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Algebra II Honors - Midterm Exam Review **All questions are NO CALCULATOR, unless specified.** Chapter 1 1. For the function y = –x2 – 6x – 7, find the vertex and axis of symmetry. Ⓐvertex (3, – 2); axis of symmetry x = 3 Ⓑvertex (–3, 2); axis of symmetry x = 4 Ⓒvertex (–3, 2); axis of symmetry x = –3 Ⓓvertex (3, –2); axis of symmetry x = –4 2. If the graph of y = ax2 + bx + c opens down, which of the following must be true? Ⓐa < 0 Ⓑa > 0 Ⓒc < 0 Ⓓc > 0 3. Which function does not have a maximum value? Ⓐy = –x2 – 5x – 6 Ⓑy = –x2 – x – 6 Ⓒy = 3x2 – 15x + 2 Ⓓy = 49 – x2 4. What is the vertex of y = –3(x – 2)2 – 4? Ⓐ(–2, –4) Ⓑ(–2, 4) Ⓒ(2, –4) Ⓓ(2, 4) 5. What are the x–intercepts of y = –2(x – 7)(x + 2)? Ⓐ–7 and 2 Ⓑ7 and –2 Ⓒ14 and –4 Ⓓ14 and –2 6. Factor the expression m2 – 4m – 21. 7. Which value of c makes the expression x2 – 5x + c a perfect square trinomial? 5 5 Ⓐ Ⓑ 2 2 25 Ⓒ Ⓓ25 4 9. Factor the expression 8x2 + 28x + 12. Ⓐ2(x + 2)(4x + 3) Ⓑ2(x + 3)(2x + 1) Ⓒ4(x + 1)(2x + 3) Ⓓ4(x + 3)(2x + 1) Ⓐ(m – 7)(m – 3) Ⓑ(m – 7)(m + 3) Ⓒ(m + 7)(m – 3) Ⓓ(m + 7)(m + 3) 8. What are the roots of the equation z2 + 11z – 42 = 0? Ⓐ–3, –14 Ⓑ3, –14 Ⓒ–3, 14 Ⓓ3, 14 10. Simplify the expression Ⓐ 4 13 Ⓒ4 2 3 2 2 3 Ⓑ2 Ⓓ 42 3 5 11. What are the solutions of the equation w2 = –9w? Ⓐ–9, 3 Ⓑ0, –9 Ⓒ0, 9 Ⓓ1, 9 12. What are the solutions of 23 = 2(x – 3)2 + 7? Ⓐ 3 Ⓑ 3 2 2 13. What are the solutions of –3 – y2 = 24? Ⓐ 3 3 Ⓑ 3i 3 Ⓒ 9 3 Ⓓ 9i 3 14. What is the standard form of the i expression ? 2i 1 2 Ⓐ i 1 Ⓑ i 1 2 3 Ⓒ 2 2 Ⓓ6 4 2 i 2 Ⓒ 1 15. What is the vertex of y = 3x2 –30x + 77? Ⓐ(–5, –2) Ⓒ(5, –2) Ⓑ(–5, 2) Ⓓ(5, 2) 1 5 16. What is the value of c if the discrimant of –2x2 – 3x + c is 41? Ⓐ–4 Ⓒ 17. How many real number solutions does the equation 7x2 – 5x + 1 = 0 have? 2 5 Ⓓ i 11 2 Ⓑ4 Ⓓ 25 4 18. A rectangular garden is 25 feet long by 10 feet wide. You have enough mulch to cover 1000 square feet. a. You would like to extend both the length and the width of the garden by x feet to use up all of the mulch. Write an equation to represent the area of the new garden. b. Solve the equation from part (a). c. Which solution do you have to reject? Explain. 19. Graph the function y = x2 – 2x + 1. Label the vertex and the axis of symmetry. 20. Tell whether the function y = x2 – 5x + 6 has a minimum value or a maximum value. Then find that value. 21. Graph the function y = 2(x – 1)2. Label the vertex and the axis of symmetry. 22. Graph the function y = –(x + 2)(x – 2). Label the vertex, the axis of symmetry and the x– intercepts. 23. Write the quadratic function y = 2(x + 3)(x – 1) in standard form. 24. Write the quadratic function y = 5(x – 2)2 – 5 in standard form. 25. Determine which number sets each quantity 26. State the domain and the range of each belongs to. function in interval notation. 3 d. p a. 4.35 c. b. 49 4 27. State the domain and the range of each function in interval notation. 28. Write the equation of the parabola with the given vertex and point. vertex (-3, 1) passing through (2, -4) Chapter 2 1. What is the simplified form of 2ab 2 c 2 3 2a Ⓒ 2 3b Ⓐ 8a 2bc 1 ? 12ab3c 2a 3b 2 c 2 2a Ⓓ 3bc Ⓑ 3. Which equation is the graph of the polynomial function shown? 2. What is (3.2 × 105)(1.4 × 10–2) written in scientific notation? Ⓐ4.48 × 103 Ⓑ4.48 × 107 Ⓒ44.8 × 103 Ⓓ44.8 × 107 4. What is the degree of the polynomial h(t) = –8t2 + 5 – 3t3? Ⓐ1 Ⓑ2 Ⓒ3 Ⓓ4 Ⓐf(x) = –2x3 + x2 – 2 Ⓑf(x) = 3x4 – x2 + 1 Ⓒf(x) = x3 – x + 7 Ⓓf(x) = –2x4 + x2 – 1 5. What is the greatest common monomial factor of 9x3y2 + 15x2y – 6xy2? Ⓐ3x2 Ⓑ3y2 Ⓒ3xy Ⓓ3x2y2 7. What is the complete factorization of 3x4 – 3x2? Ⓐ3x2(x2 – 1) Ⓑ3x2(x – 1)(x + 1) Ⓒ3x(x – 1)(x + 1) Ⓓ3(x4 – x2) 6. Given f(x) = 3x – 5 evaluate f-1(0). 8. If x + 3 is a factor of x3 – x2 – 17x – 15, what is another factor? Ⓐx + 1 Ⓑx – 1 Ⓒx + 5 Ⓓx – 3 9. If x – 2 is a factor of a polynomial f(x), which of the following statements does not have to be true? Ⓐf(2) = 0 Ⓑf(–2) = 0 Ⓒ2 is a root of f(x). Ⓓ2 is a zero of f(x) 10. Which is not a possible rational solution of f(x) = 3x3 – 11x2 + 5x – 6? 1 2 Ⓒ 2 Ⓐ 2 3 Ⓓ 6 Ⓑ 11. How many zeros does 0 = –7m3 – m4 + 1 have? 12. Use direct substitution to evaluate 2x3 – 4x2 + 8x – 3 for x = –2 13. Use synthetic substitution to evaluate 4x4 – 2x3 – 3x2 + 3x for x = 2. 14. Graph f(x) = –x4 + 1. 15. Perform the indicated operation. (x + 4)2 (x – 2) 16. Perform the indicated operation. (x3 + 2x – 1) – (2x2 +4x – 2) 17. Factor the polynomial completely using any method. x3 + 3x2 + x + 3 18. Divide. (x3 – 4x2 – 2x + 3) ÷ (x + 1) 19. Find all real zeros of f(x) = x3 – 7x – 6. 20. Graph the function f(x) = (x + 1)2(x + 4). 21. Identify the end behavior for each of the following. a. f(x) = -3x4 + 3x2 + 1 22. Which polynomial represents the volume of the cone shown? b. f(x) = 5x(3x – 1)2 2 3 20 x – x2 – 4x + 3 3 20 2 x 10 x Ⓑ 3 3 3 2 20 x Ⓒ 4x + 3 3 Ⓐ Ⓓ Chapter 3 50 4 x3 4x2 – 5x – 3 3 1. What is the value of (–243)3/5? Ⓐ–27 Ⓑ–3 Ⓒ3 Ⓓ27 2. What is the solution to 3x5 + 350 = –379? 729 Ⓐ 5 Ⓑ–3 3 729 Ⓒ3 Ⓓ5 3 3. Which expression is the simplest form of 4 3 32 3 32 4. What is the simplified expression of the length of the triangle’s hypotenuse? Ⓐ 33 4 Ⓑ 63 4 Ⓒ6 Ⓓ 16 3 2 4 3x1/2 2x3/2 5. What is the simplified form of z 2 16 z 3 3 36 z 7 ? Ⓐ z3 z Ⓑ 14z3 z Ⓐ 2 x3 2 3x 1 2 Ⓑ2x3/2 + 3x1/2 Ⓒ 4 x3 9 x Ⓓ4x3 + 9x2 6. If h(t) = t2/3 – 9 and j(t) = 3t + 5t2/3, what is h(t) – j(t)? Ⓐ–4t2/3 – 3t – 9 Ⓑ4t2/3 + 3t + 9 Ⓒ3t + 6t4/3 Ⓓ–7t7/3 – 9 Ⓒ 14z 4 z Ⓓ 92z3 z 7. What is g(f(x)) if f(x) = 3x2 and g(x) = 2x1/2? Ⓐx 6 Ⓒ6 x Ⓑ2 x 3 Ⓒx ≥ Ⓓ6x 9. Which function represents the inverse of the graph shown? 1 5 Ⓐy = –5x + 3 Ⓑy = x – 3 1 5 Ⓓy = 5x + 3 Ⓒy = x + 3 8. Given u(x) = 4 x 1 and v(x) = x – 5 what is the domain of u(v(x))? ⒶAll real numbers Ⓑx ≥ 0 1 4 Ⓓx ≥ 21 4 10. What is the inverse of the power function 8 3 g(t) = – t? 27 2 8 Ⓐh(t) = – 3 t Ⓑh(t) = – 3 t 3 27 33 3 Ⓒh(t) = – t Ⓓh(t) = – t 2 2 11. Which of the following pairs of functions are not inverses of one another? Ⓐu(x) = x – 2; v(x) = x + 2 1 1 Ⓑu(x) = 5x – 1; v(x) = x + 5 5 Ⓒu(x) = x3 + 1; v(x) = 3 x-1 Ⓓu(x) = x 2 ; v(x) = x2 + 2 14. What is (are) the solution(s) to x – 2 = 2x 1 ? Ⓐx = 1 Ⓑx = 5 Ⓒx = 1 and 5 ⒹNo solution 13. What are the domain and range of the function y = 5 x 2 ? ⒶDomain: all real numbers; range: all real numbers ⒷDomain: x ≥ 2; range: all real numbers ⒸDomain: all real numbers; range: y ≥ 0 ⒹDomain: x ≥ 2; range: y ≥ 0 16. Evaluate –274/3 without using a calculator. 15. Solve (x – 5)2/3 – 2 = 2? 17. Verify that f and g are inverse functions. f(x) = 2x + 5, g(x) = 18. Find the inverse of the function. x 5 2 f(x) = 19. Graph the function. Then state the domain and range. y=2 x2 – 2 3 x+2= 23. Solve the equation. 4x 2 4x4 - 3x2 + 3x -1 x2 - x +1 28 x 24. Let f(x) = 2x3 – 5 and g(x) = 3x2. Perform the indicated operation and state the domain. f(g(x)) 25. Identify the remainder. a. 13 x 3 – 1 2 22. Solve the equation. 2x 8 3x 5 = 2x 5 3 20. Graph the function. Then state the domain and range. y= 21. Solve the equation. 4= 12. The graph of y = x is shifted 2 units up and 3 units to the left. Which is the equation of the translated function? Ⓐy = x 2 – 3 Ⓑy = x 2 –3 Ⓒy = x 2 + 3 Ⓓy = x 3 + 2 b. (3x 5 - 4x3 + 2x - 5) ( x+1) -1 Chapter 4 1. Which function is shown in the graph? Ⓐf(x) = 2(2.3)x – 2 Ⓑf(x) = 4(2.3)x Ⓒf(x) = 4(2.3)x + 2 Ⓓf(x) = 5(2.3)x – 3 3. Which function represents exponential growth? 2 Ⓐu(t) = –7.0 3 t 2 14e x 1 Ⓐ e5 x 2 1 2 Ⓒ e9 x – x 2 4. What is the horizontal asymptote of the function y = 2(0.3)x – 1 – 4? t 3 Ⓑu(t) = –7.0 2 Ⓒu(t) = 7.0(0.8)t -t æ 10 ö Ⓓu(t) = 7.0 ç ÷ è9ø 5. What is the simplified expression of 7 e3 x 2. Gasoline costs $1.99 per gallon. If the price per gallon increases an average of 6% per month, which function models the exponential growth of the pricing? Ⓐf(x) = 1.06(1.99)x Ⓑf(x) = 1.99(1.06)x Ⓒf(x) = [1.06(1.99)]x 1.99 Ⓓf(x) = 1.06 x ? 1 2 7 Ⓓ e8 x 2 Ⓑ e8 x 7. Which expression is equivalent to x? Ⓐlog x Ⓑlog 2x x Ⓒlog 10 Ⓓlog 10x Ⓐy = – 4 Ⓒy = 2 Ⓑy = 0.3 Ⓓy = 4 6. Which function does not model exponential decay? 3 Ⓐr(x) = e –3 x 4 4 Ⓑr(x) = e –3 x 3 Ⓒr(x) = 4e–3x 3 Ⓓr(x) = e3 x 4 8. What is an equivalent expression for 2 log4 3 + log4 2? Ⓐ2 log4 6 Ⓑlog4 6 Ⓒlog4 12 Ⓓlog4 18 9. Which of the following is not equivalent to log5 8? ln 8 ln 5 Ⓒ3 log5 2 Ⓐ Ⓑ2 log5 4 Ⓓlog5 4 + log5 2 11. What is the solution to the equation log4 4x + 2 log4 x = 4? Ⓐ1 Ⓑ2 Ⓒ3 Ⓓ4 13. What is the value of x in the equation 2 x –10 1 x ? 3 = 9 15. (Calculator) Your grandparents deposited $2000 into a college savings account for you 5 years ago. If the account pays 2.5% annual interest, compounded quarterly, find the current balance of the savings account. 17. State the domain and range y = e–3x. 19. Expand the expression. a. In 16x2 ln x 2 ln 8 ln 8 Ⓒy = 2 ln x Ⓐy = 2 ln x ln 8 2 ln 8 Ⓓy = ln x Ⓑy = 12. A pheasant farmer started her farm with 120 pheasants. An analysis of her records shows that her pheasant population has increased by 15% each year. The farmer wants to determine a model of pheasant population growth using an exponential function. According to her model, what will the pheasant population be in 10 years? Ⓐ311 Ⓑ485 Ⓒ501 Ⓓ1380 14. Graph the function y = 2 • 3x + 1 – 2. State the domain and range. 16. (Calculator) You buy a computer for $1200. The value of the computer decreases by 30% each year. Find the value of the computer after 4 years. 18. Evaluate the logarithm without using a calculator. a. log5 25 b. log1/3 81 c. 10log5x d. log4 16x 20. Condense the expression. a. log3 2x + 3 log3 4x b. In 72x – 2 In 2y 2 x3 b. log5 4y 22. State the domain and range of 21. Solve. 1 a. 27(2x + 4) = 9 10. What is the inverse of the function y = 82x? ( x – 46) b. log2 (x + 8) = 4 c. log6 x + log6 (x + 16) = 2 y = log3 (x + 2) – 2. ANSWERS Ch 1 1. C 2. A 3. C 4. C 5. B 6. B 7. C 8. B 9. D 10. C 11. B 12. B 13. B 14. D 15. D 16. B 17. 0 18. a. 1000 = (10 + x)(25 + x) b. x = 15 or x = –50 c. x = –50 because you cannot have a negative length. 19. Ch 2 1. B 2. A 3. A 4. C 5. C 6. 5 3 7. B 8. A 9. B 10. A 11. 4 12. –51 13. 42 14. 15. x3 + 6x2 – 32 16. x3 – 2x2 – 2x + 1 17. (x + 3)(x2 + 1) 18. x2 – 5x + 3 19. –2, –1, 3 20. 20. minimum; 21. -1/ 4 21. a) x ® ¥, f ( x) ® -¥ x ® -¥, f ( x) ® -¥ 22. 22. A 23. y = 2x2 + 4x – 6 24. y = 5x2 – 20x + 15 25. a) R, Q b) R, Q, Z, W, N, D 26. D: -¥,¥ R: -4, 4 ( ) [ ] 27. D: ( -¥,3] R: ( -¥, 2] 28. y= - 1 (x + 3)2 + 1 5 c) R, Q d) R, I, T b) x ® ¥, f ( x) ® ¥ x ® -¥, f ( x) ® -¥ Ch 3 1. A 2. B 3. B 4. C 5. B 6. A 7. B 8. D 9. C 10. C 11. D 12. D 13. D 14. B 15. 13, -3 16. –81 17. f(g(x)) = x, g(f(x)) = x 18. f–1(x) = 19. Ch 4 1. C 2. B 3. B 4. A 5. A 6. D 7. C 8. D 9. B 10. A 11. D 12. B 13. 4 14. domain: all reals; range: y > –2 3x 5 2 domain: x ≥ –2; range: y ≥ –2 15. $2265.42 16. $288.12 17. domain: all reals; range: y > 0 20. domain: all real numbers; range: all real numbers 21. 36 22. 3 23. 7 24. 54x6 – 5; all real numbers 18. a. 2 b. -4 c. 5x d. 2x 19. a. In 16 + 2ln x b. 3log5 x + log5 2 – log5 y– log5 4 20. a. log3 128x4 b. ln 21. a. 10 22. c. 2 b. 8 domain: x>-2; range: all reals 18x y2