MATH 170, Test 2 Sample Exam All problems are taken from Sullivan Precalculus, 9th Edition. Page numbers written in standard type refer to the page where the practice problems can be found; italicized page numbers refer to the pages where the math concepts are described. Pg. 124 “Are You Prepared?” Problems 2. Find the slope of the line joining the points (2,5) and (-1,3). (pp. 19-27) Concepts and Vocabulary Problems 7. For the graph of the linear function f(x) = mx + b, m is the _____________ and b is the _____________. (pp.24, 25) 8. For the graph of the linear function H(z) = -4z + 3, the slope is __________ and the yintercept is ____________. (pp.24, 25) 9. If the slope of m of the graph of a linear function is _____________, the function is increasing over its domain.(pp. 21, 22) Skill Building Problems In problems 21 and 23, determine whether the given function is linear or nonlinear. If it is linear, determine the slope. (pp. 118, 119) 21. Linear? _______ If linear, what is the slope? __________ Answer: Linear; -3 x -2 -1 0 1 2 y = f(x) 4 1 -2 -5 -8 ______ _____ 23. Linear? _______ If linear, what is the slope? __________ Answer: Nonlinear x -2 -1 0 1 2 y = f(x) -8 -3 0 1 0 ______ _____ Application and Extension Page 125. 29. Suppose that f(x) = 4x – 1 and g(x) = -2x + 5. (pp. 54, 55) (a) (b) (c) (d) (e) Solve f(x) = 0 Solve f(x) > 0 Solve f(x) = g(x) Solve f(x) ≤ g(x) Graph y = f(x) and y = g(x) and label the point that represents the solution to the equation f(x) = g(x). Answers: See page AN11 in your textbook. “Are You Prepared?” Problems Page 143. 1. List the intercepts of the equation y = x2 – 9. (pp. 11-12) 2. Find the real solutions of the equation 2x2+7x – 4 = 0. (pp. A47-A51) 3. To complete the square of x2 – 5x, you add the number __________. (pp. A29-A30) 4. To graph y = (x – 4)2, you shift the graph of y = x2 to the _____ a distance of _____ units. (pp. 90-99) Concepts and Vocabulary Problems 5. The graph of a quadratic function is called a(n) __________. (pp. 135-137) 6. The vertical line passing through the vertex of a parabola is called the _______. (p. 136) 7. The x-coordinate of the vertex of f(x) = ax2 + bx + c, a ≠ 0, is ____________. (p. 138) Skill Building In problem 31 and 41, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing (pp. 136-142). 31. f(x) = x2 + 2x Answers: See p. AN13 in your textbook. 41. f(x) = -2x2 – 3 See p. AN13 in your textbook Page 152. Concepts described on p. 148. 7. Enclosing a Rectangular Field David has 400 years of fencing and wishes to enclose a rectangular area. (a) Express the area A of the rectangle as a function of the width w of the rectangle. Answer: A(w) = -w2 + 200w (b) For what value of w is the area the largest? Answer: A is largest when w is 100 yd. (c) What is the maximum area? Answer: 10,000yd2 Skill Building Pg. 158 For problems 8 and 10, solve each inequality. 8. x2 + 3x – 10 > 0 Answer: {x|x < -5 or x > 2} 10. x2 + 8x > 0 Answer: {x|x < -8 or x > 0} “Are You Prepared?” Problems Pg. 183 1. The intercepts of the equation 9x2 + 4y = 36 are ________________________. (pp. 11-12) Answer: x-intercepts: x = 2, x = -2; y-intercept: 9 2. Is the expression 4x3 – 3.6x2 - √2 a polynomial? If so, what is its degree? _____________________ (pp. A22-A29 Answer: Yes; degree 3 3. To graph y = x2 – 4, you would shift the graph of y = x2 _______________ a distance of ___________. Answer: down a distance of 4 units. (pp. 90-99) Skill Building Pg. 184 In the next two problems, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Refer to pp. 166 -170 for the concepts pertaining to these problems. 15. f(x) = 4x + x3 Answer: Yes; degree 3 ! 19. f(x) = 1 - ! Answer: No; x is raised to the -1 power. In the following problem, form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. 41. Zeros: -1, 1, 3; degree 3. Answer: f(x) = x3 – 3x2 – x + 3 for a = 1. In the next problem, list (a) (b) (c) (d) (e) List each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x-intercept. Determine the behavior of the graph near each x-intercept (zero). Determine the maximum number of turning points on the graph. Determine the end behavior; that is, find the power function that the graph of f resembles for the large values of |x|. Refer to pp. 171 – 180 for descriptions of the above concepts. 49. f(x) = 3(x - 7)(x + 3)2 Answer: (a) 7, multiplicity 1; -3, multiplicity 2 (b) Graph touches the x-axis at -3 and crosses it at 7. (c) Near -3: f(x) ≈ -30(x+3)2; Near 7: f(x) ≈ 300(x-7) (d) 2 (e) y = 3x3 “Are You Prepared?” Problems Pg. 196 1. True or False The quotient of two polynomial expressions is a rational expression. (pp. A36-A42) _________________ Answer: True 2. What are the quotient and remainder when 3x4 – x2 is divided by x3 – x2 + 1 ? (pp. A25-A28) ____________________________________________ Answer: Quotient is 3x + 3, Remainder is 2x2 + 3x + 3 x3 – x2 +1 Pg. 198 In the following problems, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. Refer to pp. 191-196 for the concepts covered by these problems. !! 43. R(x) = !!! Answer: vertical asymptote: x = -4; horizontal asymptote: y = 3 x3 x4-1 Answer: vertical asymptotes: x = 1, x = -1; horizontal asymptote: y = 0 47. T(x) = 51. R(x) = 6x2 + 7x – 5 3x + 5 Answer: vertical asymptote: none; oblique asymptote: y = 2x - 1 Skill Building Problems Pg. 219 Solve each inequality algebraically. 39. x + 4 ≤ 1 x–2 Answer: {x|x < 2}; (-∞, 2) 41. 3x – 5 ≤ 2 x+2 Answer: {x| -2 < x ≤ 9}; (-‐2,9] Skill Building Problems Pg. 231 In the problems below, use the Remainder Theorem when f(x) is divided by x – c. Then use the Factor Theorem to determine whether x – c is a factor of f(x). Refer to pp. 222 and 223 for the concepts covered by these problems. 11. f(x) = 4x3 – 3x2 – 8x + 4; x – 2 Answer: R = f(2) = 8; no 13. f(x) = 3x4 – 6x3 – 5x +10; x – 2 Answer: R = f(2) = 0; yes In the following problems, tell the maximum number of real zeros that each polynomial function may have. Do not attempt to find the zeros. Refer to pp. 224 and 225 for the concepts covered by these problems. 21. f(x) = -4x7 + x3 – x2 + 2 Answer: 7 23. f(x) = 2x6 - 3x2 – x + 1 Answer: 6 25. f(x) = 3x3 - 2x2 + x + 2 Answer: 3 In the following problems, Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to facto f over the real numbers. Refer to pp. 224-226 for the concepts covered by these problems. 45. f(x) = x3 + 2x2 - 5x - 6 Answer: -3, -1, 2; f(x) = (x + 3)(x + 1)(x – 2) 49. f(x) = 2x3 - 4x2 - 10x + 20 Answer: 2, √5, -√5; f(x) = 2(x - 2)(x - √5)(x + √5) Pg. 232 In the following problems, use the Intermediate Value Theorem to show that each polynomial function has a zero in the given interval. Refer to pp. 228 and 229 for the concepts covered in these problems. 77. f(x) = 8x4 - 2x2 + 5x – 1; [0,1] Answer: f(0) = -1; f(1) = 10 81. f(x) = x5 – x4 + 7x3 - 7x2 - 18x + 18; [1.4, 1.5] Answer: f(1.4) = -0.17536; f(1.5) = 1.40625