January 2025 Algebra I Regents — Answer Key & Worked Solutions

When factored, the expression $x^3 - 36x$ is equivalent to

• (A) $(x+6)(x-6)$
• (B) $(x+18)(x-18)$
• (C) $x(x+6)(x-6)$ ✓
• (D) $x(x+18)(x-18)$

First factor out the common factor x, then factor the resulting difference of squares x^2-36.

Which equation represents the line that passes through the points $(-1,8)$ and $(4,-2)$?

• (A) $y = -2x + 6$ ✓
• (B) $y = -2x + 10$
• (C) $y = -0.5x + 7.5$
• (D) $y = -0.5x + 8.5$

Compute the slope between the two given points, then use point-slope form with one of the points to find the equation.

A geometric sequence is shown below. $\frac{1}{2}, 2, 8, 32, \ldots$ What is the common ratio?

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

Divide any term by the term before it to find the common ratio of a geometric sequence.

What is the constant term of the polynomial $2x^3 - x + 5 + 4x^2$?

• (A) $5$ ✓
• (B) $2$
• (C) $3$
• (D) $4$

The constant term is the term with no variable factor, which is 5.

A landscaping company charges a set fee for a spring cleanup, plus an hourly labor rate. The total cost is modeled by the function $C(x) = 55x + 80$. In this function, what does the 55 represent?

• (A) the set fee for the cleanup
• (B) the hourly labor rate for a cleanup ✓
• (C) the profit earned by the company for one cleanup
• (D) the number of hours of labor required for one cleanup

In a linear cost function C(x)=mx+b, the coefficient m multiplying the variable (hours) represents the rate per unit, here the hourly labor rate.

Which expression is equivalent to $(5x^2 - 2x + 4) - (3x^2 + 3x - 1)$?

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

Distribute the subtraction across the second polynomial, then combine like terms.

A system of inequalities is graphed on the set of axes below. Which point is a solution to this system?

• (A) $(1,1)$
• (B) $(2,-2)$
• (C) $(1,8)$
• (D) $(4,2)$ ✓

The solution to a system of inequalities is the region where both shaded areas overlap; only (4,2) falls in that double-shaded region.

In an arithmetic sequence, the first term is 25 and the third term is 15. What is the tenth term in this sequence?

• (A) $-20$ ✓
• (B) $-25$
• (C) $70$
• (D) $75$

Find the common difference using the first and third terms, then use it to find the tenth term.

When the formula $p = 2l + 2w$ is solved for $w$, the result is

• (A) $w = \dfrac{2l+p}{2}$
• (B) $w = \dfrac{p-2l}{2}$ ✓
• (C) $w = \dfrac{p}{2} + l$
• (D) $w = l - \dfrac{p}{2}$

Isolate the term with w by subtracting 2l from both sides, then divide by 2.

Market Street Pizza kept a record of pizza sales for the month of February. The results are shown in the table below. | Type | Plain | Veggie | Meat Only | The Works | |---|---|---|---|---| | Thin Crust | 300 | 80 | 120 | 100 | | Deep-dish | 200 | 25 | 105 | 70 | Of all the pizzas sold in February, what percent were plain, deep-dish pizzas?

• (A) $20\%$ ✓
• (B) $30\%$
• (C) $40\%$
• (D) $50\%$

Divide the number of plain deep-dish pizzas (200) by the grand total of all pizzas sold (1000) to get the percentage.

When solving $-2(3x-5) = \frac{9}{2}x - 2$ for $x$, the solution is

• (A) $\frac{8}{7}$ ✓
• (B) $\frac{10}{11}$
• (C) $-\frac{16}{21}$
• (D) $-\frac{16}{3}$

Distribute the -2, collect x-terms on one side and constants on the other, then solve for x.

The expression $x^{2a+b}$ is equivalent to

• (A) $x^{2a} + x^b$
• (B) $x^a + x^{a+b}$
• (C) $x^a \cdot x^{a+b}$ ✓
• (D) $x^{a+b} \cdot x^{a+b}$

Using the product of powers rule, adding the exponents a and (a+b) gives 2a+b, matching the original expression.

The inputs and outputs of a function are shown in the table below. | x | f(x) | |---|---| | 0 | 0.0625 | | 1 | 0.125 | | 2 | 0.25 | | 3 | 0.5 | | 4 | 1 | | 5 | 2 | This function can best be described as

• (A) linear
• (B) quadratic
• (C) exponential ✓
• (D) absolute value

Since each output is a constant multiple (2) of the previous output rather than a constant difference, the function is exponential.

Stephanie is solving the equation $x^2 - 12 = 7x - 8$. Her first step is shown below. Given: $x^2 - 12 = 7x - 8$ Step 1: $x^2 - 4 = 7x$ Which property justifies her first step?

• (A) associative property
• (B) commutative property
• (C) distributive property
• (D) addition property of equality ✓

Adding the same quantity (8) to both sides of the equation to eliminate the -8 term is justified by the addition property of equality.

What is the sum of $8\sqrt{3}$ and $\sqrt{3}$?

• (A) $8\sqrt{6}$
• (B) $9\sqrt{6}$
• (C) $7\sqrt{3}$
• (D) $9\sqrt{3}$ ✓

Like radicals combine the way like terms do, by adding their coefficients while keeping the radical the same.

The dot plots below represent test scores for 20 students on a math test. The mode for this math test is 80 and the median is 85. Which dot plot correctly represents this data?

• (A) I ✓
• (B) II
• (C) III
• (D) IV

The correct dot plot must show 80 as the tallest stack (the mode) while the middle two values of the ordered 20 data points average to 85 (the median).

A function is graphed on the set of axes below. The domain of this function is

• (A) $\{x \mid x > -2\}$ ✓
• (B) $\{x \mid x \geq -2\}$
• (C) $\{x \mid x > -4\}$
• (D) $\{x \mid x \geq -4\}$

An open circle on the graph's left endpoint means that x-value is excluded from the domain, so the domain is all x greater than that value.

Which ordered pair is a solution to the equation $y - 1 = 2\left(x + \frac{1}{4}\right)$?

• (A) $(0.75, 0)$
• (B) $(1.25, 4)$ ✓
• (C) $(2.5, -6.5)$
• (D) $(4, -9.5)$

Simplify the equation to slope-intercept form y=2x+1.5, then substitute each ordered pair to find the one that satisfies the equation.

Elena's fastest time for the 50-meter dash is 7 seconds. She wants to know how fast this is in inches per minute. Which expression can Elena use for a correct conversion?

• (A) $\dfrac{7\text{ sec}}{50\text{ meters}} \cdot \dfrac{60\text{ sec}}{1\text{ min}} \cdot \dfrac{1\text{ meter}}{39.37\text{ in}}$
• (B) $\dfrac{7\text{ sec}}{50\text{ meters}} \cdot \dfrac{1\text{ min}}{60\text{ sec}} \cdot \dfrac{39.37\text{ in}}{1\text{ meter}}$
• (C) $\dfrac{50\text{ meters}}{7\text{ sec}} \cdot \dfrac{60\text{ sec}}{1\text{ min}} \cdot \dfrac{1\text{ meter}}{39.37\text{ in}}$
• (D) $\dfrac{50\text{ meters}}{7\text{ sec}} \cdot \dfrac{60\text{ sec}}{1\text{ min}} \cdot \dfrac{39.37\text{ in}}{1\text{ meter}}$ ✓

To convert meters per second to inches per minute, the units must cancel properly: seconds must cancel with seconds and meters with meters, leaving inches over minutes.

The table below shows the highest temperatures recorded in August for several years in one town. | Year | Temperature (°F) | |---|---| | 1990 | 86 | | 1991 | 78 | | 1992 | 84 | | 1993 | 95 | | 1994 | 81 | | 1995 | 77 | | 1996 | 88 | | 1997 | 93 | The interquartile range of these data is

• (A) $7$
• (B) $10$
• (C) $11$ ✓
• (D) $18$

Order the data, split into lower and upper halves to find Q1 and Q3, then subtract Q1 from Q3 to get the interquartile range.

The function $f(x) = x^2$ is multiplied by $k$, where $k < -1$. Which graph could represent $g(x) = kf(x)$?

• (A) A parabola opening upward with vertex at the origin, arms spreading outward at roughly the same rate as $y = x^2$ (a normal-width upward U-shape).
• (B) A parabola opening downward with vertex at the origin, arms spreading outward wider/flatter than $y = x^2$.
• (C) A parabola opening upward with vertex at the origin, arms rising steeply/narrower than $y = x^2$.
• (D) A parabola opening downward with vertex at the origin, arms falling steeply/narrower than $y = x^2$ (a narrow, inverted U-shape). ✓

Multiplying x^2 by a constant less than -1 reflects the parabola over the x-axis (so it opens downward) and stretches it vertically, making it narrower than the parent function, matching choice 4.

Which graph is the solution to the inequality $6.4 - 4x \ge -2.8$?

• (A) A number line from 2.1 to 2.5 with an open circle at 2.3 and an arrow/shading extending to the right (toward larger values).
• (B) A number line from 2.1 to 2.5 with a closed (filled) circle at 2.3 and an arrow/shading extending to the right (toward larger values).
• (C) A number line from 2.1 to 2.5 with an open circle at 2.3 and an arrow/shading extending to the left (toward smaller values).
• (D) A number line from 2.1 to 2.5 with a closed (filled) circle at 2.3 and an arrow/shading extending to the left (toward smaller values). ✓

Dividing both sides by -4 reverses the inequality symbol, giving x ≤ 2.3, which is graphed with a closed circle at 2.3 and shading to the left.

The number of fish in a pond is eight more than the number of frogs. The total number of fish and frogs in the pond is at least 20. If $x$ represents the number of frogs, which inequality can be used to represent this situation?

• (A) $x + 8x \ge 20$
• (B) $2x + 8 \ge 20$ ✓
• (C) $x + 8x \le 20$
• (D) $2x + 8 \le 20$

Fish equal x + 8, and the sum of fish and frogs must be at least 20, giving 2x + 8 ≥ 20.

Which graph below represents a function that is always *decreasing* over the entire interval $-3 < x < 3$?

• (A) A graph made of two line segments: one rising steeply from lower left, reaching a peak, then a shorter segment descending slightly to the right (increases then decreases).
• (B) A curve that rises from left to right the entire time, starting near the bottom left and curving upward and to the right (always increasing).
• (C) A downward-opening parabola-like curve that rises to a peak in the middle and falls on both sides (increases then decreases).
• (D) A graph consisting of a line segment falling continuously from upper left, with an open circle partway down followed by a filled dot slightly lower and the line continuing to fall steeply to the lower right (falls over the entire interval). ✓

Only the graph that falls from left to right over the whole interval, without turning upward anywhere, qualifies as always decreasing on -3 < x < 3, which is graph 4.

The graph below models Sally's drive to the store. State an interval when Sally is traveling at a constant speed. Explain your reasoning.

Constant speed corresponds to a flat (horizontal) portion of a speed-time graph; here that occurs from 5 to 9 minutes, where speed stays at 35 mph.

Graph the function $f(x) = x^2 + 4x + 3$. State the equation of the axis of symmetry of $f(x)$.

The axis of symmetry of a parabola in the form ax^2+bx+c is x = -b/2a, which gives x = -2 for this function; this is confirmed as the midpoint of the roots -1 and -3.

The function $f(x)$ is shown in the table below. | x | 0 | 3 | 2 | 6 | 1 | 5 | 4 | m | |---|---|---|---|---|---|---|---|---| | f(x) | 6 | 2 | 7 | 5 | 8 | 4 | 3 | 9 | State an appropriate value for $m$ in the table, so that $f(x)$ remains a function. Explain your reasoning.

A function cannot assign two different outputs to the same input, so m must differ from every x-value already listed (0 through 6); choosing m = 7 keeps the relation a function.

Solve $x^2 + 8x = 33$ for $x$ by completing the square.

Adding 16 (half of 8, squared) to both sides completes the square, allowing the equation to be solved by taking the square root of both sides.

If $f(x) = \dfrac{-3x-5}{2}$, algebraically determine the value of $x$ when $f(x) = -22$.

Setting f(x) equal to -22, multiplying both sides by 2, and solving the resulting linear equation gives x = 13.

Rationalize the denominator of the fraction below. Express the solution in simplest form. $\dfrac{4}{\sqrt{2}}$

Multiplying numerator and denominator by √2 removes the radical from the denominator, and simplifying gives 2√2.

Alex had $1.70 in nickels and dimes on his desk. There were 25 coins in all. Write a system of equations that could be used to determine both the number of nickels, $n$, and the number of dimes, $d$, that Alex had. Use your system of equations to algebraically determine both the number of nickels and the number of dimes that he had.

The system relates total coins (n + d = 25) and total value (0.05n + 0.10d = 1.70); substitution yields d = 9 dimes and n = 16 nickels.

The table below shows the average heart rate, $x$, and Calories burned, $y$, for seven men on an Olympic rowing team during a one-hour workout class. | Average Heart Rate (x) | 135 | 147 | 150 | 144 | 146 | 153 | 143 | |---|---|---|---|---|---|---|---| | Calories Burned (y) | 725 | 812 | 866 | 761 | 825 | 863 | 737 | Write the linear regression equation that models these data, rounding all values to the nearest tenth. State the correlation coefficient, rounded to the nearest tenth. State what the correlation coefficient suggests about the linear fit of these data.

Running linear regression on the heart rate/calories data gives the equation y = 9.1x - 527.6 with r ≈ 0.9, showing a strong, positive linear relationship, so the linear model is a good fit for the data.

Using the quadratic formula, solve $x^2 + 4x - 3 = 0$. Express your solution in simplest radical form.

Substituting a=1, b=4, c=-3 into the quadratic formula and simplifying the radical √28 = 2√7 gives x = -2 ± √7.

Solve the following system of equations algebraically for all values of $x$ and $y$. $y = x^2 - 7x + 12$ $y = 2x - 6$

Setting the two expressions for y equal produces a quadratic equation that factors to give x = 3 and x = 6; substituting back into y = 2x - 6 gives the solutions (3, 0) and (6, 6).

Anna plans to spend $30 on balloons and party hats for her daughter's birthday party. Including tax, balloons cost $2 each and party hats cost $1.50 each. The number of party hats Anna needs is twice as many as the number of balloons. If $x$ represents the number of balloons and $y$ represents the number of party hats, write a system of equations that can be used to represent this situation. Graph your system of equations on a set of axes where the x-axis shows the number of balloons from 0 to 20 and the y-axis shows the number of party hats from 0 to 20. State the coordinates of the point of intersection of your lines. Explain what each coordinate means in the context of the problem.

Substituting y = 2x into the cost equation 2x + 1.5y = 30 gives x = 6 balloons and y = 12 party hats; the intersection point (6, 12) represents the unique combination of balloons and hats that costs exactly $30 while keeping hats at twice the number of balloons.