Lyman Memorial High School. CP Pre-Calculus Prerequisite Packet. Name:

Transcription

1 Lyman Memorial High School CP Pre-Calculus Prerequisite Packet 018 Name:

2 Dear Pre-Calculus Student, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These concepts need to be reviewed and practiced throughout the summer. The completion of this review packet is very important and essential for your success in Pre-Calculus. These skills are used frequently throughout this course. Pre-Calculus is a rigorous and fast-paced course. There will be extensive use of graphing calculators which is required for this course. A TI-84 Plus graphing calculator is recommended. Any other type of graphing calculator will have to be approved by the teacher. For this prerequisite packet, calculators should be used only to check work. The Pre-Calculus prerequisite packet is due the first day of school. It will be graded and it will count as a test grade. Work must be shown to support all answers. Your test grade will reflect both, your effort (50%) which is based on attempting all problems and showing work for all problems, and accuracy (50%). The packet is broken into specific concepts. Some sections have worked out examples followed by problems for you to complete. Be sure to complete each numbered exercise included in this packet. Below are a few websites you may wish to visit for additional examples and support. Algebra1online: Algebra online: Algebra Help: Geometry: Results from the summer prerequisite work will help guide skill and concept reinforcement lessons that will take place the first few weeks of school. Have a nice summer, Lyman Math Department

3 Part 1 Lines and Coordinate Geometry Algebra Concepts Slope-intercept form of a line y = mx + b Standard form of a line Ax + By = C Point-slope form of a line y y 1 = m(x x 1 ) Slope of a line m = y y 1 x x 1 Midpoint formula ( x 1+x Geometry Concepts, y 1+y ) Distance formula d = (x 1 x ) + (y 1 y ) Perpendicular bisector a perpendicular line passing through the midpoint of a segment. Altitude of a triangle a segment from a vertex perpendicular to the opposite side. 1) Write an equation of the line in slope-intercept form that passes through (,1)and (1,6). ) Write the equation of the line parallel to the line y = x + 1 in point-slope form 3 passing through the point (-10,). 3) Write the equation of the line in slope-intercept form passing through the point (, 4) and perpendicular to the liney = 1 x 7 4) Find the distance between the points and then find the midpoint of the segment that joins them. a) (0,8) and (6,16) b) (,5) and (10,0)

4 Part Exponents & Roots Properties of Exponents a m a n = a m+n Ex: x 5 x = x 7 a m a n = am n Ex: x8 x 5 = x3 a 0 = 1 a 0 (a m ) n = a m n Ex: (x 5 ) = x 10 ( a b )m = am b m Ex: ( x )3 = 8 x 3 a n = 1 a n x Ex: = 1 1 x (ab) m = a m b m Ex: (4xy ) 3 = 64x 3 y 6 Properties of Radicals 1 = a n an Ex: 1 x = x Common Errors ab = a b a a = ( a) = a a b = a b Rationalizing the denominator: a = a b = a b b b b b Simplify the expression. Eliminate any negative exponents. 5) ( 3x y 4 ) 3 6) (3x3 ) 1 6x 7) (x y 3 ) 4 (xy 4 ) 3 x y 8) ) ) 48 3

5 Part 3 Factoring & Solving Quadratic Equations GCF AC Method Guess and check Box Method Grouping Difference of two squares Factoring Methods Sum/difference of cubes Factor each quadratic expression completely. 11) x + 8x + 7 1) x + x 4 13) x 7x ) 4x + 7x + 35 Factor each expression completely. 15) 8y 3 + 7y 16) 16x 3 + 1x + 1x ) x ) 4x 11

6 By Finding Square Roots Solving Quadratic Equations Factoring EX: Solve x 8 = 0 x = 8 x = ± 8 = ± x = and x = Quadratic Formula EX: Solve 3x 5x 1 = 0 Factor: (3x + 4)(x 3) = 0 Zero Product Property: 3x + 4 = 0 and x 3 = 0 x = 4 3, 3 By Graphing EX: Solve x + x 6 = 0 Graph y = x + x 6 EX: Solve x + 5x + 6 = 0. x = 5 ± 5 4(1)(6) (1) The x-coordinate of the vertex is b = 1, and the axis of a symmetry is x = 1. Make table of values using x values around 1 x = 5 ± 1 x = 5+1 = 4 = and x = 5 1 x =, 3 = 6 = 3 Plot the points: From the table and the graph We can see the zeros of the function Are and 3. x =, 3 Solve the equation using the indicated method. 19) Solve by finding square roots. 0) Solve by factoring. 8x 4 = 14 x 9x + 4 = 0

7 1) Solve using the Quadratic Formula. ) Solve by graphing (find the x-intercepts) 3x 11x 4 = 0 x 10x + 1 = 0 Part 4 Inequalities Solve each inequality. Graph the solution. 3) x ) 5 3x 35 Part 5 Functions & Graphs Determine if the graph represents a function. 5) 6) 7) 8)

8 Sketch the graph of the each function. If you need a reminder, use your graphing calculator to help remember transformations of functions. Keep in mind, you need to be able to graph functions without a graphing calculator. 9) f(x) = x + 30) f(x) = (x + ) 31) f(x) = x + 3 3) f(x) = (x + 1) 3 + 1

9 Combining Functions & Compositions of Functions Let f(x) = 1 and g(x) = x x Combining Functions f(x) + g(x) Ex: (f + g)x = 1 x f(x) g(x) Ex: (f g)x = 1 x Given two functions f and g, the composite function, f g, (also called the composition of f and g) is defined by (f g)(x) = f(g(x)) Find f + g and f g 33) f(x) = x 3, g(x) = x 34) Given f(x) = 6x 5 and g(x) = x find: a) f g b) g f c) f f Part 6 Polynomial Functions Polynomial Division Long Division EX: x 3 5x + x 7 x 3 Synthetic Division Quotient: x + x x 3 x + x x 3

10 Divide the polynomials using long division 35) x 3 3x x x 3 Divide the polynomials using synthetic division. 36) x 3 + x + x + 1 x + 37) x4 +3x 3 1 x+4 38) For the graph pictured at the right: a) Describe the end behavior b) Determine whether it represents an odddegree function or an even-degree function c) State the number of real zeros

11 Part 7- Right Triangles & Trigonometry For the right triangle pictured: SOHCAHTOA Pythagorean Theorem sin A = opp hyp c sin 1 ( a ) = A c a + b = c cos A = adj hyp = b c cos 1 ( b c ) = A tan A = opp adj = a b tan 1 ( a b ) = A Find the value of the trig function expressed as a fraction and find the value of the angle to the nearest degree.. 39) 40) sin S = cos S = tan S = m S = sin A = cos A = tan A = m A = Using Trigonometric Equations Example: Find the length of. YZ Write and solve a trig equation to find the value of x. 41) Solve for the value of x. Round to the nearest hundredth. x