Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. 1) f(x) = 11 + 6 x - x 3 1) A) [ - 10, 20] by [ - 50, 50] B) [ - 4, 5] by [ - 15, 25] C) [ - 10, 10] by [ - 10, 5] D) [ - 4, 5] by [ - 5, 5] Answer: B Explanation: A) B) C) D) 2) f(x) = | x 2 - 6 | 2) A) [ - 5, 5] by [ - 15, 15] B) [0, 5] by [ - 2, 10] C) [ - 5, 5] by [ - 2, 10] D) [ - 10, 10] by [ - 15, 15] Answer: C Explanation: A) B) C) D) 3) f(x) = x 2 + 1 10 cos 70 x 3) A) [ - 0.6, 0.6] by [ - 0.1, 0.6] B) [ - 0.1, 0.1] by [ - 0.1, 0.1] C) [ - 10, 10] by [ - 10, 10] D) [ - 2, 2] by [ - 1, 1] Answer: A Explanation: A) B) C) D) 4) f(x) = 7 + 6x - x 2 4) A) [ - 10, 10] by [ - 10, 5] B) [ - 10, 20] by [ - 50, 50] C) [ - 4, 5] by [ - 5, 5] D) [ - 4, 5] by [ - 15, 25] Answer: D Explanation: A) B) C) D) 1 Match the equation with its graph. 5) y = 3 x 5) A) B) C) D) Answer: C Explanation: A) B) C) D) Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. 6) f(x) = x 4 - 9 x 2 + 6 x 6) A) [ - 10, 15] by [ - 5, 5] B) [ - 5, 5] by [ - 10, 15] C) [ - 25, 15] by [ - 5, 5] D) [ - 5, 5] by [ - 25, 15] Answer: D Explanation: A) B) C) D) 2 Match the equation with its graph. 7) y = x 5 7) A) B) C) D) Answer: C Explanation: A) B) C) D) Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. 8) f(x) = x 3 - 2 x 2 - 3x + 17 8) A) [ - 20, 20] by [ - 100, 100] B) [ - 5, 25] by [ - 5, 5] C) [ - 2, 2] by [ - 10, 10] D) [ - 5, 5] by [ - 5, 25] Answer: D Explanation: A) B) C) D) 3 9) f(x) = 3 cos 60 x 9) A) [ - 10, 10] by [ - 10, 10] B) [ - 0.2, 0.2] by [ - 4, 4] C) [ - 1, 1] by [ - 4, 4] D) [ - 0.2, 0.2] by [ - 1, 1] Answer: B Explanation: A) B) C) D) 10) f(x) = x 2/3 ( 7 - x) 10) A) [ - 2, 2] by [ - 15, 15] B) [ - 4, 10 ] by [ - 10, 10] C) [ - 4, 0] by [ - 5, 5] D) [0, 10 ] by [ - 10, 10] Answer: B Explanation: A) B) C) D) 11) f(x) = x 2 - 1 x 2 + 1 11) A) [ - 5, 5] by [ - 15, 15] B) [ - 10, 10] by [ - 10, 10] C) [ - 10, 10] by [ - 2, 2] D) [ - 1, 1] by [ - 2, 2] Answer: C Explanation: A) B) C) D) 4 Match the equation with its graph. 12) y = 5 x 12) A) B) C) D) Answer: A Explanation: A) B) C) D) Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. 13) f(x) = 10 x 2 - 6 13) A) [ - 5, 5] by [ - 10, 10] B) [ - 5, 0] by [ - 10, 10] C) [ - 2, 2] by [ - 10, 10] D) [0, 5] by [ - 10, 10] Answer: A Explanation: A) B) C) D) 5 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 14) What happens if you set B = - 2 Δ in the angle sum formulas for the sine and cosine functions? Do the results agree with something you already know? 14) Answer: If B = - 2 Δ , then cos (A + B) = cos A and sin (A + B) = sin A. Because the period of both of the sine and cosine functions is 2 Δ , if B is replaced by a multiple of 2 Δ the angle sum formulas must produce the same value as the sine or cosine function. Explanation: Provide an appropriate response. 15) Derive the identity sec - 1 ( - x) = Δ - sec - 1 x by combining the following two equations: cos - 1 ( - x) = Δ - cos - 1 x sec - 1 x = cos - 1 (1/x) 15) Answer: sec - 1 ( - x) = cos - 1 ( - 1/x) = Δ - cos - 1 (1/x) = Δ - sec - 1 x Explanation: Use the addition formulas to derive the identity. 16) sin x - Δ 2 = - cos x 16) Answer: sin x - Δ 2 = sin x cos - Δ 2 + cos x sin - Δ 2 = sin x (0) + cos x ( - 1) = 0 - cos x = - cos x Explanation: 17) cos x + Δ 2 = - sin x 17) Answer: cos x + Δ 2 = cos x cos Δ 2 - sin x sin Δ 2 = cos x (0) - sin x (1) = 0 - sin x = - sin x Explanation: 6 Solve the problem. 18) Let f(x) = x - 6 and g(x) = x 2 . Graph f and g together with f H g and g H f. 18) Answer: Explanation: Use the addition formulas to derive the identity. 19) cos x - Δ 2 = sin x 19) Answer: cos x - Δ 2 = cos x cos - Δ 2 - sin x sin - Δ 2 = cos x (0) - sin x ( - 1) = 0 + sin x = sin x Explanation: 7 Solve the problem. 20) The standard formula for the tangent of the difference of two angles is tan (A - B) = tan A - tan B 1 + tan A tan B . Derive the formula. 20) Answer: tan (A - B) = sin (A - B) cos (A - B) = sin A cos B - sin B cos A cos A cos B + sin A sin B = (cos A cos B) - 1 (sin A cos B - sin B cos A) (cos A cos B) - 1 (cos A cos B + sin A sin B) = tan A - tan B 1 + tan A tan B . Explanation: Use the addition formulas to derive the identity. 21) sin x + Δ 2 = cos x 21) Answer: sin x + Δ 2 = sin x cos Δ 2 + cos x sin Δ 2 = sin x (0) + cos x (1) = 0 + cos x = cos x Explanation: 8 Solve the problem. 22) Graph the functions f(x) = x and g(x) = 3 - x together with their sum, product, two differences, and two quotients. 22) Answer: Explanation: 23) Use the angle sum formulas to derive sin (A - B) = sin A cos B - cos A sin B. 23) Answer: sin (A - B) = sin (A + ( - B)) = sin A cos ( - B) + cos A sin ( - B) = sin A cos B - cos A sin B Explanation: 24) Graph y = cos 2x and y = sec 2x together for - 3 Δ 4 K x K 3 Δ 4 . Comment on the behavior of sec 2x in relation to the signs and values of cos 2x. 24) Answer: When y = cos 2x is at a maximum point, which is at any multiple of Δ , y = sec 2x is a minimum point. Similarly, when cos (2x) is at a minimum point, which is at any odd multiple of Δ 2 , y = sec 2x is a at maximum point. Explanation: 9 25) Graph y = sin x 2 and y = csc x 2 together for - 2 Δ K x K 2 Δ . Comment on the behavior of csc x 2 in relation to the signs and values of sin x 2 . 25) Answer: When y = sin x 2 is at a maximum point, which is at x = (4n + 1) Δ for all integers n, y = csc x 2 is at a minimum point. Similarly, when y = sin x 2 is at minimum point, , which is at x = (4n - 1) Δ for all integers n, y = csc x 2 is at a maximum point. Explanation: MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Express as a single logarithm and, if possible, simplify. 26) ln ( 7 sec Ό ) + ln ( 7 cos Ό ) 26) A) ln ( 49 ) B) ln ( 49 cot Ό ) C) ln 1 D) ln ( 7 sec Ό + 7 cos Ό ) Answer: A Explanation: A) B) C) D) Provide an appropriate response. 27) If f(x) is one - to - one, is g(x) = f( - x) also one - to - one? Explain. 27) A) g(x) is a reflection of f(x) across the line y = x. It will not be one - to - one. B) g(x) is a reflection of f(x) across the y - axis. It will be one - to - one. C) There is not enough information to determine whether g(x) is one - to - one. D) g(x) is a reflection of f(x) across the x - axis. It will be one - to - one. Answer: B Explanation: A) B) C) D) One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 28) cos x = - 1 5 , x in Δ , 3 Δ 2 28) A) sin x = 2 6 5 , tan x = - 2 6 B) sin x = - 2 6 5 , tan x = - 2 6 C) sin x = - 2 6 5 , tan x = 2 6 D) sin x = 2 6 5 , tan x = 2 6 Answer: C Explanation: A) B) C) D) 10 Solve for the angle Ό , where 0 K ȱΌ K 2 Δ 29) sin 2 Ό = 3 4 29) A) Ό = 0, Δ , 2 Δ B) Ό = Δ 6 , śΔ 6 , ŝΔ 6 , 11 Δ 6 C) Ό = Δ 4 , řΔ 4 , śΔ 4 , ŝΔ 4 D) Ό = Δ 3 , ŘΔ 3 , ŚΔ 3 , śΔ 3 Answer: D Explanation: A) B) C) D) Solve the problem. 30) The accompanying figure shows the graph of y = x 2 shifted to a new position. Write the equation for the new graph. 30) A) y = x 2 - 6 B) y = (x + 6 ) 2 C) y = (x - 6 ) 2 D) y = x 2 + 6 Answer: B Explanation: A) B) C) D) 31) Suppose the consumption of electricity grows at 8.6 % per year, compounded continuously. Find the number of years before the use of electricity has tripled. Round the answer to the nearest hundredth. 31) A) 12.77 yr B) 0.13 yr C) 34.88 yr D) 1.28 yr Answer: A Explanation: A) B) C) D) 11 Use a graph to find an approximate solution to the equation. Round to the nearest thousandth. 32) 4 3 x = 6 x + 1 32) A) 2.292 B) 0. 757 C) 1.292 D) - 4.419 Answer: B Explanation: A) B) C) D) Graph the function. 33) y = ( - 6 x) 2/3 - 3 33) A) B) 12 C) D) Answer: C Explanation: A) B) C) D) Find the formula for the function. 34) Express the length d of a square's diagonal as a function of its side length x. 34) A) d = 2x B) d = x 2 C) d = x D) d = x 3 Answer: B Explanation: A) B) C) D) Find the function value. 35) cos 2 Δ 12 35) A) 2 + 3 4 B) 1 + 3 2 C) 2 - 3 4 D) 2 + 3 Answer: A Explanation: A) B) C) D) The problem tells by what factor and direction the graph of the given function is to be stretched or compressed. Give an equation for the stretched or compressed graph. 36) y = x + 1 compressed vertically by a factor of 3 36) A) y = 3 x + 3 B) y = x + 1 3 C) y = 3 x + 1 D) y = 3 x + 1 Answer: B Explanation: A) B) C) D) 13 The equation of an ellipse is given. Put the equation in standard form and sketch the ellipse. 37) 16 x 2 + 64 y 2 = 1024 37) A) x 2 64 + y 2 16 = 1 B) x 2 16 + y 2 64 = 1 C) x 2 64 + y 2 16 = 1 D) x 2 16 + y 2 64 = 1 Answer: C Explanation: A) B) C) D) 14 Graph the function in the ts - plane (t - axis horizontal, s - axis vertical). State the period and symmetry of the function. 38) s = sec t 3 38) A) Period 6 Δ , symmetric about the s - axis B) Period 6 Δ , symmetric about the t - axis C) Period 6 Δ , symmetric about the t - axis D) Period 6 Δ , symmetric about the s - axis Answer: A Explanation: A) B) C) D) 15