Algebra II Regents Exam Notes

Algebra II Regents High School Examination

General Information

The exam prohibits the use of any communication devices.
A graphing calculator and a straightedge (ruler) are required for the examination.
The exam has four parts, with a total of 37 questions.
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.
All work should be in pen, except 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 perforated sheet of scrap graph paper is provided but will not be scored.

Part I: Multiple-Choice Questions

Question 1

Topic: Survey bias.
Question: How to obtain the least biased results when surveying high school students about their exercise habits.
Options:
- Entering the gym.
- In the junior class.
- Entering the library.
- Entering the building.
Correct Approach: Surveying every fifth student entering the building would likely provide the least biased results.

Question 2

Topic: Equivalent Expressions
Question: Which expression is equivalent to $\frac{2x^3 + 3x^2 - 4x + 15}{x + 3}$ , given $x \neq -3$ ?
Options:
- $2x^3 + 9x^2 + 23x + 74$
- $2x^2 - 3x + 5 - \frac{10}{x + 3}$
- $2x^3 - 3x^2 + 5x - 10$
- $2x^2 + 9x + 23 + \frac{74}{x + 3}$
Correct Answer: $\frac{2x^3 + 3x^2 - 4x + 15}{x + 3} = 2x^2 - 3x + 5$

Question 3

Topic: Probability
Question: Probability that a fan prefers pizza given that the fan prefers football.
Data:
- Football fans who prefer wings: 14
- Football fans who prefer pizza: 20
- Football fans who prefer hot dogs: 6
- Baseball fans who prefer wings: 6
- Baseball fans who prefer pizza: 12
- Baseball fans who prefer hot dogs: 42
Solution: Probability = $\frac{20}{14 + 20 + 6} = \frac{20}{40} = \frac{1}{2}$

Question 4

Topic: Inverse Functions
Question: If $f(x) = \frac{1}{2}x - 4$ , then the inverse function $f^{-1}(x)$ is:
Options:
- $f^{-1}(x) = \frac{x + 1}{3}$
- $f^{-1}(x) = \frac{x}{3} + 1$
- $f^{-1}(x) = \frac{x + 4}{12}$
- $f^{-1}(x) = \frac{x}{12} + 4$
Solution:
- Let $y = \frac{1}{2}x - 4$
- Swap x and y: $x = \frac{1}{2}y - 4$
- Solve for y: $x + 4 = \frac{1}{2}y$
- $y = 2(x + 4) = 2x + 8$
- Therefore, $f^{-1}(x) = 2x + 8$

Question 5

Topic: System of Equations
Question: Given a graph of a quadratic function, find the solution to the system when $x + y = 4$ is drawn on the same axes.
Given: A graph of a quadratic function.
Task: Find one solution to the system.
Solution: From the graph, determine the points of intersection between the quadratic function and the line $x + y = 4$ . The correct answer is (3, 1).

Question 6

Topic: Solving Equations
Question: What is the solution of $2(3^x + 4) = 56$ ?
Options:
- $x = \log_3(28) - 4$
- $x = -1$
- $x = \log(25) - 4$
- $x = \frac{\log(56)}{\log(6)} - 4$
Solution:
- $2(3^x + 4) = 56$
- $3^x + 4 = 28$
- $3^x = 24$
- $x = \log_3(24)$

Question 7

Topic: Probability
Question: Find the probability of wanting both a large screen and a fast processor given survey data.
Given:
- 45% wanted a large screen.
- 31% wanted a fast processor.
- 58% wanted a large screen or a fast processor.
Solution:
- $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
- $0.58 = 0.45 + 0.31 - P(A \cap B)$
- $P(A \cap B) = 0.45 + 0.31 - 0.58 = 0.18$
- The probability is 18%.

Question 8

Topic: Discriminant
Question: Which quadratic functions have imaginary roots?
Given:
- $f(x)$ has a discriminant of 8.
- $g(x)$ has a discriminant of -16.
- Graphs of $h(x)$ and $j(x)$ .
Solution: A quadratic has imaginary roots when the discriminant is negative. Therefore, $g(x)$ has imaginary roots. By observing the graphs, determine if $h(x)$ or $j(x)$ have imaginary roots based on whether they intersect the x-axis. The correct answer is g(x) and h(x).

Question 9

Topic: Half-life
Question: Determine the expression for the amount of Americium remaining after t minutes, given a half-life of 25 minutes and an initial amount $A_0$ .
Options:
- $A_0 - \frac{1}{2}^{t}{25}$
- $A_0(25)^{\frac{t}{2}}$
- $25^{-\frac{1}{2}t}$
- $A_0(\frac{1}{2})^{\frac{t}{25}}$
Solution: The correct expression is $A_0(\frac{1}{2})^{\frac{t}{25}}$ .

Question 10

Topic: Y-intercepts
Question: Which function has the greatest y-intercept?
Functions:
- $f(x) = 4\sin(2x)$
- $g(x) = 3x^4 + 2x^3 + 7$
- $h(x) = 5e^{2x} + 3$
- $j(x) = 6\log_2(3x + 4)$
Solution:
- $f(0) = 4\sin(0) = 0$
- $g(0) = 3(0)^4 + 2(0)^3 + 7 = 7$
- $h(0) = 5e^{2(0)} + 3 = 5(1) + 3 = 8$
- $j(0) = 6\log2(3(0) + 4) = 6\log2(4) = 6(2) = 12$
- The function with the greatest y-intercept is $j(x)$ .

Question 11

Topic: Exponential Decay
Question: Convert a yearly decay model to a decay model based on decades.
Given:
- Yearly model: $P = 12150(0.962)^t$
Solution:
- Since $d = \frac{t}{10}$ , then $t = 10d$
- $P = 12150(0.962)^{10d} = 12150(0.962^{10})^d$
- $0.962^{10} \approx 0.679$
- The model is best represented by $P = 12150(0.679)^d$

Question 12

Topic: Trigonometry
Question: What is the value of $\tan(\theta)$ when $\sin(\theta) = \frac{2}{5}$ and $\theta$ is in quadrant II?
Solution:
- In quadrant II, cosine is negative.
- $\sin^2(\theta) + \cos^2(\theta) = 1$
- $(\frac{2}{5})^2 + \cos^2(\theta) = 1$
- $\frac{4}{25} + \cos^2(\theta) = 1$
- $\cos^2(\theta) = \frac{21}{25}$
- $\cos(\theta) = -\frac{\sqrt{21}}{5}$
- $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{2}{5}}{-\frac{\sqrt{21}}{5}} = -\frac{2}{\sqrt{21}}$

Question 13

Topic: Normal Distribution
Question: Percentage of the population that falls below 21, given a normal distribution with a mean of 23 and a standard deviation of 1.2.
Solution:
- $z = \frac{x - \mu}{\sigma} = \frac{21 - 23}{1.2} = \frac{-2}{1.2} \approx -1.67$
- Using a z-table, the percentage is 4.78%.

Question 14

Topic: Exponential Functions
Question: Describe the parameters in the formula $y = a(b)^x$ modeling the decreasing number of kindergarteners.
Solution:
- $a$ is positive because it represents the initial number of students (105).
- $b$ is positive and less than 1 because the number of students is decreasing exponentially (0 < b < 1).

Question 15

Topic: Solving equations
Question: What is the solution set for the equation 3^{\left(x^{2}+6 x
igh t\right)}=x ?
Options:
- ${6, -3}$
- ${-6, 3}$
- ${6}$
- ${-3}$
Solution: The only possible solution is ${-3}$

Question 16

Topic: Monthly Payments Calculation
Question: Calculation of monthly loan payments for a new boat using the provided formula.
Given Formula: $M = \frac{P(\frac{r}{12})(1 + \frac{r}{12})^n}{(1 + \frac{r}{12})^n - 1}$
$P = 45000$
$r = 0.0675$
$n = 5 * 12 = 60$
Solution:
- $M = \frac{45000(\frac{0.0675}{12})(1 + \frac{0.0675}{12})^{60}}{(1 + \frac{0.0675}{12})^{60} - 1}$
- $M = \frac{45000(0.005625)(1.005625)^{60}}{(1.005625)^{60} - 1}$
- $M \approx \frac{45000 * 0.005625 * 1.40013}{1.40013 - 1}$
- $M \approx \frac{35.442}{0.40013} \approx 885.76$
The monthly payment is approximately $885.76.

Question 17

Topic: Percentage Discount over time
Question: Determine the number of weeks for an item costing $50 to be sold for under half price with a 10% discount each Monday.
Solution:
- Half price is $25.
- Week 1: $50 * 0.9 = $45$
- Week 2: $45 * 0.9 = $40.50$
- Week 3: $40.50 * 0.9 = $36.45$
- Week 4: $36.45 * 0.9 = $32.805$
- Week 5: $32.805 * 0.9 = $29.5245$
- Week 6: $29.5245 * 0.9 = $26.57205$
- Week 7: $26.57205 * 0.9 = $23.914845$
After 7 weeks, the price is under half price.

Question 18

Topic: Intervals of Positive Function Values
Question: In which interval is $f(x)$ always positive, given the graph of $f(x)$ ?
Options:
- $(-2, 4)$
- $(0, 10)$
- $(-12, 5)$
- $(-10, 0)$
Solution: The correct interval is $(0, 10)$

Question 19

Topic: Real Solutions to Equations
Question: How many real solutions are there to the equation $f(x) = g(x)$ if $f(x) = (x^2 + 3x + 2)(x^2 - 4x + 3)$ and $g(x) = x^2 - 9$ ?
Solution:
- $f(x) = (x + 1)(x + 2)(x - 1)(x - 3)$
- $g(x) = (x - 3)(x + 3)$
- $(x + 1)(x + 2)(x - 1)(x - 3) = (x - 3)(x + 3)$
- If $x = 3$ , then $0 = 0$ , so $x = 3$ is a solution.
- If $x \neq 3$ , then $(x + 1)(x + 2)(x - 1) = (x + 3)$

Question 20

Topic: Factor Theorem
Question: Which expression is a factor of $x^4 - x^3 - 11x^2 + 5x + 30$ ?
Options:
- $x + 2$
- $x - 2$
- $x + 5$
- $x - 5$
Solution: Try each factor using synthetic division or the factor theorem to see which one results in a remainder of zero.

Question 21

Topic: Equivalent Expressions
Question: The expression $\frac{x^2 + 6}{x^2 + 4}$ is equivalent to which of the following?
Options:
- $\frac{6}{4}$
- $1 + \frac{10}{x^2 + 4}$
- $1 - \frac{2}{x^2 + 4}$
- $1 + \frac{2}{x^2 + 4}$
Solution: $\frac{x^2 + 6}{x^2 + 4} = \frac{x^2 + 4 + 2}{x^2 + 4} = 1 + \frac{2}{x^2 + 4}$

Question 22

Topic: Profit Modeling
Question: Modeling company profits given cost, sale price, and revenue functions.
Functions:
- $C(x) = 0.18x^3 + 0.02x^2 + 4x + 180$
- $S(x) = 95.4 - 6x$
- $R(x) = x \cdot S(x)$
Solution:
- $R(x) = x(95.4 - 6x) = 95.4x - 6x^2$
- Profit $P(x) = R(x) - C(x)$
- $P(x) = (95.4x - 6x^2) - (0.18x^3 + 0.02x^2 + 4x + 180)$
- $P(x) = -0.18x^3 - 6.02x^2 + 91.4x - 180$

Question 23

Topic: Even Functions
Question: Which function is even?
Options:
- $f(x) = x^3 + 2$
- $f(x) = x^2 + 1$
- $f(x) = |x + 2|$
- $f(x) = \sin(2x)$
Solution: An even function satisfies $f(x) = f(-x)$ .
- $f(x) = x^2 + 1$ is an even function.

Question 24

Topic: Polynomial Factors
Question: Given the graph of a cubic polynomial function $p(x)$ , which factor would appear twice?
Solution: The correct factor is $x + 2$

Part II

Question 25

Topic: Factoring Polynomials Completely
Task: Factor the expression $2x^3 - 3x^2 - 18x + 27$ completely.

Question 26

Topic: Solving System of Equations Algebraically
Task: Algebraically determine the values of x that satisfy the system of equations:
- $y = x^2 + 8x - 5$
- $y = 8x - 4$

Question 27

Topic: Solving Quadratic Equations
Task: Solve the equation $3x^2 + 5x + 8 = 0$ . Write the solution in $a + bi$ form.

Question 28

Topic: Graphing Cosine Functions
Task: Sketch at least one cycle of a cosine function with:
- Midline at $y = -2$
- Amplitude of 3
- Period of $\frac{\pi}{2}$

Question 29

Topic: Simplifying Complex Numbers
Task: Given i is the imaginary unit, simplify $(5xi^3 - 4i)^2$ as a polynomial in standard form.

Question 30

Topic: Parabolas and Directrix
Task: Consider the parabola given by $y = \frac{1}{4}x^2 + x + 8$ with vertex $(-2, 7)$ and focus $(-2, 8)$ . Explain how to determine the equation of the directrix.

Question 31

Topic: Rational Exponents
Task: Write $\frac{\sqrt[3]{x}}{\sqrt{x} \cdot \sqrt[5]{x^3}}$ as a single term in simplest form with a rational exponent.

Question 32

Topic: Average Rate of Change
Task: A fruit fly population can be modeled by the equation $P = 10(1.27)^t$ , where P represents the number of fruit flies after t days. What is the average rate of change of the population, rounded to the nearest hundredth, over the interval [0, 10.5]?

Question 33

Topic: Logarithmic Functions
Task: Sketch $p(x) = 2\log_2(x + 3) + 2$ on the axes below. Describe the end behavior of $p(x)$ as $x \to -3$ and as $x \to \infty$ .

Part III

Question 34

Topic: Solving Rational Equations Algebraically
Task: Solve for x algebraically: $\frac{1}{x-6} + \frac{x}{x-2} = \frac{4}{x^2-8x+12}$

Question 35

Topic: Solving System of Equations
Task: Solve the following system of equations algebraically for x, y, and z.
- $2x + 4y - 3z = 12$
- $3x - 2y + 2z = 29$
- $2x + y - 3z = 0$

Question 36

Topic: Statistical Significance
- Determine an interval containing the middle 95% of the simulation results. Round your answer to the nearest hundredth.
- Does the interval indicate that the difference between the classes’ grades is significant? Explain.

Question 37

Topic: Compound Interest and Continuous Compounding
Task: The Manford family started savings accounts for their twins, Abby and Brett, on the day they were born. They invested $8000 in an account for each child.
- Abby’s account pays 4.2% annual interest compounded quarterly.
- Brett’s account pays 3.9% annual interest compounded continuously.
- Write a function, A(t), for Abby’s account and a function, B(t), for Brett’s account that calculates the value of each account after t years.
- Determine who will have more money in their account when the twins turn 18 years old, and find the difference in the amounts in the accounts to the nearest cent.
- Algebraically determine, to the nearest tenth of a year, how long it takes for Brett’s account to triple in value.

High School Math Reference Sheet

Conversions

1 inch = 2.54 centimeters
1 meter = 39.37 inches
1 mile = 5280 feet
1 mile = 1760 yards
1 mile = 1.609 kilometers
1 kilometer = 0.62 mile
1 pound = 16 ounces
1 pound = 0.454 kilogram
1 kilogram = 2.2 pounds
1 ton = 2000 pounds
1 cup = 8 fluid ounces
1 pint = 2 cups
1 quart = 2 pints
1 gallon = 4 quarts
1 gallon = 3.785 liters
1 liter = 0.264 gallon
1 liter = 1000 cubic centimeters

Geometry

Triangle: Area $(A = \frac{1}{2}bh)$
Parallelogram: Area $(A = bh)$
Circle: Area $(A = \pi r^2)$
Circle: Circumference $(C = \pi d)$ or $(C = 2\pi r)$
General Prisms: Volume $(V = Bh)$
Cylinder: Volume $(V = \pi r^2h)$
Sphere: Volume $(V = \frac{4}{3}\pi r^3)$
Cone: Volume $(V = \frac{1}{3}\pi r^2h)$
Pyramid: Volume $(V = \frac{1}{3}Bh)$
Pythagorean Theorem: $(a^2 + b^2 = c^2)$

Algebra

Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Arithmetic Sequence: $(an = a1 + (n - 1)d)$
Geometric Sequence: $(an = a1r^{n - 1})$
Geometric Series: $(Sn = \frac{a1 - a_1r^n}{1 - r})$ where $r \neq 1$

Radian and Degree Measure

Radians: 1 radian = $(\frac{180}{\pi})$ degrees
Degrees: 1 degree = $(\frac{\pi}{180})$ radians

Exponential Growth/Decay

$(A = A0e^{k(t - t0)} + B_0)$