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PAP Algebra 2 with Trig Mid-Term Review Topics First 9-weeks Review Equations & Inequalities Solve absolute value inequalities using interval notation Literal equations Classification of real and complex numbers Polynomials – Lessons 3-1, 3-2 and 3-4 Add, subtract, and multiply polynomials Factoring polynomials using multiple step Radicals – Lessons 5-6, 5-7, 5-8 Simplify, add, subtract, multiply & divide Rationalize the denominator including use of conjugates Simplify expressions with rational exponents Simplify, add, subtract and divide complex numbers Classification of real & complex numbers Polynomial identities to complex numbers Solve simple radical equations Additional Topics Relation vs. Function Representations of functions/relations Graph linear functions Write the equation of a linear function Rate of Change Application Problems Second 9-weeks Review Parent Graphs & Transformations – Lesson 1-1, 1-2, 2-1 Graph Absolute Value Graph Quadratics – parabolas Graph Square Roots Graph Cubic Graph Rational Identify characteristics of each graph ( vertex, intercepts, domain, range, max/min, increasing & decreasing intervals, end behavior Graph inequalities Quadratics – Lessons 2-2 to 2-9 Solve by graphing, factoring, completing the square or quadratic formula Graph standard form of a quadratic function Solve projectile and other application problems Literal Equations & Absolute Value Inequalities Review Solve the literal equation for the given variable 1. A r kt for t 4. V 4 r 2 h for r 2. 3. S 3xh 2 yh for h 4 x 2a for y 5 y 3c BONUS: A cone has a height two times its radius. Express the volume of the cone in terms of the height using the formula for the volume of a cone 1 3 is 𝑉 = 𝜋𝑟 3 . Solve each inequality. Write the solution in interval notation. Graph the solution showing the endpoints. 11. x 5 7 18 13. x5 4 12. 3 x 2 4 23 6 14. 2 x 17 9 Polynomials Review Part 1: Simplify each expression 1. 5𝑥 4 𝑦 3 (4𝑥 2 𝑦 5 ) 4. 4𝑥 3 (3𝑥 2 − 5𝑥 + 7) 2. 24𝑥 6 𝑦 3 𝑧 4 16𝑥 2 𝑦5 𝑧 5. (3x - 4)(2x +7) 2 3. (4𝑎2 𝑏3 )(2𝑎3 ) 32𝑎5 𝑏2 6. (3x +2)(4x2 - 5x + 7) Part 2: Factor Completely. Show necessary steps for full credit. 7. 9m2 - 49 8. 3y3 + 21y2 + 36y 9. 4x2y - 36y 10. 3x2 - 3xy + 6x - 6y 11. 2x5 - 32x 13. 6x2 + 7x - 3 14. x5 - 9x3 + 8x2 -72 12. 8x3 + 125 15. Which is a perfect square trinomial? a. 4x2 +18x + 81 b. 9x2- 25 c. 4x2 – 36x + 81 d. 5x2 – 12x + 36 16. What is the factored form of the polynomial functions with x-intercepts at x = 3, x = 0 and x = - 5? a. f(x) = (x + 3)(x – 5) c. f(x) = x(x – 3)(x + 5) b. c. f(x) = x(x + 3)(x – 5) d. f(x) = (x – 3)(x + 5) Part 4: Graphing Calculator 17. For the functions below, use a graphing calculator to: a) graph the function (draw a rough sketch on the grid) b) find the zeros of the function c) write the function in factored form 17. f(x) = 2x3 - 3x2 - 11x + 6 Radical Expressions and Rational Exponent Expressions Part 1: Radical Expressions – Simplify Each (5 points) 3 1. √9𝑥 2 𝑦 8 2. √24𝑥 3 𝑦 11 3. √(𝑥 − 8)6 4. 3𝑎 √16𝑎9 5. √12𝑥𝑦 ∙ √3𝑥 5 𝑦 8 6. 4 7 √2 Use the window for x: [-10, 10] and y: [-10 , 10] 7. 5√27 + 2√12 − 4√75 8. 3 9. √𝑥 6𝑛 𝑦 10𝑛 2+√5 4−√3 10. 4 5 √𝑥 2 Part 2: Translations (3 points) 11.Write in simplest radical form 2 3 7 12. Write in simplest rational exponent form 4 √𝑥 5 Part 3: Simplifying using rational exponents (5 points) 13. 15. 3 2 81 3 4 14. 1 5 (𝑥 ) (𝑥 ) 16. 3 1 4 3 (𝑥 ) − 64 2 3 Parent Functions & Transformations Part 1: Multiple Choice – Answer each of the following. Write the letter to the correct answer in the blank. 1. In the function 𝑓(𝑥) = 3(𝑥 − 4)2 + 5, what is the vertex? a. (4, 5) b. (3, 5) c. (- 4, 5) d. (4, - 5) 2. At what point do the asymptotes of the function, a. (5, - 3) b. (0, 0) c. (- 5, - 3) 𝑓(𝑥) = 2 𝑥+5 − 3, intersect? d. (- 5, 3) 3. Describe the transformation from the parent function, 𝑓(𝑥) = 𝑥 3 , that would be used to graph the function, 1 3 𝑓(𝑥) = (𝑥 + 4)3 − 5. a. b. c. d. Vertical compression of 1/3, shift to the left 4, shift down 5 Vertical stretch of 1/3, shift to the right 4, shift down 5 Vertical compression of 1/3, shift to the left 4, shift up 5 Vertical stretch of 1/3, shirt to the left 5, shift up 4 4. What is the range for the 𝑓(𝑥) = |𝑥 + 4| − 3? a. [-4, ∞) b. [-3, ∞) c. (-4, -3) d. (-∞, -3] 1 5. Which of the following statements are true for the function 𝑓(𝑥) = 2 |𝑥 − 3| + 1? I. The axis of symmetry occurs at x = - 3. II. The vertical compression or “slope” is ½. III. The vertex occurs at (3, 1). a. I and II b. II and III c. I, II and III d. None of them Part 2: Short Answer – Answer each of the following questions, use complete sentences where appropriate. 6. Write the equation of the square root function 7. Write the equation of the graph below. with a reflection over the x-axis, shifted to the left 8 units, and shift down 8 units. 8. 9. Graph 𝑦 > |𝑥 − 1| − 4 Graph 𝑓(𝑥) = √𝑥 + 2 + 3 10. Sketch the graph of 𝑓(𝑥) = 2(𝑥 − 1)2 + 3 and list the following characteristics Graph Concavity Intervals of Increase and/or Decrease Vertex End Behavior Domain & Range 11. Sketch the graph of 𝑓(𝑥) = Graph 12. Graph 𝑓(𝑥) = (𝑥 − 2)3 + 3 Graph 1 𝑥−2 + 1 and list the following characteristics Vertical Asymptote Horizontal Asymptote Domain Range Point of Inflection End Behavior Concave Up Concave Down