 14 6 V. Solve for x in each of the following equations: 1) 5 x 6 7 = x ) 8 6 4 = x + x 7 ) x + 4 = 6 4) = x x + 1 4 1 5) ( x + 1) = 4 6) x = 5 1
VI. omplete the following. 1. a) Give the equation of a line with a slope of 0 and a y-intercept of (0, 1).. a) Give the equation of a line that contains the points (-, ) and (-6, -5).. a) Give the equation of a line with a slope of - and a y-intercept of (0, 5). 4. a) Give the equation of a line perpendicular to x 4y = and passing through (1, 1). VII. Multiply the polynomials and expand. 1) ( x 9 )( x + 8) ) ( 8) 4) ( 1)( x + 5) x 5) x + y ( ) x ) x + ( ) 6) ( x )( 4 + x x ) VIII. Solve the following equations for x by factoring: 1) x x 7 = 0 ) x + 9x 5 = 0 ) 4x 6x + 7 = 0 4) x 16x + 64 = 0 5) x 64 = 0 6) x 4 1x + 6 = 0 IX. Solve the following equations for x by using the quadratic formula (remember to give all solutions in two ways: exactly, using radicals and an approximation using your calculator): 1) x + x 5 = 0 ) x 4x + 7 = 0 X. Solve the following systems of equations: 1) 5x + 4y = 6 x y = 1 ) x + y = 8 y = x XI. For each of the following functions: a) Graph the function b) State the domain of the function using interval notation. Example: [, ) or (, 7) c) State the range of the function using interval notation 1) f ( x) = x + 4 ) ( x) = x + 4 + f ) f ( x) = ( x ) 1 4) f ( x) = x + 6x + 1 5) f ( x) = x 4 6) f ( x) = x 7) f ( x) = x + 8) f ( x) = x + 9) f ( x) = x + 5
XII. For each of the following inequalities, sketch the set of points in the xy -plane that satisfies the inequality: 1) y x + 1 ) y < x + 4 ) y 4 4) x > 5) y < x 6) y > x XIII. Simplify the following expressions: 1) ( + 4x 7) + ( x 7x + 8) x ) 4 4 ) ( 9 4a + a a 7) ( 10a + a a a + 8) 64 x a 4) x z( x z) 5 y - 16 x y + x y 8 x y 5 5) xy ( x y) 6) ( + x 1)( x ) x 7) 7 10 a b c 5 7 5 a bc 4 5 8) ( 8 )( b a b a ) 9) ( x y z) 4 10) ( 15a b c) 0 XIV. x y 11) - 5 6 x y Solve for x in each of the following equations: 1) x = 8 ) x - 5 = x + 4 ) - x = 4 4) x - 4= XV. 1. Let parallel lines and be intersected by line XY at the points on and Q on in such a way that and are on one side of XY and and are on the other. nswer the following questions using this figure: X Q Y 1 a. If m X = x +, m YQ = x 8, find the measure of each of these angles. b. If m Q x x m Q x x = + + 1, = + 7 + 9, find the measure of each of these angles. c. What is the measure of the angle formed by the intersection of the angle bisector of Q and the angle bisector of Q?
. In Δ, if the ratio of m : m : m = :4:5, find the measure of each angle of the triangle.. In Δ, extend side past to the point. If m = x + 5x 5, m = 5x x, and m = 10, find the measure of each of these three angles. (Give all solutions that work.) 4. In Δ, suppose that the bisector of angle meets side at point E. If = 1, = 14, and = 18, find E and E. 1 14 E 18 5. Given right triangle with right angle at, altitude is drawn to the hypotenuse of the triangle. If = 1,and = 4, find,, and. 6. In parallelogram, = 1, = 0, = 5x + y, and = x + y. Find the lengths of the sides of the parallelogram. 7. If is a rhombus with diagonal = 10, and diagonal = 4, find the perimeter of the rhombus. 8. If is a rectangle and is any point in its interior, prove that + = +. 9. In trapezoid with bases and, if = 10 and =, find the length of the median (also known as midsegment) of the trapezoid. 4
10. Given trapezoid with bases and, draw diagonals and. Let E be the midpoint of and F the midpoint of. a. rove that E and F lie on the midsegment of the trapezoid. b. If = 10 and =, find EF. 11. Suppose isosceles trapezoid has = = 1, = 10, and =. Find the area of the trapezoid. 10 1 1 XVI. 1. Given circle O with points,,, and on the circle, answer the following: a. If m O = 60 and O is 8, determine the area of sector O. b. If m O = 0 and O = 10, determine the area of the segment formed by chord and arc. c. If = 16 and O is 10, how far is chord from the center O.. Let chords and of circle O intersect at point E. If E = x, E = x 1, E = x 1, and E = 4x, find the lengths of these chords.. From a point outside of circle O, let R be tangent to the circle at R and let secant M intersect the circle at M and N (with M between and N). If M = 9 and MN =, find R. R O M N 4. From a point outside of circle O, let R be tangent to the circle at R. Find O if the radius of the circle is 5 and R = 1. R O 5
XVII. roofs: 1. Use a truth table to prove the validity of [( ~ Q) Q] ( Q). rove that the base angles of an isosceles triangle are congruent three times in three different ways. XVIII. Given the indicated measures of angles and lengths of sides, solve the triangles below for the missing parts. 1. Given right triangle, m = 56 o, a = 4km, c = 51km.. m = 4, m = 6, a = 9cm. a = 1.1m, b = 4.6m c= 1.0m 6