CP1 Math 2 Cumulative Exam Review

Transcription

1 Name February 9-10, 2016 If you already printed the online copy of this document, there are answer corrections on pages 4 and 8 (shaded). Deductive Geometry (Ch. 6) Writing geometric proofs Triangle congruence and its uses Parallel lines and their uses Quadrilaterals and their properties CP1 Math 2 Cumulative Exam Review Exponents & Radicals (Ch. 1) Arithmetic with Radicals Adding, subtracting, multiplying, dividing includes rationalizing the denominator Simplifying radicals Solve simple equations ( = k 3) Classify numbers (real, rational, irrational, integer, non-real, counting) Properties of Exponents (use in both directions) Solving exponential equations (change to same base and solve) Quadratics & Polynomials (Ch. 2 & 3) Solving Quadratic Equations 3 methods to solve quadratics equations: Factoring o Types of quadratics we factored: difference of 2 perfect squares, monics and nonmonics (with and without GCF s) o Methods: factor out GCF; sums and products, splitting the middle, z-substitution Completing the square Quadratic formula Connections between equations and Graphs of Quadratics x-intercepts, roots, zeros are solutions to the quadratic when equal to 0 y-intercept point on y-axis (i.e. f(0)) line of symmetry average of the roots vertex maximum or minimum of parabola, (avg roots, f(avg roots)) or complete the square other points use symmetry of graph to find additional point(s) Formats for quadratic functions & ability to move between forms: Standard Form: y = ax 2 + bx + c Factored Form: y = a(x-r 1 )(x r 2 ) Vertex Form: y = a(x h) 2 + k Standard vertex (complete square); Standard factored (factor) factored or vertex standard (multiply and simplify) Applications of quadratics Types: Projectile Motion & Maximizing area

2 Part I Deductive Geometry (Chapter 6) 1. is a right angle. Find. 2. Find the value of : 3. ABC is isosceles with.. The bisectors of and meet at D. Find the measure of. A D 4. : X C B 5. : is an isosceles trapezoid with legs and. 6. : ; 2

3 7. : is an altitude of A bisects is the median to B D C 8. : Parallelogram with diagonals intersecting at point. 9. Ronda and Stephanie are working to prove that is parallel to in the diagram to the right. Ronda s argument is: because: and by the exterior angle sum theorem. Therefore. Since these are alternate interior angles and they are congruent we know by AIP that the lines are parallel. Where is the incorrect step in Ronda s proof? 10. : is a parallelogram; are midpoints. a. Using geometric relationships explain why is not necessarily a rhombus. b. What must be true in order for to be a rhombus? 3

4 Answers 1. 2a. b VAT SAS CPCTC AIP is an isosceles trapezoid with 1. legs and Definition of a trapezoid The base angles of an isosceles trapezoid are congruent PAI Transitive property ; Addition Property Reflexive Property ASA CPCTC is an altitude of 1. bisects Definition of altitude 3. are right angle 3. Perpendicular lines form right angles All right angles are congruent Definition of angle bisector Reflexive Property ASA CPCTC 9. D is the midpoint of 9. Definition of midpoint 10. is the median to 10. Definition of median 4

5 8. 1. Parallelogram with diagonals 1. intersecting at point. 2. and bisect each other 2. The diagonals of a parallelogram bisect each other Definition of segment bisector VAT Opposite sides of a parallelogram are parallel (Definition of parallelogram) PAI ASA CPCTC 9. The step that is incorrect is: Therefore Though it is true that and, making by the transitive property. It is not true that any of the individual angles need to be equal to each other. For example = is true. Both sums equal 90, but none of the numbers in the equation are equal. 10. a. The opposite sides and angles are equal in a parallelogram, so halves of equal sides would also be equal. This would make by SAS so that by CPCTC. Also, by SAS so that by CPCTC. This does not mean that which would have to be true to make the STAR a rhombus. b. In order for STAR to be a rhombus, In other words, JUMP must be a rectangle. Part II Exponents (Chapter 1) 1. For each equation, find the value of k that satisfies the equation. a. b. c. d Write each expression as a single power of x. Simplify numerical exponents when the exponent is 4 or less. a. b. c. d. e. f. 3. If, find and Simplify your answers as much as possible. 5

6 4. Write each radical in simplified or standard form. a. b. c. d. 5. If, find 6. Evaluate each expression. a. b. c. d. 7. Determine whether each of the following numbers is rational or irrational. Explain your answer. a b. c. d. Answers 1. a. k = 7 b. k = 5 c. k = 6 d. k = 7 2. a. b. c. d. 27 e. f a. 11 b. c. d a. 4 b. 5 c. 3 d. 7. a. rational, can be written as b. rational, can be written as c. rational, can be written as d. irrational, non-perfect square under radical. 6

7 Part III - Quadratics (Chapters 2 & 3) Directions Problems 1-5 should be completed WITHOUT a calculator. 1. Here is a quadratic in vertex form: Sketch a graph clearly labeling: The vertex The zeros The axis of symmetry The y-intercept The other point on the graph with the same y-value as the y-intercept. 2. Here is a quadratic in factored form: Sketch a graph clearly labeling: The vertex The zeros The axis of symmetry The y-intercept The other point on the graph with the same y-value as the y-intercept. 3. A parabola has x-intercepts at 4 and. The y-intercept is at (0, 6). Write an equation in factored form to represent the function described. Hint for finding a: You know the zeros so plug them in, and then figure out the a value by temporarily plugging in the given y-intercept. 4. Solve by any method you choose. a. b. c. d. e. 5. Write an equation in normal form for a quadratic function with roots of and Directions You may use a calculator to complete problems Solve by completing the square: 7. Put in vertex form and identify the vertex: a. b. 7

8 8. A ball is thrown upward from a flat surface. Its height in feet as a function of time since it was thrown (in seconds) is given by the equation. a. Evaluate, and explain what it means in the context of this problem. b. What was the highest the ball got, and when did it reach that height? c. At what time does the ball land? 9. The diagram at the right is a graph of f(x) = 2x x 32. ABC has vertices at the x intercepts and vertex of the parabola. a. Find the area of ABC. b. Find the location of point D that makes quadrilateral ABCD a rhombus. c. Find the quadratic function whose vertex, point E, would make quadrilateral ABCE a kite. Answers: 1. Vertex: Zeros: and 7 Line of symmetry: y-intercept: Other point: 2. Vertex: Zeros: and 2.5 Line of symmetry: y-intercept: Other point: a. 4, -4 Solving directly is the easiest way b. -2 This is a perfect square already. c. 0, 4 Common factor factoring works d. 6, -2 Factor out a GCF and then use sum/product e. First divide by 3, then square root of both sides, remembering the ± a.. Vertex at b. Vertex at 8. a. 43 feet, height of ball after 1 second b seconds, feet c seconds 9. a. 54 sq. units b. (5, 18) c. y = 2(x 5) 2 3 (y-value of vertex may vary) 8