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Test : (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 gn are graphed in the figure above. Which of the following is an
expression for the n thterm of the geometric sequence?
(A)
n
g n
(B)
1
n
gn
−
(C)
2
n
gn
−
(D)
2
n
gn
2. Selected values of the terms of an arithmetic sequence an are graphed in the figure above. Which of the following is
an expression for the n thterm of the arithmetic sequence?
(A) ( )
a n = + n − (B) ( )
a n = + n − (C) ( )
a n = + n + (D) ( )
a n = + n −
n 1 5 7 8 15
an b 32 26 c 2
3. The table above contains selected values of an arithmetic sequence an , where b and c are constants. What is the
value of b + c?
(A) 49 (B) 58 (C) 67 (D) 76
4. Let gn be a geometric sequence with g 1 = 8 and g 4 = 27. Which of the following is the value of g 2?
(A)
(B) 12 (C)
(D) 18
5. Selected values of the function g are given in the table above. Which of the following statements about g is most
appropriate?
(A) g could be linear because over equal-length input-value intervals, the output values change at a constant rate.
(B) g could be linear because the output values change linearly as the input values change proportionally.
(C) g could be exponential because over equal-length input-value intervals, the output values change proportionally.
(D) g could be exponential because the output values change linearly as the input values change proportionally.
x g ( x )
9. Let f be an exponential function with the following end behavior.
lim ( ) x
f x
→− = andlim (^) ( ) 0 x
f x
→
Which of the following could be the graph of f?
(A) (B)
(C) (D)
- Let ( ) 1
x
m x
−
. Which of the following statements about m is correct?
(A) m is increasing and the graph of m is concave up.
(B) m is increasing and the graph of m is concave down.
(C) m is decreasing and the graph of m is concave up.
(D) m is decreasing and the graph of m is concave down.
- Let f be an exponential function of the form f (^) ( x (^) ) = abx , where a and b are constants. If the function g is the
result of a horizontal dilation to the graph of f , then which of the following equations could represent the transformation
from f to g?
(A) g (^) ( x (^) ) = 2 f (^) ( x ) (B) g (^) ( x (^) ) = f (^) ( x + (^2) ) (C) g (^) ( x ) (^) = f (^) ( x^2 ) (D) ( ) ( ( )) 2
g x = f x
- Which of the following functions is an equivalent form of the function h x ( (^) ) = 100 9 x? (A) h x ( (^) ) = 10 3^2 x (B) h x ( (^) ) = 10 9^2 x (C) h x ( (^) ) = 100 3^2 x (D) ( ) ( 2 )
x
h x =
- Which of the following functions is an equivalent form of the function k (^) ( x (^) ) = 4 9^2 x? (A) k (^) ( x (^) ) = 362 x (B) ( ) 4 3 x
k x =
(C) ( ) 4 81 x
k x =
(D) ( ) 16 81 x
k x =
- Which of the following functions is an equivalent form of the function ( ) 2
x
p x
−
(A) ( ) ( 16 ) x
p x = −
(B) ( ) 2 x
p x =
(C) ( )
x
p x
(D) ( )
x
p x
- A quadratic regression was used to develop a model for a given set of data. The figure above shows a graph of the residuals of the quadratic regression. Which of the following statements about the quadratic regression is true? (A) The quadratic model is not appropriate because there is a clear pattern in the graph of the residuals. (B) The quadratic model is not appropriate because there are more positive residual values than negative residual values. (C) The quadratic model is appropriate because there is a linear pattern in the graph of the residuals. (D) The quadratic model is appropriate because none of the residuals fall outside the pattern in the graph of the residuals. 1 9. 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 = 5.7 −2.9 x
(B)
2
y = − 1.3 x + 2.1 x +0.
(C) 0.3 1.72( ) x
y = −
(D) y = −0.3 +2.4 log x
2 0. After Mr. Passwater discovers what he thought was a new radioactive element, he names one particular isotope Boogeranium-317. He notices that Boogeranium-317 has a half-life of 9.7 seconds. If Mr. Passwater wants to find a function that models the amount of Boogeranium-317 left in a particular sample, which of the following model types is best? (A) linear (B) quadratic (C) cubic (C) exponential
Test : (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.
- The largest chess streamer on YouTube is GothamChess (Levy Rozman), who currently has over 4 million
subscribers. The number of Gotham subscribers can be modeled by the arithmetic sequence an , where an is the number
of Gotham subscribers (in millions) for month n ( n = 1 represents January 2021 ).
If GothamChess had 1.66 million subscribers in month 9 and an + 1 − an =0.143for n 1 , how many subscribers (in
millions) does the arithmetic model predict GothamChess will have for month 13? (A) 1.803 (B) 2.232 (C) 2.833 (D) 3.
- Dommaraju Gukesh, better known as Gukesh D, is one of the best young chess players in the world. At only 17 years old, Gukesh D is already the 8th^ highest rated chess player in the world and recently surpassed the former world chess champion Vishy Anand as the highest rated chess player in India. When he first began playing chess, Gukesh’s rating could be modeled using a geometric sequence, where the first term represents his rating during month 1 (December 2013 ). Gukesh had a rating of 1475 during month 9 (August 201 4 ) and a rating of 1700 during month 15 (February 2015 ). According to this model, what was Gukesh’s rating during month 25 (December 2015)? Round your answer to the nearest whole number. (A) 2075 (B) 2154 (C) 3950 (D) 7031
Test : (Topics 2 .1- 2 .6) Free Response Questions Name: Section 2 : Part A - 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.
- The purchasing power of the US currency is the amount of goods and services that can be bought with one US dollar.
The function P gives the purchasing power on one US dollar based on the value of a dollar in the year 2020. The table
above presents values for the function P at selected values of t , where t is the number of years since 1933.
P t ( ) can be modeled by the exponential regression 21.682 0.963( ) t
y = , where t is the number of years since 1933.
(A) (i) Find the time t predicted by the given exponential regression model when P t ( ) (^) =5. Show the setup that leads to your answer. (ii) Find the value of P ( 54 )predicted by the exponential regression model.
(iii) Find the residual value of the purchasing power of the US currency when t = 54. Does this residual value
indicate that our model overestimated or underestimated the purchasing power of the US currency when
t = 54? Give a reason for your answer.
(B) (i) The function P t ( )can also be modeled with a linear model. Let y = a + bx be the linear regression model for P t ( ). Use the data in the table to find the linear regression model for P t ( ). (ii) Find the value of P ( 30 )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 appropriate for the data. Explain your reasoning.
t
(years since 1933 )
P t ( ) (in dollars)
2. The table above presents values for a sequence f at selected values of n , where k is a constant. The data in the table
can be used to find two models for fn : an arithmetic model (^) ( an )and a geometric model (^) ( gn ).
(A) (i) The arithmetic model an for fn can be written as an = a 0 + dn , where d is the common difference between
successive terms. Use the data in the table to find an expression for an.
(ii) Using the arithmetic model an , find the value of the constant k in the table above.
(iii) According to the arithmetic model, for what value of n will f n =66? Show the work that leads to your
answer.
(B) (i) The geometric model gn for fn can be written as g n = abn , where a and b are constants. Use the given
data to write two equations that can be used to find the values for constants a and b in the expression for gn.
(ii) Find the values for a and b.
(iii) Using the geometric model gn , find the value of the constant k in the table above.
(C) (i) For n 0 , the value of fn + 1 − fn increases as n increases. Based on this, is an or gn a better model for fn?
Give a reason for your answer.
n 2 4 5
fn 3 k 24
