Algebra II - Regents High School Examination
Algebra II Regents High School Examination
General Information
The exam prohibits the use of any communications device. Possession or use of such a device will invalidate the examination.
A graphing calculator and a straightedge (ruler) must be available for use during the examination.
The examination has four parts, with a total of 37 questions.
All questions must be answered.
Answers for Part I (multiple-choice) are recorded on a separate answer sheet.
Answers for Parts II, III, and IV are written directly in the examination booklet.
Work should be written in pen, except for graphs and drawings, which should be done in pencil.
Necessary steps, including formula substitutions, diagrams, graphs, and charts, should be clearly indicated.
Diagrams are not necessarily drawn to scale.
A sheet of formulas is provided at the end of the examination booklet.
Scrap paper is not permitted, but blank spaces in the booklet may be used as scrap paper. A perforated sheet of scrap graph paper is provided for graphing purposes but will not be scored.
Upon completion of the examination, the statement at the end of the answer sheet must be signed, indicating no unlawful knowledge of the questions or answers and no assistance given or received during the examination.
The examination is administered on Thursday, January 23, 2025, from 1:15 to 4:15 p.m.
Part I - Multiple Choice Questions
Each correct answer in Part I receives 2 credits. No partial credit is allowed.
Question 1
The exact value of \sin\left(\frac{3}{8}\pi\right) is:
(1) \frac{\sqrt{2}}{1}
(2) \frac{\sqrt{2}}{2}
(3) \frac{\sqrt{3}}{2}
(4) \frac{\sqrt{3}}{1}
Question 2
A teacher divides students into two groups, grades homework for one group, and compares test scores. This data collection method is best described as:
(1) an experiment
(2) an unbiased survey
(3) a simulation
(4) an observational study
Question 3
Which expression is equivalent to (x^2 + 2)^2 + 27(x^2 + 2) + 90?
(1) (x + 30)(x + 3)
(2) (x + 28)(x + 5)
(3) (x - 30)(x - 3)
(4) (x - 2)(x + 25)(x - 90)
Question 4
Given f(x) = 2x + 5 and g(x) = x^3, what are the solutions to f(x) = g(x)?
(1) (0.75, 4) or (-2,-1.5)
(2) x = 0.75 or x = -2
(3) y = -1.5 or y = 4
(4) (-2, 0.75)
Question 5
Given f(x) = 2x^3 - 3x^2 - 5x - 12 and g(x) = x - 3, the quotient of \frac{f(x)}{g(x)} is:
(1) 2x^2 + 3x + 4
(2) 2x^3 + 3x^2 + 4x
(3) 2x^2 - 9x + 22 - \frac{78}{x - 3}
(4) 2x^3 - 9x^2 + 22x - 78
Question 6
Abby is told that each day there is a 50% chance it will rain. Which simulation can Abby perform to determine the likelihood of it raining for the next seven days?
(1) Flip a coin seven times, count how many heads, and repeat 50 times.
(2) Roll a die seven times, count how many twos, and repeat 50 times.
(3) Roll a pair of dice, count totals of seven, and repeat 50 times.
(4) Flip a coin 50 times and count how many heads.
Question 7
What are the solutions to 4x^2 - 7x - 2 = -10?
(1) \frac{-4 \pm i\sqrt{1}}{2}
(2) \frac{7 \pm i\sqrt{79}}{8}
(3) \frac{8 \pm \sqrt{6}}{6}
(4) \frac{7 \pm i\sqrt{143}}{8}
Question 8
If x - 5 is a factor of p(x) = ax^4 + bx^3 + cx^2 + dx + e, then which statement must be true?
(1) p(25) = 0
(2) p(25) \neq 0
(3) p(5) = 0
(4) p(5) \neq 0
Question 9
In a small city, there are 22 gas stations. The mean price for a gallon of regular gas was \$2.12 with a standard deviation of \$0.05. The distribution of the data was approximately normal. Given this information, the middle 95% of the gas stations in this small city likely charge
(1) \$1.90 to \$2.34 for a gallon of gas
(2) \$1.97 to \$2.27 for a gallon of gas
(3) \$2.02 to \$2.22 for a gallon of gas
(4) \$2.07 to \$2.17 for a gallon of gas
Question 10
The expression \frac{x^2 - 1}{4x^2 - 5} is equivalent to
(1) 4 - \frac{2}{x^2 - 1} + 1
(2) 4 + \frac{2}{x^2 - 1} + 1
(3) 4 - \frac{x^2 - 1}{9}
(4) 4 - \frac{x^2 - 1}{4}
Question 11
For all positive values of x, which expression is equivalent to x \cdot \sqrt[4]{x^{11}}?
(1) x^{\frac{19}{22}}
(2) x^{\frac{11}{8}}
(3) x^{\frac{13}{4}}
(4) x^{\frac{2}{11}}
Question 12
The expression i^2(5x - 2i)^2 is equivalent to
(1) -25x^2 + 20xi - 4
(2) -25x^2 + 20xi + 4
(3) 25x^2 + 20xi + 4
(4) 25x^2 + 4
Question 13
Functions f and g are given below.
f(x) = -7x^2 - 5x + 11
g(x) = 3x^2 - 7x + 25
When 2f(x) is subtracted from g(x), the result is
(1) 4x^2 - 3x - 3
(2) -4x^2 + 3x + 3
(3) 4x^2 - 17x - 47
(4) -4x^2 - 17x + 47
Question 14
A manufacturer claims that the number of ounces of a beverage dispensed by one of its automatic dispensers is normally distributed with a mean of 8.0 ounces and a standard deviation of 0.04 ounces. To the nearest tenth of a percent, what percent of the cups filled by this company’s dispenser will contain between 7.9 and 8.11 ounces?
(1) 99.5
(2) 99.4
(3) 99.1
(4) 97.6
Question 15
What is the value of x in the solution of the system of equations below?
5x + 2y - z = -14
7y - z = 31
5y + 4z - 5x = -23
(1) -17
(2) -2
(3) -1
(4) -7
Question 16
The graph below shows the amount of a radioactive substance left over time.
The daily rate of decay over an 8-day interval is approximately
(1) 23%
(2) 95%
(3) 5%
(4) 77%
Question 17
If 4(10^{5x-2})=12, then x equals
(1) 2.3
(2) 3 + (\log_{40}12 \cdot 5)
(3) \frac{\log(3) + 2}{5}
(4) \frac{ \log 4 + \log 12 + 2}{5}
Question 18
A random sample of 152 students was surveyed on a particular day about how they got to school. The survey results are summarized in the table below.
Which statement is best supported by the data?
(1) The probability of being late given that a student walked is greater than the probability that a student walked given that the student was late.
(2) The probability of being late given that a student walked is less than the probability that a student walked given that the student was late.
(3) The probability of being late given that a student walked is equal to the probability that a student walked given that the student was late.
(4) The probability of being late given that a student walked cannot be determined.
Question 19
If f(x) = \sqrt[3]{x} + 4, then f^{-1}(x) equals
(1) \sqrt{x-4}^3
(2) (x-4)^3
(3) x^3 + 4
(4) \sqrt[3]{x} - 4
Question 20
Given the equation S(x) = 1.7 \sin(bx) + 12, where the period of S(x) is 12, what is the value of b?
(1) \frac{\pi}{6}
(2) 24\pi
(3) 12\pi
(4) 6\pi
Question 21
Jin solved the equation \sqrt{4-x} = x + 8 by squaring both sides. What extraneous solution did he find?
(1) -5
(2) -12
(3) 3
(4) 4
Question 22
The expression (x^2 + y^2)^2 is not equivalent to
(1) (x^2 - y^2)^2 + (2xy)^2
(2) (x + y)^4 + 2(xy)^2
(3) x^2(x^2 + 2y^2) + (y^2)^2
(4) (2x^2 + y^2)^2 - (3x^4 + 2x^2 y^2)
Question 23
The height of a running trail is modeled by the quartic function y = f(x), where x is the distance in miles from the start of the trail and y is the height in feet relative to sea level.
If this trail has a minimum height of 16 feet below sea level, which function(s) could represent a running trail whose minimum height is half of the minimum height of the original trail?
I. y = f(\frac{1}{2}x)
II. y = f(x) + 8
III. y = \frac{1}{2} f(x)
(1) I, only
(2) II, only
(3) I and III
(4) II and III
Question 24
The crew aboard a small fishing boat caught 350 pounds of fish on Monday. From that Monday through the end of the week on Friday, the weight of the fish caught increased 15% per day. The total weight, in pounds, of fish caught is approximately
(1) 411
(2) 612
(3) 1748
(4) 2360
Part II
Answer all 8 questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs,
charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For all questions in this part, a correct
numerical answer with no work shown will receive only 1 credit. All answers should be written in pen, except for graphs and drawings, which should be done in pencil. [16]
Question 25
Describe the translations that map f(x) = \log x to g(x) = \log(x + 3) - 5.
Question 26
Solve algebraically for x:
\frac{2x + 2}{6} = \frac{5}{x-3}
Question 27
Given \cos \theta = -\frac{\sqrt{2}}{7} with \theta in Quadrant II, find the exact value of \sin \theta.
Question 28
Given a > 1, use the properties of rational exponents to determine the value of x for the equation below.
\frac{a^{10}}{5} = (a^3)^{\frac{1}{2}} a^x
Question 29
Graph at least one cycle of y = 5\sin(4x) - 3 on the set of axes below.
Question 30
The cost of a brand-new electric-hybrid vehicle is listed at \$33,400, and the average annual depreciation for the vehicle is 15%. The car’s value can be modeled by the function
V(x) = 33,400(0.85)^x, where x represents the years since purchase.
Julia and Jacob have each written a function that is equivalent to the original.
Jacob’s function: V(x) = 33,400(0.1422)^{\frac{x}{12}}
Julia’s function: V(x) = 33,400(0.9865)^{12x}
Whose function is correctly rewritten to reveal the approximate monthly depreciation rate? Justify your answer.
Question 31
Write a recursive formula for the sequence 8, 20, 50, 125, 312.5,…
Question 32
A grocery store orders 50 bags of oranges from a company’s distribution center. The bags have a mean weight of 3.85 pounds per bag. The company claims that their bags of oranges have a mean
weight of 4 pounds. The grocery store ran a simulation of 50 bags, 2500 times, assuming a mean of 4 pounds. The results are shown below.
Is the mean weight of the grocery store’s sample unusual? Explain using the results of the simulation.
Part III
Answer all 4 questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs,
charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For all questions in this part, a correct
numerical answer with no work shown will receive only 1 credit. All answers should be written in pen, except for graphs and drawings, which should be done in pencil. [16]
Question 33
At the Lakeside Resort, the probability that a guest room has a view of the lake is 0.24. The probability that a guest room has a queen-size bed is 0.74. Let A be the event that the guest room
has a view of the lake, and let B be the event that the guest room has a queen-size bed. Events A and B are found to be independent of each other.
Determine the exact probability that a randomly selected guest room has a view of the lake and a queen-size bed.
Determine the exact probability that a randomly selected guest room has a view of the lake or a queen-size bed.
Question 34
Which function has a greater average rate of change on the interval [-1,4]? Justify your answer.
Given p(x) = 3x + 1
Question 35
Determine an equation for the parabola with focus (-2, 4) and directrix y = 10. (The use of the grid below is optional.)
Question 36
Algebraically find the zeros of c(x) = x^3 + 2x^2 - 16x - 32.
On the axes below, sketch y = c(x).
Part IV
Answer the question in this part. A correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc.
Utilize the information provided to determine your answer. Note that diagrams are not necessarily drawn to scale. A correct numerical answer with no work shown will receive only 1
credit. All answers should be written in pen, except for graphs and drawings, which should be done in pencil. [6]
Question 37
The populations of honeybees in two different colonies are studied for four months. During this time, the colony population can be approximated by P(t) = P_0 e^{rt}, where P(t) is the colony
Question: The exact value of \sin\left(\frac{3}{8}\pi\right) is:
Answer: (3) \frac{\sqrt{3}}{2}
Explanation: The value of \sin\left(\frac{3}{8}\pi\right) can be found using the unit circle and trigonometric identities.
Question: A teacher divides students into two groups, grades homework for one group, and compares test scores. This data collection method is best described as:
Answer: (4) an observational study
Explanation: The teacher simply observes and compares outcomes without manipulating factors, making it observational.
Question: Which expression is equivalent to (x^2 + 2)^2 + 27(x^2 + 2) + 90?
Answer: (2) (x + 28)(x + 5)
Explanation: Expand the options to check which one matches the given expression.
Question: Given f(x) = 2x + 5 and g(x) = x^3, what are the solutions to f(x) = g(x)?
Answer: (2) x = 0.75 or x = -2
Explanation: Set the functions equal to each other and solve for x.
Question: Given f(x) = 2x^3 - 3x^2 - 5x - 12 and g(x) = x - 3, the quotient of \frac{f(x)}{g(x)} is:
Answer: (3) 2x^2 - 9x + 22 - \frac{78}{x - 3}
Explanation: Perform polynomial long division to find the quotient.
Question: Abby is told that each day there is a 50% chance it will rain. What simulation can Abby perform to determine the likelihood of it raining for the next seven days?
Answer: (1) Flip a coin seven times, count how many heads, and repeat 50 times.
Explanation: Coin flips represent a binary outcome mimicking the 50% chance of rain.
Question: What are the solutions to 4x^2 - 7x - 2 = -10?
Answer: (4) \frac{7 \pm i\sqrt{143}}{8}
Explanation: Rearrange the equation and use the quadratic formula to find the solutions.
Question: If x - 5 is a factor of p(x) = ax^4 + bx^3 + cx^2 + dx + e, then which statement must be true?
Answer: (3) p(5) = 0
Explanation: A factor means that the corresponding root evaluates to zero.
Question: In a small city, there are 22 gas stations. The mean price for a gallon of regular gas was \$2.12 with a standard deviation of \$0.05. What price range likely encompasses the middle 95%?
Answer: (1) \$1.90 to \$2.34 for a gallon of gas
Explanation: Utilize the empirical rule for normal distributions (mean ± 2 standard deviations).
Question: The expression \frac{x^2 - 1}{4x^2 - 5} is equivalent to:
Answer: (4) 4 - \frac{x^2 - 1}{4}
Explanation: Simplify each option to find equivalences with the original expression.
Question: For all positive values of x, which expression is equivalent to x \cdot \sqrt[4]{x^{11}}?
Answer: (3) x^{\frac{13}{4}}
Explanation: Combine exponents accordingly when multiplying.
Question: The expression i^2(5x - 2i)^2 is equivalent to:
Answer: (2) -25x^2 + 20xi + 4
Explanation: Expand and apply the property i^2 = -1 to simplify.
Question: Functions f and g are given below. When 2f(x) is subtracted from g(x), the result is:
Answer: (2) -4x^2 + 3x + 3
Explanation: Combine the functions and simplify the result.
Question: A manufacturer claims that the number of ounces dispensed is normally distributed with a mean of 8.0 ounces and a standard deviation of 0.04 ounces. What percent of cups will contain between 7.9 and 8.11 ounces?
Answer: (1) 99.5
Explanation: Use the normal distribution properties to evaluate the probability of this range.
Question: What is the value of x in the solution of the system of equations below?
Answer: (2) -2
Explanation: Solve the system algebraically and determine the value of x.
Question: The graph shows the amount of a radioactive substance left over time. What is the daily rate of decay over an 8-day interval?
Answer: (3) 5%
Explanation: Analyze the given graph data to extract the decay rate.
Question: If 4(10^{5x-2})=12, then x equals:
Answer: (3) \frac{\log(3) + 2}{5}
Explanation: Apply logarithmic properties to isolate x.
Question: A random sample of 152 students was surveyed about how they got to school. Which statement is best supported by the data?
Answer: (2) The probability of being late given that a student walked is less than the probability that a student walked given that the student was late.
Explanation: Evaluate conditional probabilities from the survey results.
Question: If f(x) = \sqrt[3]{x} + 4, then f^{-1}(x) equals:
Answer: (2) (x-4)^3
Explanation: Solve for the inverse function algebraically.
Question: Given S(x) = 1.7 \sin(bx) + 12, where the period is 12, what is the value of b?
Answer: (1) \frac{\pi}{6}
Explanation: Use the period formula for sine functions to solve for b.
Question: Jin solved the equation \sqrt{4-x} = x + 8 by squaring both sides. What extraneous solution did he find?
Answer: (4) 4
Explanation: Extraneous solutions can arise from squaring both sides; validate which is extraneous.
Question: The expression (x^2 + y^2)^2 is not equivalent to:
Answer: (4) (2x^2 + y^2)^2 - (3x^4 + 2x^2 y^2)
Explanation: Expand the expressions to find non-equivalence.
Question: If this trail has a minimum height of 16 feet below sea level, which function(s) could represent a running trail whose minimum height is half of the minimum height? I, II, III?
Answer: (4) II and III
Explanation: Examine translations that halve the height correctly