Quest REVIEW 2.1 - 2.6 Exponential Functions and Models PDF

Quest Review: (Topics 2.1-1.6) Exponential Functions and Modeling Name: Section 1: Part A - No Calculators Allowed Directions: Read the following problems carefully. Work out each question and circle the best answer. Be sure to write your answer choices on the separate answer she...

Quest Review: (Topics 2.1-1.6) Exponential Functions and Modeling Name: Section 1: Part A - No Calculators Allowed Directions: Read the following problems carefully. Work out each question and circle the best answer. Be sure to write your answer choices on the separate answer sheet as well. 1. Values of the terms of a geometric sequence are graphed in the figure given. Which of the following is an expression for the term of the geometric sequence? 1 𝑛−2 (A) 𝑔𝑛 = 4 ( ) 2 𝑛−1 (B) 𝑔𝑛 = 8(2) 1 𝑛 (C) 𝑔𝑛 = 8 ( )2 1 𝑛−1 (D) 𝑔𝑛 = 16 ( ) 2 2. An arithmetic sequence 𝑎𝑛 with initial term 𝑎0 has a third term of 6 and a fifth term of 2. Which of the following gives the general term of the sequence? (A) 𝑎𝑛 = 6 − 4(𝑛 − 3) (B) 𝑎𝑛 = 6 − 2(𝑛 − 3) (C) 𝑎𝑛 = 6 − 4(𝑛 − 2) (D) 𝑎𝑛 = 6 − 2(𝑛 − 2) 3. The table below contains selected values of an arithmetic sequence , where and are constants. What is the value of 4 (A) 4 (B) 48 (C) 40 (D) 23 b 26 8 2 c 𝑔10 4. Let be a geometric sequence with 𝑔0 = 6 and 𝑔1 =− 12 What is 𝑔9 ? 1 (A) -18 (B) -6 (C) -2 (D) − 2 5. The table below gives values of a function 𝑓(𝑥) for select values of x. For what value of b would 𝑓(𝑥) be best modeled by an exponential function and why? (A) b=3, as this would make the outputs change at a constant rate. (B) b=3, as this would make the outputs change proportionally. (C) b=3.75, as this would make the outputs change proportionally (D) b=3.75, as this would make the outputs change at a constant rate. 𝑥 6. Let 𝑔 be the function given by 𝑔(𝑥) =− 5(1. 05). Which of the following pairs of limit statements could describe the end behaviors for the graph of ? (A) and (B) and (C) and lim 𝑔(𝑥) =− ∞ 𝑥→∞ (D) lim 𝑔(𝑥) = ∞ and 𝑥→−∞ 𝑥 7. The function 𝑘(𝑥) = 𝑎𝑏 , can be used to model the price 𝐾 of a Broadway show ticket price weeks after openting. If the graph of 𝐾contains the output 𝐾(𝑥) = 1. 1∙1. 1∙1. 1∙80, which of the following is true? (A) a=80 and b=3, and the price of the ticket increases by $1.10 each week. (B) A=80 and b=1.1, and the price of the ticket increases by 10% each week. (C) A=80 and b=1.1, and the price of the ticket increases by $80 each week. (D) A=3 and b=1.1, and the price of the ticket increases by 10% each week. 3 𝑥−1 8. Let 𝑚(𝑥) = ( ) 2. Which of the following statements about is correct? (A) is increasing and the graph of is concave (C) is decreasing and the graph of is concave up. up. (B) is increasing and the graph of is concave (D) is decreasing and the graph of is concave down. down.  𝑥 𝑥 9. How do the graphs of 𝑓(𝑥) = 3 and 𝑔(𝑥) = 9∙3 relate to one another? (A) The graph of g is the graph of f with a horizontal translation of -9. (B) The graph of g is the graph of f with horizontal translation of -2. (C) The graph of g is the graph of f with a horizontal trasnlation of +2. (D) The graph of g is the graph of f with a horizontal translation of +9. 1 𝑥+ 2 10. Which of the following functions is an equivalent form of the function 𝑘(𝑥) = 3 · 25 ? 𝑥 (A) 𝑘(𝑥) = 8 · 25 𝑥 (B) 𝑘(𝑥) = 15 · 25 𝑥 (C) 𝑘(𝑥) = 3 · 5 𝑥 (D) k(𝑥) = 15 · 5 11. Let be an exponential function with the following end behavior. lim 𝑓(𝑥) =− ∞ and 𝑥→−∞ Which of the following could be the graph of ? (A) (B) (C) (D) 𝑥 12. Which of the following is true about 𝑓(𝑥) = 2 ? (A) A vertical dilation of the graph of the f by a factor (C) A vertical dilation of the graph of f by a factor of 4 of 4 is equivalent to a horizontal translation of the graph is equivalent to a horizontal translation of the graph of f of f by -4 by -2 (B) A vertical dilation of the graph of f by a factor of 4 (D) A vertical dilation of the graph of f by a factor of 4 is equivalent to a horizontal translation of the graph of f is equivalent to a horizontal translation of the graph of f by +2 by +4 13. A green energy company’s profits doubled every fiscal quarter (of a year) for the first 3 years after it’s incpetion. The 𝑞 function 𝑃(𝑞) = 10∙2 can be used to model the company’s profits, in thousands of dollars. Which of the following equivalent forms 𝑃(𝑞) would reveal the growth rate of the company’s profit each month since its inception? ( ) 𝑞 ( ) 𝑞 4𝑞 ( ) 1 3 4 1/3 (3𝑞) ( ) ( ) ( ) 3 4 (A) 𝑃(𝑞) = 10 2 (B) 𝑃(𝑞) = 10 2 (C) 𝑃(𝑞) = 10 2 (D) 𝑃(𝑞) = 10 2 4 14. A quadratic regression was used to model a data set, and a graph of the residuals from the regression is shown above. Which of the following is true? (A) The quadratic model is appropriate, and the majority of the preditions are underestimates. (B) The quadratic model is appropriate, and the majority of the predictions are overestimates. (C) The quadratic model is not appropriate, and the majority of the preditions are underestimates. (D) The quadratic model is not appropriate, and the majority of the predictions are overestimates. 15. A set of data was used to create a linear, a quadratic, and an exponential regression model. The residual plots for the three models are shown above. Based on the three residual plots, which of the following could be an appropriate model for the data? (A) 𝑦 = 3 + 2𝑥 2 (B) 𝑦 = 𝑥 + 2𝑥 + 3 𝑥 (C) 𝑦 = 3(2) (D) 𝑦 = 3 + 2 log 𝑙𝑜𝑔 𝑥 Quest Review: (Topics 2.1-1.6) Exponential Functions and Modeling Name: Section 1: Part B - Calculator Required Directions: The following problems require the use of a graphing calculator. Read each problem carefully. Do the work for each question and then circle the appropriate answer. Be sure to mark your answers on the answer sheet as well. 16. The largest chess streamer on YouTube is Gotham Chess (Levy Rozman), who currently has over 4 million subscribers. The number of Gotham subscribers can be modeled by the arithmetic sequence , where is the number of Gotham subscribers (in millions) for month ( represents January 2021). If GothamChess had 1.66 million subscribers in month 9 and the difference between two successive terms is 0.143 for , how many subscribers (in millions) does the arithmetic model predict Gotham Chess will have for month 13? (A) 1.803 (B) 2.232 (C) 2.833 (D) 3.519 𝑥 17. The function 𝑓(𝑥) = 4∙𝑏 contains the input-output pair (5,8). The function f can be rewritten using the natural base e 𝑘𝑥 into the function 𝑓(𝑥) = 4∙𝑒 for some value of k. What is k? (A) 0.139 (B) 0.693 (C) 2.079 (D) 3.466 18. The table above presents values for a function f at selected values of. An exponential regression 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) 47.8 (D) 47.917 19. Over the years 1950-2020, the total world population can be modeled by a linear function. Selected values for the total world population P, in billions, are given in the table above, where t represtnts the number of years since 1950. Use the regression capabilities on your calculator to find a linear model of the form y=a+bx for the world population (in billions) x years since 1950. According to the model, what was the world population in 1979? (A) 4. 527 (B) 4. 801 (C) 8. 054 (D) 5. 929 20. A marketing team is analyzing two competing models to predict sales, where x represents the number of customers. How many customers make Model A and B equivalent? 2 Model A: 𝑓(𝑥) = 200𝑥 − 5000 Model B: 𝑔(𝑥) = 0. 03𝑥 + 500 (A) x= 22. 29 (B) 𝑥 = 72. 31 (C) 𝑥 = 27. 61 (D) 𝑥 = 65. 45 Quest Review: (Topics 2.1-2.6) Free Response Questions Name: Section 2: Calculator Required You may use your graphing calculator for this portion of the exam. Directions: The following page contains one free response problem. While you may write on the following page, the only work and answers that will be graded are what you write in the appropriate spaces of the free response answer booklet. When completing the free response problem, consider the following: Show all of your work. Your work will be scored on the correctness and completeness of your responses, including your supporting work and answers. Answers without supporting work may not receive credit in cases where supporting work is required. For the following questions, you are expected to use your graphing calculator for tasks such as producing graphs and tables, evaluating functions, solving equations, and performing computations. Avoid rounding intermediate computations on the way to the final result. Unless otherwise specified, any decimal approximations reported in your work should be accurate to three places after the decimal point. It may be helpful to use your graphing calculator to store information such as computed values for constants, functions you are working with solutions to equations, and any intermediate values. Computations with the graphing calculator that use the stored information help to maintain as much precision as possible and ensure the desired accuracy in final answers. 1. An air compressor pressurizes air for tools like nail guns. Though the compressor has a tank that is supposed to keep the air compressed even when it is not on, it inevitably loses pressure over time when it isn’t turned on, especially if there is any defect. The table below shows the amount of pressure in pounds per square inch (psi) for an air compressor’s tank t hours after being shut off. (A) (i) An exponential function P(t) can be constructed to model the amount of pressure in the tank P after t hours. Perform exponential regression and write the expression for P(t) that is the result of the regression. Include correct variables. (ii) After how many hours does the model from (i) predict the tank will have less than 125 psi of pressure? (iii) The function from (i) models the pressure in the tank after t hours. Write an equivalent expression of (i) that models the pressure in the tank after t days. (B) (i) The function can also be modeled with a linear model. Let be the linear regression model for. Use the data in the table to find the linear regression model for. (ii) Find the value of predicted by the linear regression model. (C) (i) The graphs of the residuals for the exponential and linear regression models are below. Use the graphs of the residuals to determine which model was most appropriate for the data. Explain your reasoning.