Algebra I - Regents High School Examination - June 4, 2024

General Instructions

• The exam prohibits the use of communication devices.

• A graphing calculator and straightedge (ruler) are required.

• The exam has four parts and 35 questions.

• Answer all questions.

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

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

• Use a pen for all work, except for graphs and drawings, which should be done in pencil.

• Show all necessary steps, including formula substitutions, diagrams, graphs, charts, etc.

• Note that diagrams are not necessarily drawn to scale.

• Formulas are provided at the end of the examination.

• Scrap paper is not permitted, but blank spaces in the booklet may be used. A perforated sheet of scrap graph paper is provided.

• Any work on the scrap graph paper will not be scored.

• Sign the statement at the end of the answer sheet to confirm no unlawful knowledge or assistance.

Part I: Multiple Choice Questions

Question 1

• A ball's height is recorded each second after being launched.

• The data is provided in a table:

  • Time (sec): 0, 1, 2, 3, 4

  • Height (ft): 11, 59, 75, 59, 11

• The question asks for a valid conclusion based on the data.

• The correct answer is (4): The ball reaches its maximum height at 2 seconds.

Question 2

• A tour bus can seat at most 48 passengers.

• Adult ticket costs $18 and child ticket costs $12.

• The bus company must collect at least $650 to make a profit.

• $a$ represents the number of adult tickets, and $c$ represents the number of child tickets sold.

• The question asks for a system of inequalities that models this situation.

• The correct answer is (4):

  • $a + c \le 48$

  • $18a + 12c \ge 650$

Question 3

• The question asks which equation is always true.

• The correct answer is (1): $x^2 \cdot x^3 = x^5$

Question 4

• The question asks for the equivalent expression to $2(x^2 - 2x + 1) + (3x^2 + 3x - 5)$.

• Expanding and simplifying:

  • $2x^2 - 4x + 2 + 3x^2 + 3x - 5$

  • $5x^2 - x - 3$

• The correct answer is (1) $5x^2 - x - 3$

Question 5

• The question asks which sum is irrational.

• (1) $\sqrt{225} + \sqrt{100} = 15 + 10 = 25$ (Rational)

• (2) $\sqrt{4} + \sqrt{3} = 2 + \sqrt{3}$ (Irrational because $\sqrt{3}$ is irrational)

• (3) $\sqrt{25} + \sqrt{64} = 5 + 8 = 13$ (Rational)

• (4) $\sqrt{49} + 3\sqrt{121} = 7 + 3(11) = 7 + 33 = 40$ (Rational)

• The correct answer is (2): $\sqrt{4} + \sqrt{3}$

Question 6

• The question asks for the solution to $4(x - 5) + 2 = 14$.

• Solving the equation:

  • $4x - 20 + 2 = 14$

  • $4x - 18 = 14$

  • $4x = 32$

  • $x = 8$

Question 7

• A rare breed of rabbit doubled its population each month for two years.

• The question asks which type of function best models the increase in population.

• The correct answer is (3): exponential growth.

Question 8

• The question asks for the degree of the polynomial $2x - x^2 + 4x^3$.

• The degree of the polynomial is the highest power of the variable.

• The correct answer is (3): 3.

Question 9

• The question asks for the zeros of the function $f(x) = x(x - 5)(3x + 6)$.

• The zeros are the values of $x$ that make $f(x) = 0$.

• Setting each factor to zero:

  • $x = 0$

  • $x - 5 = 0 \Rightarrow x = 5$

  • $3x + 6 = 0 \Rightarrow 3x = -6 \Rightarrow x = -2$

• The zeros are 0, 5, and -2.

• The correct answer is (2): 0, 5, and -2.

Question 10

• The question asks for the y-intercept of the line that passes through the points (-1, 5) and (2, -1).

• First, find the slope (m):

  • $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1} = \frac{-1 - 5}{2 - (-1)} = \frac{-6}{3} = -2$

• Using the point-slope form with the point (-1, 5):

  • $y - 5 = -2(x - (-1))$

  • $y - 5 = -2(x + 1)$

  • $y - 5 = -2x - 2$

  • $y = -2x + 3$

• The y-intercept is 3.

• The correct answer is (3): 3.

Question 11

• Nancy has four choices for collecting her annual salary over eight years, represented by the functions:

  • $a(t) = 2t + 25$

  • $b(t) = 10t + 75$

  • $c(t) = 400t + 80$

  • $d(t) = 2(t + 1)^2 - 10t + 50$

• We need to find which plan gives the highest salary in her eighth year (t = 8).

  • $a(8) = 2(8) + 25 = 16 + 25 = 41$

  • $b(8) = 10(8) + 75 = 80 + 75 = 155$

  • $c(8) = 400(8) + 80 = 3200 + 80 = 3280$

  • $d(8) = 2(8 + 1)^2 - 10(8) + 50 = 2(9)^2 - 80 + 50 = 2(81) - 30 = 162 - 30 = 132$

• Plan c(t) gives the highest salary in her eighth year.

• The correct answer is (3): c(t).

Question 12

• The third term in a sequence is 25, and the fifth term is 625.

• We need to find a possible common ratio of the sequence.

• Let the sequence be denoted by $a<em>n = a</em>1 \cdot r^{n-1}$, where $a<em>n$ is the nth term, $a</em>1$ is the first term, and $r$ is the common ratio.

  • $a<em>3 = a</em>1 \cdot r^{3-1} = a_1 \cdot r^2 = 25$

  • $a<em>5 = a</em>1 \cdot r^{5-1} = a_1 \cdot r^4 = 625$

• Divide the fifth term by the third term:

  • $\frac{a<em>5}{a</em>3} = \frac{a<em>1 \cdot r^4}{a</em>1 \cdot r^2} = \frac{625}{25}$

  • $r^2 = 25$

  • $r = \pm 5$

Question 13

• The box plot summarizes the amount of snowfall in inches during the winter of 2021 for 12 locations.

• The question asks for the interquartile range (IQR).

• From the box plot, Q3 (the upper quartile) appears to be 110, and Q1 (the lower quartile) appears to be 30.

  • $IQR = Q3 - Q1 = 110 - 30 = 80$

• The correct answer is (3): 80.

Question 14

• Four quadratic functions are represented, and the question asks which function has the smallest minimum value.

• I: $a(x) = (x - 3)^2 - 7$. The vertex is (3, -7), so the minimum value is -7.

• II: The vertex of $b(x)$ is at x = 4, meaning the minimum is -2.

• III: The graph goes through (-1, -4), this is an absolute value function with vertex and minimum value of $x=0$, $y=-5$.

• IV: $c(x) = x^2 + 6x + 3$. To find the vertex, use $x = -\frac{b}{2a} = -\frac{6}{2(1)} = -3$. Then, $c(-3) = (-3)^2 + 6(-3) + 3 = 9 - 18 + 3 = -6$.

• Comparing the minimum values: -7, -5, -6 and -2.

• Smallest minimum value is -7. This is for first function I.

• The correct answer is (1): I.

Question 15

• The question asks for the equation that represents the sequence -2, 1, 4, 7, 10, …

• This is an arithmetic sequence with first term $a_1 = -2$ and common difference $d = 1 - (-2) = 3$.

• The formula for the nth term of an arithmetic sequence is $a<em>n = a</em>1 + (n - 1)d$.

• So, $a_n = -2 + (n - 1)3 = -2 + 3n - 3 = 3n - 5$.

• Testing: If n=1, a_1 = 3(1)-5 = -2 (correct)

• Alternative form shown in the choices is $a_n = -2 + (3)(n-1)$

• The correct answer is (4): $a_n = -2 + (3)(n - 1)$

Question 16

• The dot plot shows the number of goals Jessica scored in each lacrosse game last season.

• We need to determine which statement about the dot plot is correct.

• From the dot plot (assuming the data is 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6):

  • Mode (most frequent) = 2

  • Median (middle value) = 3

  • Mean = (1+1+2+2+2+3+3+4+4+5+6)/11 = 33/11 = 3

• Mode < Median, Mean = Median.

• The correct answer is (2): mean = median.

Question 17

• The students in Mrs. Smith’s algebra class were asked to describe the graph of $g(x) = 2(x - 3)^2$ compared to the graph of $f(x) = x^2$.

• The transformation $x \rightarrow (x - 3)$ shifts the graph to the right by 3 units.

• The factor of 2 makes the graph narrower.

• The correct answer is (4): Don said that the graph of g(x) is narrower and shifted right 3 units.

Question 18

• Dave took a long bike ride, and the graph models his trip.

• We need to find Dave’s average rate of change in miles per hour.

• From the graph, at 0 hours, he traveled 0 miles, and at 5 hours, he traveled 58 miles.

• The average rate of change is the slope of the line connecting these two points.

  • Average rate of change = $\frac{58 - 0}{5 - 0} = \frac{58}{5} = 11.6$ miles per hour.

• The correct answer is (3): 11.6.

Question 19

• The question asks which expression is equivalent to $(x - 5)(2x + 7) - (x + 5)$.

• Expanding and simplifying:

  • $2x^2 + 7x - 10x - 35 - x - 5$

  • $2x^2 - 4x - 40$

• The correct answer is (4): $2x^2 - 4x - 40$

Question 20

• The functions $f(x)$ and $g(x)$ are graphed on the set of axes.

• The question asks for the solution to the equation $f(x) = g(x)$.

• The solutions are the x-coordinates where the two graphs intersect.

• From the graph, the intersection points are at x = 1 and x = 5.

• The correct answer is (1): 1 and 5.

Question 21

• Nicole charges an hourly rate and an additional charge for gas when babysitting.

• The function $C(h) = 6h + 5$ determines how much to charge.

• We need to identify what the constant term of this function represents.

• The term $6h$ represents the hourly rate multiplied by the number of hours.

• Therefore, the constant term 5 represents the additional charge for gas.

• The correct answer is (1): the additional charge for gas.

Question 22

• The question asks: When solved for x in terms of a, the solution to the equation $3x - 7 = ax + 5$ is:

• Solve for x:

  • $3x - ax = 5 + 7$

  • $x(3 - a) = 12$

  • $x = \frac{12}{3 - a}$

• The correct answer is (1): $\frac{12}{3-a}$

Question 23

• Wayde van Niekerk ran 400 meters in 43.03 seconds to set a world record.

• We need to find the calculation that determines his average speed in miles per hour.

• We need to convert meters to miles and seconds to hours.

  • 1 mile = 1609 meters = 1.609 kilometers approximately, or 0.62 miles = 1000 meters (approx)

  • 1 hour = 3600 seconds

• The correct setup is:
  $\frac{400 \text{ m}}{43.03 \text{ sec }} \cdot \frac{3600 \text{ sec}}{1 \text{ hr }} \cdot \frac{0.62 \text{ mi}}{1000 \text{ m}}$

• The correct answer is(3):
  $\frac{400 \text{ m}}{43.03 \text{ sec }} \cdot \frac{3600 \text{ sec}}{1 \text{ hr }} \cdot \frac{0.62 \text{ mi}}{1000 \text{ m}}$

Question 24

• We need to find which function has a domain of all real numbers and a range greater than or equal to three.

• (1) $f(x) = 2x + 3$: This is a linear function, so the domain and range are all real numbers.

• (2) $g(x) = x^2 + 3$: This is a quadratic function. The domain is all real numbers, and the range is $y \ge 3$.

• (3) $h(x) = 3x$: This appears to be typo, assumes $3^x$. Domain is all real numbers, Range is y>0.

• (4) $m(x) = |x + 3|$: This is an absolute value function. The domain is all real numbers, and the range is $y \ge 0$.

• The correct answer is (2): $g(x) = x^2 + 3$.

Part II: Constructed Response Questions

Question 25

• Solve $5(x - 2) \le 3x + 20$ algebraically.
  $5x - 10 \le 3x + 20$
  $2x \le 30$
  $x \le 15$

Question 26

• Given $g(x) = x^3 + 2x^2 - x$, evaluate $g(-3)$.

• $g(-3) = (-3)^3 + 2(-3)^2 - (-3) = -27 + 2(9) + 3 = -27 + 18 + 3 = -6$

Question 27

• Given the relation $R = {(-1, 1), (0, 3), (-2, 4), (x, 5)}$.

• State a value for $x$ that will make this relation a function.

• A relation is a function if each x-value has only one y-value. Since -1, 0, and -2 are already used as x-values, x cannot be any of these numbers.

• Any value other than -1, 0 or -2 will work. For instance, x=1 makes this a function.

• Explanation: Because 1 is not already the first element in an ordered pair. If $x=1$, the relation is a function because no two ordered pairs have the same first element.

Question 28

• A survey of 150 students was taken.

• $\frac{2}{3}$ of the students play video games: $\frac{2}{3} (150) = 100$

• 85 of the students who play video games also use social media.

• Of the students who do not play video games, 20% do not use social media: $0.20 (150 - 100) = 0.20(50) = 10$

• Complete the two-way frequency table:

Question 29

• Use the method of completing the square to determine the exact values of x for the equation $x^2 + 10x - 30 = 0$.

• $x^2 + 10x = 30$

• $x^2 + 10x + (\frac{10}{2})^2 = 30 + (\frac{10}{2})^2$

• $x^2 + 10x + 25 = 30 + 25$

• $(x + 5)^2 = 55$

• $x + 5 = \pm \sqrt{55}$

• $x = -5 \pm \sqrt{55}$.

Question 30

• Factor $20x^3 - 45x$ completely.

• First find the greatest common factor (GCF): $5x$

• Factor out $5x$ $5x(4x^2 - 9)$

• Recognize the difference of squares: $4x^2 - 9=(2x-3)(2x+3)$

• The fully factored result is $5x(2x-3)(2x+3)$.

Part III: Constructed Response Questions

Question 31

• Graph the following system of equations:

  • $y = x^2 - 3x - 6$

  • $y = x - 1$
    State the coordinates of all solutions.
    Solution:

• Intersection Points: (-2, -3), (5, 4)

Question 32

• The table shows the amount of money a popular movie earned, in millions of dollars, during its first six weeks in theaters.

• Find the linear regression equation, rounding all values to the nearest hundredth.

• State the correlation coefficient to the nearest hundredth.

• State what this correlation coefficient indicates about the linear fit of the data.

• Linear Regression Equation $\hat{y} =-35.26x + 204.57$

• Correlation Coefficient: r = -0.99
  The data has a strong negative correlation indicating a linear fit for the data

Question 33

• Use the quadratic formula to solve the equation $3x^2 - 10x + 5 = 0$.

• Express the answer in simplest radical form.
  Quadratic Formula: $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$, where $a=3, b=-10, c=5$
  Substitute variables:
  $x=\frac{-(-10) \pm \sqrt{(-10)^2-4(3)(5)}}{2(3)}$
  $x=\frac{10 \pm \sqrt{100-60}}{6}$
  $x=\frac{10 \pm \sqrt{40}}{6}$
  Simplify the radical: $\sqrt{40} = \sqrt{4}\sqrt{10}=2\sqrt{10}$
  $x=\frac{10 \pm 2\sqrt{10}}{6}$
  Divide everything by 2:
  $x=\frac{5 \pm \sqrt{10}}{3}$

Question 34

• Graph the system of inequalities:

  • $3y + 2x \le 15$

  • y - x > 1

• State the coordinates of a point in the solution to this system.

• Justify your answer.

Solution:

• graph inequalites.
  A point is on the plane is (1, 2). A valid points must satisfy the following equations:
  $3(2) + 2(1) \le 15$
  $6 + 2 \le 15$
  $8 \le 15$
  2 - 1 > 1
  1>1 This can not be. So the answer would need to be (0,2) or (1,3) (check the graph). The point must satisfiy both.

• coordinates = (0, 2)

• Justification: The coordinates (0, 2) are on the plane and solve the above two equations in a true statement.

Part IV: Constructed Response Question

Question 35

• Write a system of equations that can be used to model the situation.

• Courtney thinks that one latte costs $2.75 and one donut costs $2.25. Is Courtney correct? Justify your answer.

• Use your equations to determine algebraically the exact cost of one latte and the exact cost of one donut.

• Define variables

• $x$ = cost of one latte

• $y$ = cost of one donut

• System of Equations:

  • $4x + 2y = 15.50$

  • $3x + 5y = 18.10$

• To prove it you substitute Courtney's figures into the equations to check:
  Equation 1:
  Substituting x = 2.75 and y= 2.25, it becomes
  $4(2.75) + 2(2.25) = 15.50$
  $11 + 4.50 = 15.50$
  $15.50 = 15.50$
  Equation 2:
  Substituting x = 2.75 and y= 2.25, it becomes
  $3(2.75) + 5(2.25) = 18.10$
  $8.25 + 11.25 = 18.10$
  $19.5 \ne 18.10$
  The equation gives a not true value, therefore, Courtney is not correct.

• Solve the equations:

  • Multiply first equation by 5: $20x + 10y = 77.50$

  • Multiply second equation by -2: $-6x - 10y = -36.20$
    Now you can add
    $14x = 41.3$
    Solving for $x$: $x = \frac{41.3}{14}$ or $x = 2.95$

• Substitue $x = 2.95$ into the first equation $4(2.95) + 2y = 15.50$
  $11.8 + 2y = 15.50$
  Solve for $y$: $2y = 15.50 - 11.8$ or $2y = 3.7$
  $y = 1.85$
  $x = 2.95$
  $y = 1.85$

• One latte costs $2.95 and one donut costs $1.85.