Algebra II Exam Notes

Algebra II Regents High School Examination

• Instructions:

  • A graphing calculator and a straightedge (ruler) must be available.

    • Explanation: These tools are necessary for solving problems and drawing graphs.

  • Do not open the examination booklet until the signal is given.

    • Explanation: Prevents cheating and ensures everyone starts at the same time.

  • Print your name and school name on the provided lines.

    • Explanation: Identifies your work.

  • Answer all 37 questions in the examination.

    • Explanation: Completing all questions maximizes your score.

  • Record answers to Part I multiple-choice questions on the separate answer sheet.

    • Explanation: Multiple-choice answers are graded separately.

  • Write answers to Parts II, III, and IV directly in the booklet.

    • Explanation: Free-response answers require showing your work.

  • All work should be written in pen, except graphs and drawings which should be done in pencil.

    • Explanation: Pen ensures permanence, while pencil allows for corrections on graphs.

  • Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc.

    • Explanation: Showing work is crucial for partial credit.

  • Utilize the information provided for each question to determine the answer.

    • Explanation: Each question provides necessary clues.

  • Note that diagrams are not necessarily drawn to scale.

    • Explanation: Do not rely on visual estimations.

  • The formulas that you may need are found at the end of the examination.

    • Explanation: Use the reference sheet to your advantage.

  • Scrap paper is not permitted, but you may use blank spaces in the booklet as scrap paper.

    • Explanation: Manage your space efficiently.

  • A perforated sheet of scrap graph paper is provided for graphing purposes but will not be scored.

    • Explanation: Use it for preliminary graphing.

  • Sign the statement at the end of the answer sheet to indicate you had no unlawful knowledge and did not give or receive assistance.

    • Explanation: An honesty pledge.

Part I

• Question 1: Which expression is equivalent to $(x^{\frac{1}{3}})^2 + 4(x^{\frac{1}{3}}) - 5$?

  • The correct answer is (2) $(x^{\frac{1}{3}} + 5)(x^{\frac{1}{3}} - 1)$

    • Explanation: Factor the quadratic expression by recognizing $x^{\frac{1}{3}}$ as a single variable. Calculator hack: Substitute a value for x into the original expression and the answer choices to check for equivalence.

• Question 2: Which value, to the nearest tenth, is an approximate solution for the equation $f(x) = g(x)$, if $f(x) = x^{\frac{2}{3}} + 5$ and $g(x) = 2(1.3)^x$?

  • The correct answer is (2) 3.9

    • Explanation: Solve by graphing both functions and finding the intersection point. Calculator hack: Graph $y = x^{\frac{2}{3}} + 5$ and $y = 2(1.3)^x$ in your calculator and find the intersection.

• Question 3: Which equation is graphed in the diagram below?

  • The correct answer is (4) $y = -5\cos(\frac{2}{3}\pi x) + 3$

    • Explanation: Identify the amplitude, period, and vertical shift from the graph to match the correct cosine function. Calculator hack: Graph the equation to confirm if it matches the provided diagram.

• Question 4: Which point is in Quadrant III and is a solution to the system below?-

  • $y = x^2 - 24$

  • $y = x - 12$

  • The correct answer is (2) (-3, -15)

    • Explanation: Substitute each point into both equations to see which satisfies both. Also, verify that the point is in Quadrant III. Calculator hack: solve the system of equations using the calculator.

• Question 5: For which equations will the value $s = 4$ make the statement an identity?-

  • I. $(2x - 3)^2 = 4x^2 - 3sx + 9$

  • II. $(x - 2)^3 = (x - 2)(x^2 + sx + s)$

  • The correct answer is (2) II, only

    • Explanation: Substitute s = 4 into each equation and check if both sides are equal for all x. Calculator hack: Substitute s=4 and plot left and right and see if they are the same

• Question 6: The number of bacteria in a sample, which can be modeled by an exponential regression, is shown in the table. Assuming this trend continues, approximately how many bacteria would be present 8 hours after the observation began?-

  • The correct answer is (3) 168

    • Explanation: Perform an exponential regression on the data to find the equation, then evaluate at t = 8. Calculator hack: Use the calculator's exponential regression function to find the curve of best fit and then evaluate for $t = 8$.

• Question 7: Factored completely, $x^4 + 4x^3 - 9x^2 - 36x$ is equivalent to:-

  • The correct answer is (2) $x(x + 3)(x - 3)(x + 4)$

    • Explanation: Factor out common terms and then factor the remaining polynomial. Calculator hack: Graph the original equation and the choices to find a match

• Question 8: The solution set for the equation $\sqrt{x + 1} + 5 = \sqrt{4x + 25}$ is:-

  • The correct answer is (2) {6}

    • Explanation: Solve the radical equation by squaring both sides and isolating x. Check for extraneous solutions. Calculator hack: Input the the original equation into the calculator to check for the solution

• Question 9: If f(x) is represented by the graph below, which translation of f(x) would have imaginary roots?-

  • The correct answer is (3) f(x) + 5

    • Explanation: Imaginary roots occur when the graph does not intersect the x-axis. Vertical shifts upward can cause this. Calculator hack: Graph f(x) and the translation to see when the roots become imaginary.

• Question 10: The solutions to the equation $3x^2 - 4x + 2 = 2x - 3$ are:-

  • The correct answer is (2) $\frac{1}{3} \pm \frac{2}{3}i$

    • Explanation: Use the quadratic formula to solve for x. Look for complex solutions. Calculator hack: use the calculator to solve the quadratic and confirm the answers

• Question 11: A culture of 1000 bacteria triples every 10 hours. Which expression models the number of bacteria in the sample after t hours?-

  • The correct answer is (4) $1000(3)^{\frac{t}{10}}$

    • Explanation: Exponential growth model with a base of 3 and a time factor of t/10. Calculator hack: Input the choices into the calculator and see if it follows the description.

• Question 12: An initial investment of $5000 in an account earns 3.5% annual interest. Which function correctly represents a recursive model of the investment after n years?-

  • The correct answer is (4)

  • $a_0 = 5000$

  • $a<em>n = a</em>{n-1}(1.035)$

    • Explanation: Recursive model increases the amount by 3.5% each year beginning with the initial amount. No calculator hacks needed.

• Question 13: Which expression is equivalent to $\frac{2xy^2}{\sqrt[3]{x^2y}}$

  • The correct answer is (4) $2x^{\frac{1}{3}}y^{\frac{5}{3}}$

    • Explanation: Simplify the expression using exponent rules. Calculator hack: Substitute values for $x$ and $y$ and compare with the original to check equivalence.

• Question 14: Which statements must be true about the polynomial function $k(x) = -2x^3 - 11x^2 - 12x + 9$?-

  • I. $(x - 3)$ is a factor of $k(x)$

  • II. $k(0) = 9$

  • III. $k(x)$ divided by $(x + 2)$ has a remainder of 5

  • The correct answer is (3) II and III

    • Explanation: Use the factor theorem, evaluate k(0), and use synthetic division to find the remainder when divided by (x+2). Calculator hack: Use the calculator the evaluation and synthetic division.

• Question 15: To prepare for lacrosse tryouts, Kole is increasing the amount of time he spends at the gym. This week he is spending 150 minutes there and he plans to increase this amount by 2% each week. The amount of time, in minutes, that he plans to spend at the gym t weeks from now is given by the function $A(t) = 150(1.02)^t$. In terms of a daily growth rate, the amount of time Kole is planning to spend at the gym can best be modeled by the function:-

  • The correct answer is (4) $A(t) = 150(1.00283)^{7t}$

    • Explanation: Convert the weekly growth rate to a daily growth rate. Calculator hack: Evaluate both equations to check the equivalence.

• Question 16: Mr. Zachary posts review assignments on the Betamath website for his students. On his last test, 49% of his students used Betamath and passed. Overall, 68% of his students used Betamath. Approximately what percentage of Mr. Zachary’s students passed, given that they used Betamath?-

  • The correct answer is (4) 72%

    • Explanation: Conditional probability calculation. P(Passed | Betamath) = P(Passed and Betamath) / P(Betamath).

• Question 17: Given $\sin(\theta) = \frac{7}{25}$ and $\theta$ terminates in Quadrant II, what is the value of $\tan(\theta)$?-

  • The correct answer is (1) $\frac{-7}{24}$

    • Explanation: Use trigonometric identities and the quadrant to determine the sign. Calculator hack: Use $sin^{-1}(\frac{7}{25})$ and make sure it's not in the second quadrant, then use the trig identities to find the answers and make sure that is also in the second quadrant.

• Question 18: A family-owned grocery store in New Hartford, NY employs 49 people whose ages are approximately normally distributed with a mean of 36 years and a standard deviation of 6.2 years. Ryan has been hired to work at this store. He is 30 years old. How many people who work at this store would you expect to be younger than Ryan?-

  • The correct answer is (2) 7

    • Explanation: Calculate the z-score for Ryan's age and find the corresponding percentile. Multiply by the total number of employees. Calculator hack: Use the normalcdf to calculate the percentile of people younger than ryan and multiply that by the number of employees.

• Question 19: For the function $d(x) = \sqrt[3]{x} + 2$, the inverse function, $d^{-1}(x)$, equals-

  • The correct answer is (4) $(x - 2)^3$

    • Explanation: To find the inverse, switch x and y and solve for y. Calculator hack: Graph the original function and the inverse and observe the reflection over $y = x$.

• Question 20: Given $f(x) = x^4 + x^3 - 3x^2 + 9x - 108$ and $f(3) = 0$, which values satisfy $f(x) = 0$?-

  • The correct answer is (4) $\pm 3i, -4, 3$

    • Explanation: Use synthetic division to reduce the polynomial, then solve for the remaining roots. Calculator hack: you can see if these are roots by inputting the equation into the calculator and observing them as roots.

• Question 21: The profit function, p(x), is found by subtracting the cost function, c(x), from the revenue function, r(x). Which function below represents the cost function given $p(x) = -15x^2 + 600x + 60$ and $r(x) = -0.4x^2 + 130x + 1200$?-

  • The correct answer is (3) $c(x) = 14.6x^2 - 470x + 1140$

    • Explanation: Since $p(x) = r(x) - c(x)$, then $c(x) = r(x) - p(x)$.

• Question 22: What is the remainder when $4x^3 - 3x + 3$ is divided by $x - 2$?-

  • The correct answer is (4) 29

    • Explanation: Use the remainder theorem or synthetic division. Calculator hack: Use the calculator to synthically divide the expression to find the remainder

• Question 23: The parabola with equation $\frac{1}{2}(y + 1) = (x - 4)^2$ has-

  • The correct answer is (3) a directrix $y = -4$

    • Explanation: Convert to standard form and identify the vertex and p-value to find the directrix. Calculator hack: identify certain values or properties of the equation by using the calculator to graph it.

• Question 24: Jay is training for a bike race over fifteen weeks. At the end of the first week, he has ridden ten miles, and he is planning to increase his weekly distance by nine percent each week. Approximately how many miles total will he have ridden from the beginning of his training to the end of the fifteenth week?-

  • The correct answer is (4) 293.609

    • Explanation: Geometric series problem. Calculator hack: summing a geometric series in the calculator.

Part II

• Question 25: Given x is a real number, write the expression in simplest a + bi form: $(x + 2i)(3 - 2xi) + 2x^2i$-

  • Solution:

  • $(x + 2i)(3 - 2xi) + 2x^2i = 3x - 2x^2i + 6i - 4xi^2 + 2x^2i$

  • $= 3x + 6i + 4x$

  • $= 7x + 6i$

    • Explanation: Multiply and combine like terms, remembering that $i^2 = -1$.

• Question 26: Solve $3.8e^{1.5t} = 16$ algebraically for t to the nearest hundredth.-

  • Solution:

  • $e^{1.5t} = \frac{16}{3.8}$

  • $1.5t = \ln(\frac{16}{3.8})$

  • $t = \frac{\ln(\frac{16}{3.8})}{1.5} \approx 1.01$

    • Explanation: Isolate the exponential term and use logarithms to solve for t.

• Question 27: In an attempt to get the student body’s opinion of a new dress code, members of the statistics class surveyed the students of the first period computer science class. Explain a statistical bias in the method of data collection.-

  • Solution: There is statistical bias because surveying only students in the first period computer science class does not represent the entire student body. Students in that class may have different opinions or demographics compared to the rest of the school.

    • Explanation: The sample is not representative of the entire population.

• Question 28: Sketch a graph of polynomial P(x), given the criteria below:-

  • $P(x)$ has zeros only at -5, 1, and 4

  • As $x \to \infty$, $P(x) \to \infty$

  • As $x \to -\infty$, $P(x) \to -\infty$

  • Solution: Sketch a graph with x-intercepts at -5, 1, and 4. The graph should rise to the right and fall to the left.

    • Explanation: The end behavior indicates a positive leading coefficient and odd degree.

• Question 29: The height, above ground, of a Ferris wheel car can be modeled by the function $h(t) = -103.5\cos(\frac{2\pi t}{5}) + 108.5$ where h is measured in feet and t is measured in minutes. State the period of the function and describe what the period represents in this context.-

  • Solution:

  • Period: $T = \frac{2\pi}{\frac{2\pi}{5}} = 5$

  • The period of 5 minutes represents the time it takes for the Ferris wheel to complete one full rotation.

    • Explanation: The period is found using the coefficient of t in the cosine function.

• Question 30: Solve algebraically for all values of x: $\frac{x + 5}{8} = \frac{2}{\sqrt{x + 3}} = 5$-

  • Solution:

  • $\frac{x+5}{8} = \frac{7}{1}$

  • $x + 5 = 56$

  • $x = 51$

    • Explanation: I am unable to proceed with this question since it seems to be improperly formatted, so an explanation here would not be helpful.

• Question 31: The transportation methods used by the upperclassmen at Calhoun High School are summarized in the table. Are the events “being a junior” and “driving to school” independent? Using statistical evidence, justify your answer.-

  • Solution:

  • $P(Junior) = \frac{Junior}{total} = \frac{58 + 75 + 12}{58+75+12+81+39+12} = \frac{145}{277}$

  • $P(drive) = \frac{drive}{total} = \frac{58 + 81}{58+75+12+81+39+12} = \frac{139}{277}$

  • $P(Junior \cap drive) = \frac{58}{277}$

  • $P(Junior) * P(drive) = (\frac{145}{277}) * (\frac{139}{277}) = \frac{20155}{76729} = 0.2626$

  • Since $P(Junior \cap drive) \neq P(Junior) * P(drive)$. The events are not independent.

    • Explanation: Check if $P(A \cap B) = P(A) * P(B)$ to determine independence.

• Question 32: Can $f(x) = x^3 + 7$ be classified as an odd function? Justify your answer.-

  • Solution: Test for odd function: $f(-x) = (-x)^3 + 7 = -x^3 + 7$

  • $-f(x) = -(x^3 + 7) = -x^3 - 7$

  • Since $f(-x) \neq -f(x)$, the function is not odd.

    • Explanation: An odd function satisfies $f(-x) = -f(x)$.

Part III

• Question 33: Solve the following system of equations algebraically for all values of x, y, and z:-

  • $3x - 8y + 2z = -60$ (1)

  • $-2x - 7y - 5z = -31$ (2)

  • $-6x + 2y - 4z = 36$ (3)

  • Solution:

  • Multiply equation (1) by 2: $6x - 16y + 4z = -120$ (4)

  • Add equation (3) and (4):

  • $-14y = -84$

  • $y =6$

  • Substitute y = 6 into equation 1 and 2:

  • $3x - 8(6) + 2z = -60$

  • $-2x - 7(6) - 5z = -31$

  • Simplify:

  • $3x + 2z = -12$ (5)

  • $-2x - 5z = 11$ (6)

  • Multiply equation (5) by 5 and equation (6) by 2:

  • $15x + 10z = -60$ (7)

  • $-4x - 10z = 22$ (8)

  • Add equation (7) and (8):

  • $11x = -38$

  • $x = -\frac{38}{11}$

  • From equation 5:

  • $3(-\frac{38}{11}) + 2z = -12$

  • $z = -\frac{1}{22}(132 + 114)$

  • $z = -\frac{246}{22} = -\frac{123}{11}$

    • Explanation: Use elimination or substitution to solve the system. Note: Calculator can solve a 3x3 system of equations.

• Question 34: In the town of Skaneateles, New York, house prices since 2008 have changed based on the function $H(t) = 200000(1.045)^t$, where t is the number of years since 2008 and H(t) is the median house price. Determine the average rate of change for the median house price in Skaneateles from 2010 to 2018 to the nearest dollar per year.-

  • $H(2) = 200000(1.045)^2 = 218405$

  • $H(10) = 200000(1.045)^{10} = 307009$

  • AverageRateOfChange = \frac{307009-218405}{10-2} = \frac{88604}{8} = 11075\frac{$}{year}

  • Explanation: From 2010 to 2018, then median house price in Skaneateles increased by approximately $11,075 per year.

    • Explanation: Average rate of change is calculated as the change in H(t) divided by the change in t.

• Question 35: A sporting goods manufacturer is trying to determine if they should continue to produce multiple types of hockey pucks. The company surveyed 50 randomly chosen customers and asked them if they purchased both game regulation pucks and lighter training pucks. Of those surveyed, 40 of them said that they purchase both types of pucks. A simulation that was run 100 times based on the survey results produced the approximately normal results below.-

  • a) Determine an interval containing the middle 95% of plausible values that estimates the proportion of all customers who would purchase both types of pucks from the company.

  • $0.795 \pm 2(0.085) = [0.625, 0.965]$

  • b) The company will continue to manufacture both types of hockey pucks if it is reasonable to assume that the true proportion of customers who buy both types of hockey pucks is above 0.60. Using the interval from part a, explain whether or not the company should continue to produce both types of hockey pucks.

  • Yes, the company should continue to manufacture both types of hockey pucks because the entire interval of plausible values is above 0.60.

    • Explanation: The 95% confidence interval is calculated using the mean and standard deviation from the simulation results.

• Question 36: Graph $y = f(x)$, where $f(x) = \log_2(x - 1) + 3$ on the set of axes below.-

  • State the equation of the asymptote of $f(x)$.

  • $x = 1$

  • When $f(x)$ is reflected over the line $y = x$, a new function is formed: $g(x) = 2^{x - 3} + 1$. State the equation of the asymptote of $g(x)$.

  • $y = 1$

    • Explanation: The asymptote of $f(x)$ occurs when the argument of the logarithm is zero, and the asymptote of $g(x)$ is a horizontal line determined by the vertical shift. Calculator hack: You can confirm the asymptote by graphing it on the calculator.

Part IV

• Question 37: Megan is performing an experiment in a lab where the air temperature is a constant 73°F and the liquid is 237°F. One and a half hours later, the temperature of the liquid is 112°F.

• Newton’s law of cooling states $T(t) = T<em>a + (T</em>0 - T_a)e^{-kt}$ where:

  • $T(t)$: temperature, °F, of the liquid at t hours

  • $T_a$: air temperature = 73°F

  • $T_0$: initial temperature of the liquid = 237°F

  • k: constant, t = 1.5 hours T(1.5) = 112°F

• Determine the value of k, to the nearest thousandth, for this liquid.

  • $112 = 73 + (237 - 73)e^{-k(1.5)}$

  • $39 = (164)e^{-1.5k}$

  • $\frac{39}{164} = e^{-1.5k}$

  • $ln(\frac{39}{164}) = -1.5k$

  • $k = -\frac{1}{1.5}ln(\frac{39}{164})$

  • $k = 0.917$

    • Explanation: Substitute given values into the formula and solve for k.

• Determine the temperature of the liquid using your value for k, to the nearest degree, after two and a half hours.

  • $T(2.5) = 73 + (237 - 73)e^{-0.917(2.5)}$

  • $T(2.5) = 73 + 164e^{-2.2925} = 73 + 164(0.101) \approx 89$ °F

    • Explanation: Substitute t = 2.5 and the calculated k into the formula to find T(2.5).

• Megan needs the temperature of the liquid to be 80°F to perform the next step in her experiment. Use your value for k to determine, to the nearest tenth of an hour, how much time she must wait since she first began the experiment.

  • $80 = 73 + (237 - 73)e^{-0.917t}$

  • $7 = 164e^{-0.917t}$

  • $\frac{7}{164} = e^{-0.917t}$

  • $ln(\frac{7}{164}) = -0.917t$

  • t = -\frac{1}{0.917}ln(\frac{7}{164}) \approxx 3.4 hours

    • Explanation: Substitute T(t) = 80 and the calculated k into the formula and solve for t.