Precalculus Exam 2023

Section I: Multiple-Choice PART A No calculator is allowed for this part of the exam. 1. The polynomial function p is given by p ( x ) = −4 x 5 + 3x 2 + 1. Which of the following statements about the end behavior of p is true? (A) The sign of the leading term of p is positive, and the degree of the leading term of p is even; therefore, lim p( x ) = ∞ and lim p( x ) = ∞. x→−∞ x→ ∞ (B) The sign of the leading term of p is negative, and the degree of the leading term of p is odd; therefore, lim p( x ) = ∞ and lim p( x ) = − ∞. x→−∞ x→ ∞ (C) The sign of the leading term of p is positive, and the degree of the leading term of p is odd; therefore, lim p( x ) = − ∞ and lim p( x ) = ∞. x→−∞ x→ ∞ (D) The sign of the leading term of p is negative, and the degree of the leading term of p is odd; therefore, lim p( x ) = − ∞ and lim p( x ) = ∞. x→−∞ x→ ∞ 2023 2. The depth of water, in feet, at a certain place in a lake is modeled by a function W. The graph of y = W (t ) is shown for 0 ≤ t ≤ 30, where t is the number of days since the first day of a month. What are all intervals of t on which the depth of water is increasing at a decreasing rate? (A) ( 3, 6 ) only (B) ( 3, 12 ) (C) ( 0, 3) and (18, 30 ) only (D) ( 0, 6 ) and (18, 30 ) 3. 4. Which of the following functions has a zero at x = 3 and has a graph in the xy-plane with a vertical asymptote at x = 2 and a hole at x = 1 ? (A) h( x ) = x 2 − 4x + 3 x 2 − 3x + 2 (B) j ( x ) = x 2 − 5x + 6 x 2 − 3x + 2 (C) k( x ) = x−3 x − 3x + 2 (D) m( x ) = x−3 x 2 − 4x + 3 2 The polynomial function p is an odd function. If p( 3) = − 4 is a relative maximum of p, which of the following statements about p( −3) must be true? (A) p( −3) = 4 is a relative maximum. (B) p( −3) = − 4 is a relative maximum. (C) p( −3) = 4 is a relative minimum. (D) p( −3) = − 4 is a relative minimum. 2023 5. The function g is given by g ( x ) = x 3 − 3x 2 − 18 x, and the function h is given h( x ) by h( x ) = x 2 − 2 x − 35. Let k be the function given by k ( x ) = . What is g(x ) the domain of k ? (A) all real numbers x where x ≠ 0 (B) all real numbers x where x ≠ −5, x ≠ 7 (C) all real numbers x where x ≠ −3, x ≠ 0, x ≠ 6 (D) all real numbers x where x ≠ −5, x ≠ −3, x ≠ 0, x ≠ 6, x ≠ 7 6. The figure shown is the graph of a polynomial function g . Which of the following could be an expression for g ( x ) ? (A) 0.25( x − 5 )( x − 1)( x + 8 ) (B) 0.25( x + 5 )( x + 1)( x − 8 ) (C) 0.25( x − 5 )2 ( x − 1)( x + 8 ) (D) 0.25( x + 5 )2 ( x + 1)( x − 8 ) 2023 7. x −8 −4 −2 −1 0 3 f (x ) 87 55 5 −4 −7 20 The table gives values for a polynomial function f at selected values of x. Let g ( x ) = af (bx ) + c, where a, b, and c are positive constants. In the xy-plane, the graph of g is constructed by applying three transformations to the graph of f in this order: a horizontal dilation by a factor of 2, a vertical dilation by a factor of 3, and a vertical translation by 5 units. What is the value of g ( −4 ) ? (A) 266 (B) 170 (C) 28 (D) 20 8. Let k, w, and z be positive constants. Which of the following is equivalent to kz log10  2  ? w  (A) log10 ( k + z ) − log10 ( 2w ) (B) log10 k + log10 z − 2 log10 w 1 log w 2 10 (D) log10 k − log10 z + 2 log10 w (C) log10 k + log10 z − 9. Values of the terms of a geometric sequence g n are graphed in the figure. Which of the following is an expression for the nth term of the geometric sequence? 1 (n− 2) (A) g n = 4 2 () (B) g n = 8( 2 )(n−1) ( 12 ) 1 = 16( ) 2 (C) g n = 8 (D) g n n (n−1) 2023 x g(x ) −2 4 0 1 2 3 −2 4 3 36 9 10. The table gives values of the function g for selected values of x. The function f is given by f ( x ) = 3x + x 2 . What is the value of f ( g ( 3)) ? (A) −72 37 9 (C) 9 (B) (D) 97 11. A food vendor developed a new sandwich type for sale. The vendor made estimates about the sales of the new sandwich type over time. A linear regression was used to develop a model for the sales over time. The figure shows a graph of the residuals of the linear regression. Which of the following statements about the linear regression is true? (A) The linear model is not appropriate, because there is a clear pattern in the graph of the residuals. (B) The linear model is not appropriate, because the graph of the residuals has more points above 0 than below 0. (C) The linear model is appropriate, because there is a clear pattern in the graph of the residuals. (D) The linear model is appropriate, because the positive residual farthest from 0 and the negative residual farthest from 0 are about the same distance, although more points are above 0 than below 0. 2023 12. The value, in millions of dollars, of transactions processed by an online payment platform is modeled by the function M. The value is expected to increase by 6.1% each quarter of a year. At time t = 0 years, 54 million dollars of transactions were processed. If t is measured in years, which of the following is an expression for M (t ) ? (Note: A quarter is one fourth of a year.) (A) 54 ( 0.061)(t 4 ) (B) 54 ( 0.061)( 4t ) (C) 54 (1.061)(t 4 ) (D) 54 (1.061)( 4t ) 13. Iodine-131 has a half-life of 8 days. In a particular sample, the amount of iodine-131 remaining after d days can be modeled by the function h given by h( d ) = A0 ( 0.5 )(d /8) , where A0 is the amount of iodine-131 in the sample at time d = 0. Which of the following functions k models the amount of iodine-131 remaining after t hours, where A0 is the amount of iodine-131 in the sample at time t = 0 ? (There are 24 hours in a day, so t = 24d.) (A) k(t ) = A0 ( 0.5 )(t /24 ) ( (B) k(t ) = A0 0.5(1 ( ( 24 ) (t (C) k(t ) = A0 0.5 ( (D) k(t ) = A0 0.5(1 ) 24 ) ( 8t ) ) 8) ) 192 ) t ( ) 14. What are all values of x for which ln x 3 − ln x = 4 ? (A) x = −2 and x = 2 (B) x = −e 2 and x = e 2 (C) x = e 2 only (D) x = e 4 2023 15. Let f ( x ) = 1 + 3sec x and g ( x ) = −5. In the xy-plane, what are the x-coordinates of the points of intersection of the graphs of f and g for 0 ≤ x < 2π ? π 5π (A) x = and x = 3 3 (B) x = π 5π and x = 6 6 (C) x = 2π 4π and x = 3 3 (D) x = 7π 11π and x = 6 6 16. The figure shows the graph of a sinusoidal function g . What are the values of the period and amplitude of g ? (A) The period is 4, and the amplitude is 3. (B) The period is 8, and the amplitude is 3. (C) The period is 4, and the amplitude is 6. (D) The period is 8, and the amplitude is 6. 2023 17. Which of the following is the graph of the polar function r = f (θ ), where f (θ ) = 3cos θ + 2, in the polar coordinate system for 0 ≤ θ ≤ 2π ? (A) (B) 2023 (C) (D) 2023 18. What are all values of θ , −π ≤ θ ≤ π , for which 2cos θ > −1 and 2sin θ > 3 ? (A) − (B) π 5π <θ < only 6 6 (C) − (D) 5π 5π <θ < 6 6 2π 2π <θ < only 3 3 2π π <θ < only 3 3 19. A polar function is given by r = f (θ ) = −1 + sin θ . As θ increases on the π interval 0 < θ < , which of the following is true about the points on the graph 2 of r = f (θ ) in the xy-plane? (A) The points on the graph are above the x-axis and are getting closer to the origin. (B) The points on the graph are above the x-axis and are getting farther from the origin. (C) The points on the graph are below the x-axis and are getting closer to the origin. (D) The points on the graph are below the x-axis and are getting farther from the origin. 2023 PART B A graphing calculator is required for some questions on this part of the exam. 20. The temperature, in degrees Celsius (°C), in a city on a particular day is 75t 3 − 836t 2 + 3100t − 4185 , 14t 2 + 10t − 35 where t is measured in hours from 12 p.m. for 2 ≤ t ≤ 9. Based on the model, how many hours did it take for the temperature to increase from 0°C to 5°C ? modeled by the function T defined by T (t ) = (A) 7.701 (B) 5.420 (C) 4.114 (D) 2.280 x f (x ) −2 10 −1 15 1 40 2 56 21. The table presents values for a function f at selected values of x. An exponential regression y = ab x is used to model these data. What is the value of f (1.5 ) predicted by the exponential function model? (A) 46.767 (B) 47.342 (C) 47.800 (D) 47.917 2023 22. The number of minutes of daylight per day for a certain city can be modeled by 2π (t − 172 ) + 729, where t is the day the function D given by D (t ) = 160cos 365 of the year for 1 ≤ t ≤ 365. Which of the following best describes the behavior of D (t ) on day 150 ? ( ) (A) The number of minutes of daylight per day is increasing at a decreasing rate. (B) The number of minutes of daylight per day is decreasing at a decreasing rate. (C) The number of minutes of daylight per day is increasing at an increasing rate. (D) The number of minutes of daylight per day is decreasing at an increasing rate. 23. The function g is given by g ( x ) = sin x − cos x and has a period of 2π . In order to define the inverse function of g , which of the following specifies a restricted domain for g and provides a rationale for why g is invertible on that domain? (A) 0 ≤ x ≤ π , because all possible values of g ( x ) occur without repeating on this interval. π 3π ≤x≤ , because all possible values of g ( x ) occur without repeating 4 4 on this interval. (B) − (C) 0 ≤ x ≤ π , because the length of this interval is half of the period. (D) − π 3π ≤x≤ , because the length of this interval is half of the period. 4 4 2023 24. A theme park thrill ride involves a tower and a carriage that rapidly moves passengers up and down along a vertical axis, as shown in the figure. The carriage is lifted to the top of the tower, then released to move down the tower. The ride involves 10 controlled bounces from the highest point to the lowest point, and back to the highest point. A point X is located on the bottom of the carriage. The height of X above the ground, in feet, can be modeled by a periodic function H . At time t = 0 seconds, X is at its highest point of 120 feet. The lowest point for X is at a height of 20 feet. The next time X is at its highest point is at time t = 8 seconds, which is the end of the first bounce. Which of the following can be an expression for H (t ), where t is the time in seconds? ( π4 t ) + 70 π (B) 50cos ( t ) + 70 4 π (C) 50sin ( t ) + 70 8 π (D) 50cos ( t ) + 70 8 (A) 50sin 2023