NY Regents Algebra I — January 2026 Exam: Full Answers & Walkthrough of All 35 Questions

Answer: (1) x = 3 and (3, −4)

Rules: the axis of symmetry of a parabola is a vertical line, so its equation has the form x = h — never y = …. That immediately eliminates (2) and (4).

The vertex lies on the axis of symmetry, so its x-coordinate matches the axis. Reading the graph, the lowest point of the parabola sits at (3, −4), so the axis is x = 3.

Answer: (1) an irrational number

Simplify: √25 · √2 = 5 · √2 = 5√2.

Rule: a nonzero rational (5) times an irrational (√2) is always irrational. Since 5√2 ≈ 7.07… has a non-terminating, non-repeating decimal, it cannot be written as a ratio of integers.

Answer: (3) 3

Fastest method: test the choices. Plug each into both functions.

Algebraic check: solving |4x + 2| = 3x + 5 gives two cases:

Only x = 3 appears in the answer set.

Answer: (1) (x + 4)(x − 30)

Find two numbers whose product is −120 and whose sum is −26.

Verify by FOILing: (x + 4)(x − 30) = x² − 30x + 4x − 120 = x² − 26x − 120. ✓

Answer: (3) 3 + 4√5

Combine like radical terms only:

The rational term (3) and the radical term (4√5) are unlike — you cannot combine them into a single number, so the answer stays in two parts.

Answer: (4) 4x² + 2x³ − 6

Rewrite each choice in standard form (highest degree first) and check three properties: degree, leading coefficient, and constant term.

Only (4) passes all three checks. Watch out for choice (3) — it’s a classic trap where the exponent and leading coefficient are right, but the constant is +4 instead of −6.

Answer: (1)

Vertical Line Test: a graph represents a function if and only if every vertical line crosses it at most once. Any graph where a vertical line hits two or more points is not a function.

In this problem, choice (1) — a set of discrete points where no two share the same x-coordinate — passes the test. The other three choices contain curves (circles, sideways parabolas, or vertical segments) where a single x-value maps to two y-values, breaking the definition of a function.

Answer: (3) $500

Original price means t = 0 years after purchase.

Any nonzero number raised to the 0 power equals 1, so the growth factor drops out and we’re left with the leading coefficient a = 500 — that’s always the initial value in an exponential model y = a · b^t.

Answer: (1) h = (S − 2πr²) / (2πr)

Isolate h step by step:

Choice (3) has the numerator reversed — that would give a negative height for realistic cylinders.

Answer: (3) 2x + 5(3x − 10) = 1

Isolate y in the first equation:

Substitute that expression for y into the second equation:

Choice (4) drops the correct sign after solving for y and would represent solving for the wrong intersection.

Answer: (3) — open circle at ½, shading to the right.

Solve algebraically:

Since the inequality is strict (>), the endpoint ½ is not included — use an open circle. The arrow points to the right (all values larger than ½).

Answer: (1) addition property of equality

What changed? The −8 on the left disappeared, and the right side went from 7x + 10 to 7x + 18 — that means +8 was added to both sides.

Whenever you add the same quantity to both sides of an equation, you are invoking the addition property of equality — the rule that keeps equations balanced. Distributive would have expanded 3(2x + 5), commutative would swap the order of terms, and multiplication property would multiply both sides.

Answer: (2)

Test for exponential: consecutive outputs share a constant ratio. Check column (2):

The common ratio is 0.9, which is between 0 and 1 — that means exponential decay (values shrink by 10% each step).

Column (1) is linear (constant difference of −3), column (3) is linear with different rates — not exponential, and column (4) is neither linear nor exponential (it’s roughly quadratic).

Answer: (2) 7

Substitute x = 3:

Common trap: forgetting the square root and computing 3 + 1 + 5 = 9 (choice 1). The radical only applies to what’s under it.

Answer: (2) g(x) = (x + 8)² − 2

Horizontal shift rule: to shift a graph left by 3, replace x with (x + 3) — the sign is opposite of what most students expect.

A left shift bumps x-values down (the vertex moves from x = −5 to x = −8). Choice (3) confuses the constant term with a horizontal shift.

Answer: (4) 0, 6, and −6

Factor completely: x² − 36 is a difference of squares.

Zero-product property: set each factor equal to 0:

Choice (3) forgets that x = 0 is also a factor.

Answer: (2) −4 or 3

Substitute y = 6 into the equation:

Both values are valid because a parabola can pass through the same y-height at two different x-values (once on each side of the vertex).

Answer: (4) 0.145

Joint relative frequency = (the specific cell) ÷ (grand total). The cell for “pizza AND water” is 58; the grand total is 400.

Choice (1) uses just the pizza row total (58/182 ≈ 0.32); (2) is 140/400 (just the water column); (3) is 182/400 (just pizza). Always identify whether the question wants a joint frequency (uses grand total) or a conditional frequency (uses a row or column total).

Answer: (1) feet per second

Track each unit as it cancels:

A quick sanity check: the original was distance per time, so the final rate must also be distance per time. That eliminates choices (3) and (4).

Answer: (4) −8x⁶

Rule: (ab)ⁿ = aⁿbⁿ and (a^m)ⁿ = a^mn. Distribute the exponent 3 to every factor inside the parentheses, including the −2:

Choices (1) and (2) forget to cube the −2; choices (1) and (3) add exponents (2 + 3) instead of multiplying (2 · 3).

Answer: (4) −51.8

Average rate of change = (change in output) ÷ (change in input). Only the endpoints of the interval matter.

The sign matters — the substance is decaying, so the rate of change is negative. Choices (1) and (2) drop the sign.

Answer: (3) −x² + 5x − 9

Rewrite carefully: “A is subtracted from B” means B − A.

The most common error is forgetting to flip all three signs inside the second polynomial when distributing the subtraction.

Answer: (3) (x − 3)² = 33

Complete the square: take half of the middle coefficient, square it, and add to both sides.

The perfect-square trinomial x² − 6x + 9 factors as (x − 3)² — not (x − 6)². The half-coefficient becomes the constant inside the parentheses.

Answer: (4) 54

Geometric sequence formula: a_n = a₁ · rⁿ⁻¹ with a₁ = −2, r = −3, n = 4.

Alternate check by listing terms: −2, 6, −18, 54. Multiply by −3 each step. The signs alternate because the ratio is negative.

Answer: x = 1/4

Always show the distribution step; a bare final answer without work gets only 1 of the 2 credits.

Answer: exponential growth curve passing through (−1, 1.5), (0, 3), (1, 6), and (2, 12).

Strategy: plug in the four integer inputs plus the left endpoint to get a clean set of plotting points.

The graph is a smooth, always-increasing curve. Because the base b = 2 > 1, values double each time x increases by 1 — a classic exponential growth pattern.

Answer: −12x³ − 8x² + 13x − 3

Distribute each term of the binomial across the trinomial:

Standard form means highest-degree term first. The middle-term combinations are where students lose credit — keep them stacked in columns like above.

Answer: Five-number summary → min = 70, Q1 = 83, median = 88, Q3 = 94, max = 98.

Step 1 — sort:

Step 2 — median (middle value of 13 numbers is the 7th one): 88.

Step 3 — Q1 (median of the lower half 70, 72, 83, 83, 85, 87): the middle two are 83 and 83, so Q1 = 83.

Step 4 — Q3 (median of upper half 90, 93, 94, 94, 95, 98): the middle two are 94 and 94, so Q3 = 94.

Box plot: draw a box from 83 to 94 with a line inside at 88, then whiskers extending to 70 (minimum) and 98 (maximum) on the number line.

Answer: y = (2/3)x − 1

Start with point-slope form and convert to y = mx + b:

Check by plugging in (6, 3): y = (2/3)(6) − 1 = 4 − 1 = 3. ✓

Answer: 5 ride tickets (maximum).

Set up an inequality using n = number of tickets:

Since Abby cannot buy a fractional ticket, we round down: the maximum is 5 tickets. Rounding up to 6 would put her over $20 — a common trap.

Answer: Reaches maximum height at t = 2.5 s. Maximum height = 100 ft.

Vertex (max height) — use t = −b / (2a) with a = −16, b = 80:

Compute the height at t = 2.5:

Zeros for the sketch: the rocket leaves the ground at t = 0 and returns to h = 0 when −16t(t − 5) = 0, giving t = 0 or t = 5. Parabola opens downward with vertex (2.5, 100) and roots (0, 0) and (5, 0).

Answer: x = (2 ± √10) / 2 (equivalently, x = 1 ± ½√10).

Identify: a = 2, b = −4, c = −3.

Apply the quadratic formula:

Simplify the radical: √40 = √(4 · 10) = 2√10.

Common error: losing a sign on −4ac. Since both a and c multiply to a negative product, −4ac becomes positive 24, not negative.

Answer: y = −56.97x + 2352.22; r ≈ −0.98; the data have a strong negative linear relationship.

On a TI-84:

1. Press STAT → 1: Edit. Enter ages (16, 17, 18, 18, 21, 22, 30) into L1 and costs (1452, 1332, 1284, 1320, 1200, 1188, 600) into L2.
2. Turn on diagnostics: 2nd → 0 (CATALOG) → scroll to DiagnosticOn → ENTER twice.
3. Press STAT → CALC → 4: LinReg(ax+b) with Xlist: L1, Ylist: L2.

Calculator output: a ≈ −56.9688, b ≈ 2352.2245, r ≈ −0.9821. Rounded to hundredths: y = −56.97x + 2352.22 and r ≈ −0.98.

Interpretation: since r ≈ −0.98 is very close to −1, there is a strong negative linear correlation — as driver age increases, insurance cost tends to decrease, and a linear model fits the data very well.

Answer: No, (0, 3) is NOT in the solution set. It satisfies the first inequality but fails the second.

Rewrite each in slope-intercept form:

Test the point (0, 3):

The point (0, 3) lies exactly on the dashed line for the second inequality. Because the inequality is strict (>, not ≥), points on the boundary are excluded. So (0, 3) fails and is NOT in the solution set S.

Answer: System: 30x + 10y = 3700 and 15x + 20y = 3575. Jacob is NOT correct. Actual prices: running shoes = $85, basketball shoes = $115.

Part A — Write the system. Each purchase becomes one linear equation:

Part B — Check Jacob’s claim (x = 80, y = 130):

Jacob’s values satisfy the January purchase but fail the March purchase — they overshoot by $225. So Jacob is incorrect. A solution must satisfy both equations, not just one.

Part C — Solve algebraically (elimination is fastest). Multiply the second equation by 2:

Interpretation: a pair of running shoes costs $85 and a pair of basketball shoes costs $115. Verify both equations: