Algebra II Trig Semester 1 Final Review 2022-2023 PDF

Name: _____________________________________________________________ Date: ______________ Period: ____ Algebra II Trig – Semester 1 Final Review 2022-2023 Algebra II Review – Unit 1 1.) Solve and write your solution using interval notation 2|𝑥 + 3| > 6...

Name: _____________________________________________________________ Date: ______________ Period: ____ Algebra II Trig – Semester 1 Final Review 2022-2023 Algebra II Review – Unit 1 1.) Solve and write your solution using interval notation 2|𝑥 + 3| > 6 1 2.) Solve the absolute value equation 2 |𝑥 − 1| − 7 = 2. Find the sum and product of your solutions. 3.) Evaluate 𝑓(−3) for 𝑓(𝑥) = −𝑥 3 + 4𝑥 2 − 5𝑥 + 10 3𝑥 − 𝑦 + 5 = 0 _________ 4.) What is the product of the solutions to the following system of equations? { 2𝑥 + 3𝑦 − 4 = 0 5.) Write the equation of each line in slope-intercept form with the following characteristics: 2 a.) Thru the point (−4,2) and perpendicular to the line 𝑦 = − 𝑥 + 2 3 b.) Thru the points (15,20) and (−12,29) Quadratic Functions – Unit 2 & 3 _________ 6.) 𝑓(𝑥) goes through the points (−3, 0), (5, 0) 𝑎𝑛𝑑 (1, −32). Which of the following represents 𝑓(𝑥)? A.) 𝑓(𝑥) = −2(𝑥 − 3)(𝑥 + 5) B.) 𝑓(𝑥) = −2(𝑥 + 3)(𝑥 − 5) C.) 𝑓(𝑥) = 2(𝑥 − 3)(𝑥 + 5) D.) 𝑓(𝑥) = 2(𝑥 + 3)(𝑥 − 5) _________ 7.) Determine the equation of the axis of symmetry for the parabola that passes through points ( − 4, 0) & (6, 0). A.) 𝑥 = −2 B.) 𝑥 = 2 C.) 𝑥 = −1 D.) 𝑥 = 1 1 _________ 8.) Which represents the quadratic function 𝑦 = − 2(𝑥 + 2)(𝑥 − 1) A.) 𝑦 = −2𝑥 2 + 4 B.) 𝑦 = −2𝑥 2 − 2𝑥 − 4 C.) 𝑦 = −2𝑥 2 − 2𝑥 + 4 D.) 𝑦 = −2𝑥 2 + 2𝑥 + 4 _________ 9.) Which quadratic function opens upwards and has a vertex at (0 , 3)? A.) 𝑦 = − (𝑥 − 3)2 B.) 𝑦 = (𝑥 − 3)2 C.) 𝑦 = − 𝑥 2 + 3 D.) 𝑦 = 𝑥 2 + 3 _________ 10.) Which quadratic function has an axis of symmetry of 𝑥 = 2 and a minimum value of – 4? A.) 𝑦 = 3(𝑥 + 2)2 − 4 B.) 𝑦 = −3(𝑥 + 2)2 + 4 C.) 𝑦 = 3(𝑥 − 2)2 − 4 D.) 𝑦 = −3(𝑥 − 2)2 + 4 _________ 11.) What is the y-intercept of the quadratic function 𝑦 = 2(𝑥 − 2)2 + 2? A.) (0, −6) B.) (0, 6) C.) (0, −10) D.) (0, 10) _________ 12.) Which of quadratic functions has the narrowest graph? 1 1 A.) 𝑦 = − 3𝑥 2 B.) 𝑦 = −2𝑥 2 C.) 𝑦 = − 3 𝑥 2 D.) 𝑦 = − 2 𝑥 2 _________ 13.) Solve the following quadratic equation: 𝑥 2 − 6𝑥 = −15 A.) −3 ± 2𝑖√6 B.) 3 ± 𝑖√6 C.) 3 ± √6 D.) −3 ± 2√6 _________ 14.) The equation 𝑎𝑥 2 − 𝑏𝑥 = 0 ______________ has the algebraic solution 𝑥 = 0. A.) always B.) sometimes C.) never 2 15.) Write the equation, 𝑦 = −𝑥 2 − 8𝑥 − 7, in vertex form by completing the square. 16.) Write as a complex number in standard form: 3−2𝑖 a.) (3 + 𝑖)(7 − 4𝑖) b.) 5+4𝑖 17.) For the quadratic function 𝑦 = 2𝑥 2 − 3𝑥 + 5, find the value of the discriminant and interpret the meaning. 18.) Using the graph to the right, answer the following: a) Write the equation in intercept form. b) Then, rewrite in standard form. 19.) Multiply the following binomials: a) (2𝑥 − 3)(3𝑥 + 2) 20.) The equation of a circle is 𝑥 2 − 2𝑥 + 𝑦 2 + 6𝑦 = −3. Find the center and the radius of the circle. 3 21.) Solve using any method: a) 3𝑥 2 − 𝑥 − 2 = 0 c) 2(𝑥 − 2)2 + 3 = −6 b) 2𝑥 2 + 3𝑥 − 1 = 0 d) 𝑥 2 − 2𝑥 − 5 = 0 Polynomial Functions – Unit 4 22.) Simplify: a) (5𝑥 2 𝑦 4 + 𝑥 3 𝑦 3 − 7𝑥𝑦 2 ) − (7𝑥 3 + 2𝑥𝑦 2 + 2𝑥 2 𝑦 4 ) b) (6𝑘 2 + 𝑘 − 6)(2𝑘 + 3) 23.) Divide 𝑥 4 − 3𝑥 2 + 2 by (𝑥 + 2). a) Based on your result, is (𝑥 + 2) a factor of is 𝑥 4 − 3𝑥 2 + 2? 24.) Given 𝑥 = 4 is a zero of 𝑓(𝑥) = 7𝑥 3 − 33𝑥 2 + 15𝑥 + 20, find the remaining zeros. 4 25.) Given 𝑔(𝑥) = 2𝑥 4 + 6𝑥 3 − 18𝑥 2 − 54𝑥 a) How many zeros does 𝑔(𝑥) have? b) Find all zeros of 𝑔(𝑥). List their multiplicities. c) Sketch a graph of 𝑔(𝑥) _________ 26.) Which of the following is NOT a zero of 𝑓(𝑥) = 2𝑥 3 − 5𝑥 2 − 14𝑥 + 8 given 𝑓(4) = 0? 1 A.) 8 B.) 2 C.) −2 D.) 4 𝑐 _________ 27.) When 2𝑥 4 − 𝑥 3 + 4 is divided by 𝑥 + 1, the result can be expressed as 2𝑥 3 − 3𝑥 2 + 𝑎𝑥 + 𝑏 + 𝑥+1. What is the sum of 𝑎, 𝑏 and 𝑐? A.) −1 B.) 7 C.) 1 D.) 13 _________ 28.) In the 𝑥𝑦-plane, how many zeros (with multiplicities) does the following equation have? 𝑓(𝑥) = 𝑥 2 (𝑥 + 6)(𝑥 − 1)2 A.) 6 B.) 5 C.) 4 D.) 3 _________ 29.) The function ℎ is defined by a polynomial. If ℎ(−2) = 0, which of the following is NOT true? A.) ℎ(𝑥) has a root of −2 C.) ℎ(𝑥) has a factor of (𝑥 − 2) B.) The graph of ℎ(𝑥) has an x-intercept at −2 D.) The remainder when ℎ(𝑥) is divided by −2 is 0 _________ 30.) Which of the following could be the function 𝑓, as shown in the graph to the right? A.) 𝑔(𝑥) = 𝑥 2 (𝑥 + 2)2 (𝑥 − 1) C.) 𝑔(𝑥) = 𝑥 2 (𝑥 + 2)2 (𝑥 − 1)2 B.) 𝑔(𝑥) = 𝑥 2 (𝑥 + 2)2 (𝑥 + 1)2 D.) 𝑔(𝑥) = 𝑥 2 (𝑥 − 2)2 (𝑥 + 1) _________ 31.) If (𝑥 + 5) is a factor of the polynomial 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 − 23𝑥 + 60, which of the following are the other factors of 𝑝(𝑥)? A.) (𝑥 + 4)(𝑥 − 4) B.) (𝑥 + 4)(𝑥 + 3) C.) (𝑥 − 3)(𝑥 − 4) D.) (𝑥 + 3)(𝑥 − 3) 5 _________ 32.) If 𝑥 = −6 is a root of 𝑓(𝑥) = 𝑥 3 + 6𝑥 2 + 5𝑥 + 30, which of the following is also a root of 𝑓(𝑥)? A.) 𝑥 = −5𝑖 B.) 𝑥 = −5 C.) 𝑥 = −𝑖√5 D.) 𝑥 = √5 _________ 33.) Based on the graph to the right, which of the following is FALSE? A.) The multiplicity at 𝑥 = −4 is even B.) The least degree of the polynomial is 4 C.) (𝑥 − 2) is a factor D.) As 𝑥 → −∞, 𝑓(𝑥) → ∞ Important Formulas to Study for the Exam 𝑦 −𝑦 Slope: 𝑚 = 𝑥2 −𝑥1 2 1 Linear Equations: Slope Intercept Form: 𝑦 = 𝑚𝑥 + 𝑏 Standard Form: 𝐴𝑥 + 𝐵𝑦 = 𝐶 Point Slope Form: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) Quadradic Equations: Standard Form: 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Intercept Form: 𝑦 = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞) Vertex Form: 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 𝑏 2 −𝑏±√𝑏2 −4𝑎𝑐 Complete the square: 𝑐 = (2) Quadratic Formula: 𝑥 = 2𝑎 Discriminant: 𝑑 = 𝑏 2 − 4𝑎𝑐 Equation of Circle: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 Polynomial equation: ℎ(𝑥) = 𝑎𝑛 𝑥 𝑛 + ⋯ + 𝑎1 𝑥 + 𝑎0 6