June 2025 Algebra I Regents — Answer Key & Worked Solutions

The expression $\frac{10}{\sqrt{2}}$ is equivalent to

• (A) $5$
• (B) $20$
• (C) $5\sqrt{2}$ ✓
• (D) $10\sqrt{2}$

Multiplying numerator and denominator by root 2 rationalizes the denominator and simplifies to 5 root 2.

A parabola is graphed on the set of axes below. Over which interval is the parabola only increasing?

• (A) $[1,4]$
• (B) $[3,\infty)$ ✓
• (C) $(-\infty,3]$
• (D) $[-1,1]$

The graph shows the vertex at x = 3; since the parabola opens upward, the function is increasing only for x >= 3.

Which scenario represents an exponential relationship?

• (A) Kirsten's New Year's resolution is to lose one pound each week.
• (B) Sarah wants to increase her grade by 5 points each quarter.
• (C) Tommy wants to reduce his spending by $50 each month.
• (D) Dylan hopes to grow his business by 5% each month. ✓

A percent increase each month is multiplicative growth, the defining feature of an exponential relationship; the other three scenarios describe constant (linear) changes.

The geometry test scores for Andrea and Joe are shown in the table below. Which statement about their test scores is correct?

• (A) Both the mean and standard deviation of Andrea's test scores are higher than Joe's.
• (B) Both the mean and standard deviation of Joe's test scores are higher than Andrea's.
• (C) The mean of Andrea's test scores is higher than Joe's, but Joe's standard deviation is higher than Andrea's. ✓
• (D) The mean of Joe's test scores is higher than Andrea's, but Andrea's standard deviation is higher than Joe's.

Andrea's mean (85.75) is higher than Joe's (82.5), but Joe's scores are much more spread out, giving Joe a higher standard deviation.

Which polynomial has a degree of 3 and a leading coefficient of 2?

• (A) $2x^2+3x+1$
• (B) $6x^3+3x^2-2x$
• (C) $3x^2+2x+2$
• (D) $2x^3+x^2+4x$ ✓

2x^3+x^2+4x has degree 3 (highest exponent) and leading coefficient 2 (coefficient of the x^3 term).

The expression $(-3x^2+9)-(7x^2-5x+4)$ is equivalent to

• (A) $-10x^2+5x+5$ ✓
• (B) $-10x^2+5x+13$
• (C) $-10x^2-5x+5$
• (D) $-10x^2-5x+13$

Distribute the negative sign across the second polynomial, then combine like terms.

The function h(x) is used to calculate the average height, in inches, of a tomato plant x weeks after it is transplanted. These data are represented in the table below. Between weeks 4 and 12, the average rate of change, in inches per week, is

• (A) $6$ ✓
• (B) $8$
• (C) $48$
• (D) $58$

Average rate of change is the change in height divided by the change in weeks, from week 4 to week 12.

Chloe is solving the equation $x^2+5x=3x+3$. Her first step is shown below. Given: $x^2+5x=3x+3$ Step 1: $x^2+2x-3=0$ Which property justifies this step?

• (A) the zero product property
• (B) the commutative property
• (C) the distributive property
• (D) the subtraction property of equality ✓

Both 3x and 3 were subtracted from each side of the equation, which is the subtraction property of equality.

Which function represents the graph of $w(x) = |x|$ shifted 2 units to the right?

• (A) $g(x) = |x+2|$
• (B) $h(x) = |x-2|$ ✓
• (C) $q(x) = |x|+2$
• (D) $r(x) = |x|-2$

Shifting a graph to the right replaces x with (x - 2), giving h(x) = $|x - 2|$.

Given the system of equations: $y + 4x = 5$ $2x - 3y = 10$ A step in solving this system by using the substitution method would be

• (A) $2(5 - 4x) + 4x = 5$
• (B) $2(5 + 4x) + 4x = 5$
• (C) $2x - 3(5 - 4x) = 10$ ✓
• (D) $2x - 3(5 + 4x) = 10$

Solving the first equation for y gives y = 5 - 4x; substituting into the second equation gives 2x - 3(5 - 4x) = 10.

Which equation is equivalent to $x^2 - 6x = 27$?

• (A) $(x-3)^2 = 27 - 9$
• (B) $(x-3)^2 = 27 + 9$ ✓
• (C) $(x-3)^2 = 27 + 36$
• (D) $(x-3)^2 = 27 - 36$

Completing the square requires adding (b/2)^2 = 9 to both sides, giving (x-3)^2 = 27 + 9.

The box plots below summarize the ages of athletes on the swim team and the track team. Based on the box plots, which statement must be true?

• (A) The IQR of both teams is the same. ✓
• (B) There are more athletes on the swim team than on the track team.
• (C) The median age of the swim team is less than the median age of the track team.
• (D) The range of ages of the swim team is smaller than the range of ages of the track team.

Both box plots show an IQR (distance from Q1 to Q3) of about 3 years, so the IQR of both teams is the same.

The graph of f(x) is shown below. The domain of this function is

• (A) $[0,12]$
• (B) $[15,45]$
• (C) $0 < x \le 12$ ✓
• (D) $15 < x \le 45$

The graph starts with an open circle at x = 0 (excluded) and ends with a closed point at x = 12 (included), so the domain is 0 < x <= 12.

The sum of 3 and $\sqrt{5}$ is

• (A) rational, since the sum can be expressed as an integer
• (B) rational, since the sum can be expressed as a nonterminating decimal
• (C) irrational, since the sum can be expressed as a terminating decimal
• (D) irrational, since the sum cannot be expressed as a terminating or repeating decimal ✓

Root 5 is irrational, and a rational number plus an irrational number is always irrational; irrational numbers cannot be written as terminating or repeating decimals.

Which expression is equivalent to $a^8 - b^6$?

• (A) $(a^4-b^3)^2$
• (B) $(a^6-b^4)^2$
• (C) $(a^4)^2-(b^3)^2$ ✓
• (D) $(a^6)^2-(b^4)^2$

a^8 = (a^4)^2 and b^6 = (b^3)^2, so a^8 - b^6 = (a^4)^2 - (b^3)^2.

The sum of $2\sqrt{27}$ and $4\sqrt{12}$ is

• (A) $14\sqrt{3}$ ✓
• (B) $34\sqrt{3}$
• (C) $6\sqrt{39}$
• (D) $8\sqrt{39}$

Simplify each radical to like terms of root 3, then add the coefficients.

The sum of Tim's age and Jack's age is 44. Tim's age is 4 less than 7 times Jack's age, x. An equation that could be used to model this scenario is

• (A) $(7x - 4) + x = 44$ ✓
• (B) $(4 - 7x) + x = 44$
• (C) $7x - 4 = 44$
• (D) $4 - 7x = 44$

Tim's age is (7x - 4) and Jack's age is x; their sum equals 44.

Given the function $g(x) = \frac{2^{x+3}}{x^2-2}$, what is the value of $g(-2)$?

• (A) $1$ ✓
• (B) $\frac{1}{3}$
• (C) $-1$
• (D) $-\frac{1}{3}$

Substitute x = -2 into the function and simplify.

Four graphs are shown below, labeled A, B, C, and D. Which of the graphs represent(s) a function?

• (A) A, only
• (B) A and B, only ✓
• (C) A, B, and C, only
• (D) A, B, C, and D

Graph A (a single curve) and graph B (a single horizontal line) each pass the vertical line test; graph C (a single vertical line) and graph D (a circle) do not.

The formula to calculate kinetic energy is $K = \frac{1}{2}mv^2$, where K is kinetic energy, m is mass, and v is velocity. When m is written in terms of K and v, the equation is

• (A) $m = \frac{2K}{v^2}$ ✓
• (B) $m = 2Kv^2$
• (C) $m = \sqrt{2Kv^2}$
• (D) $m = \frac{K}{2v^2}$

Multiply both sides by 2 and divide by v^2 to isolate m.

The solution to the equation $\frac{2(3x-1)}{3} = x + 2$ is

• (A) $\frac{1}{3}$
• (B) $\frac{2}{3}$
• (C) $\frac{4}{3}$
• (D) $\frac{8}{3}$ ✓

Clear the fraction by multiplying both sides by 3, then solve the resulting linear equation.

Which equation represents the sequence 12, 6, 3, $\frac{3}{2}$, ..., where $a_1 = 12$?

• (A) $a_n = 12\cdot\left(\frac{1}{2}\right)^{n-1}$ ✓
• (B) $a_n = 12\cdot\left(\frac{1}{2}\right)^{n}$
• (C) $a_n = 12\cdot(2)^{n-1}$
• (D) $a_n = 12\cdot(2)^{n}$

Each term is half the previous term, so the common ratio is 1/2 and a1 = 12.

The axis of symmetry is x = 2 for which quadratic function?

• (A) A graph of f(x), an upward-opening parabola with vertex near x = 1
• (B) $j(x) = 2x^2 + 8x$
• (C) A table of values for g(x): x = -2, -1, 0, 1, 2 with g(x) = 6, 3, 2, 3, 6
• (D) $h(x) = x^2 - 4x - 5$ ✓

For h(x) = x^2 - 4x - 5, the axis of symmetry is x = -b/2a = 2. The graph (choice 1) appears to have vertex near x = 1; j(x) has axis x = -2; the table (choice 3) is symmetric about x = 0.

Each day, a freight train passes by Anna's house. This freight train travels at 49 miles per hour. Each railroad car is 56 feet long. Which expression represents the number of railroad cars that pass by Anna's house per minute?

• (A) $\frac{49\text{ mi}}{1\text{ hr}}\cdot\frac{1\text{ mi}}{5280\text{ ft}}\cdot\frac{1\text{ hr}}{60\text{ min}}\cdot\frac{1\text{ car}}{56\text{ ft}}$
• (B) $\frac{49\text{ mi}}{1\text{ hr}}\cdot\frac{1\text{ mi}}{5280\text{ ft}}\cdot\frac{60\text{ min}}{1\text{ hr}}\cdot\frac{1\text{ car}}{56\text{ ft}}$
• (C) $\frac{49\text{ mi}}{1\text{ hr}}\cdot\frac{5280\text{ ft}}{1\text{ mi}}\cdot\frac{1\text{ hr}}{60\text{ min}}\cdot\frac{1\text{ car}}{56\text{ ft}}$ ✓
• (D) $\frac{49\text{ mi}}{1\text{ hr}}\cdot\frac{5280\text{ ft}}{1\text{ mi}}\cdot\frac{60\text{ min}}{1\text{ hr}}\cdot\frac{1\text{ car}}{56\text{ ft}}$

Units must cancel correctly: miles cancel with miles (5280 ft/mi, not inverted), and hours cancel with hours (1 hr/60 min, not inverted), leaving cars per minute; only choice (3) sets up all conversion factors so the units cancel properly.

A survey was taken to determine whether students preferred to watch videos or listen to music. Of the 100 students surveyed, 44 were seniors. Of the 65 students who preferred to watch videos, 42 were juniors. Use this information to complete a frequency table with rows Watch Videos, Listen to Music, and Total, and columns Juniors, Seniors, and Total.

Use the total of 100 students and 44 seniors to find 56 juniors, then subtract the given cell values from the row and column totals to fill in the rest of the two-way frequency table.

Solve the inequality for y: $5(2-y) > -11y-8$

Distribute the 5, collect the y-terms on one side, and divide by a positive coefficient (6) to isolate y (no need to flip the inequality since dividing by a positive number).

Express $(5x-3)(-2x+7)$ as a trinomial in standard form.

Use the distributive property (FOIL) to multiply the two binomials, then combine the like x-terms.

The first and fourth terms in an arithmetic sequence are given below. $-20, \_\_\_\_, \_\_\_\_, -2$ Determine the eighth term.

Find the common difference using the first and fourth terms (three intervals apart), then use it to extend the sequence to the eighth term.

Write an equation in slope-intercept form for the line that passes through $(-2,5)$ and has a slope of $-3$.

Use point-slope form with the given point and slope, then solve for y to get slope-intercept form.

Factor the expression $x^3 - 36x$ completely.

Factor out the GCF x, then factor the resulting difference of squares x^2 - 36.

Graph $f(x) = -3x$ and $g(x) = x^2 + 2$ on the set of axes below. State the values of x that satisfy the equation $f(x) = g(x)$.

Set the linear and quadratic expressions equal, solve the resulting quadratic by factoring; the graphs of the line and parabola intersect at these two x-values (at points (-1,3) and (-2,6)).

Using the quadratic formula, solve $6x^2 + 2x - 1 = 0$. Express the answer in simplest radical form.

Substitute a, b, c into the quadratic formula, simplify the radical (root 28 = 2 root 7), then reduce the resulting fraction to simplest radical form.

The table below shows the price of a new cell phone and the length of time, in months, since its release. Time Since Release, in Months (x): 0, 3, 6, 9, 12 Price, in Dollars (y): 1200, 1150, 1100, 1000, 920 State the linear regression equation for this set of data. Round all values to the nearest hundredth. State the correlation coefficient for this data set, to the nearest hundredth. State what the correlation coefficient indicates about the linear fit of the data.

Using linear regression (by calculator or by-hand least squares) on the five data points gives slope of about -23.67 and y-intercept of about 1216.00. The correlation coefficient of about -0.99 shows a strong negative linear correlation, meaning the linear model is an excellent fit (price decreases steadily and predictably over time).

Solve the following system of equations algebraically for all values of x and y. $y = x^2 + 9x + 4$ $y - 2x = -6$

Solve the linear equation for y, substitute into the quadratic equation, solve the resulting quadratic by factoring, then back-substitute each x-value to find the corresponding y-value.

Sarah earns $6 per hour babysitting and $12 per hour tutoring. Her goal is to earn at least $120 per week. Sarah is allowed to work a maximum of 14 hours per week doing both jobs. If x represents the number of hours Sarah babysits and y represents the number of hours she tutors, write a system of inequalities that could model this situation. On the set of axes below, graph the system of inequalities that you wrote. State a combination of hours babysitting and tutoring that would satisfy this situation. Justify your answer.

Translate the earning goal and the hour limit into two inequalities and graph their overlapping (feasible) region on the given axes. Any point in that shaded region works; for example, 0 hours babysitting and 10 hours tutoring earns exactly $120 while using only 10 of the allowed 14 hours, satisfying both constraints.