Algebra I Regents Exam January 2026 Paper with Step-by-Step Solutions

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Question No. 01 | Algebra 1 Regents — Parabola Problem

Question No. 01

A parabola is graphed on the set of axes. The parabola opens upward (U-shape) with its lowest point (vertex) visibly located in the fourth quadrant.

What are the equation of the axis of symmetry and the coordinates of the vertex of this parabola?

(1) x = 3 and (3, −4)

(2) y = 3 and (3, −4)

(3) x = −4 and (−4, 3)

(4) y = −4 and (−4, 3)

✅ Final Answer

Choice (1): x = 3 and Vertex = (3, −4)

📌 Regents Tip:
For any parabola that opens up or down:

The axis of symmetry is always a vertical line → equation form is x = h
The vertex is the point (h, k) — the tip of the parabola
In the standard form y = a(x − h)² + k, the vertex is directly readable as (h, k)
Quick elimination tip: If the parabola opens up/down and the answer says "y = ..." for the axis of symmetry — eliminate it immediately!

Question No. 02 | Algebra 1 Regents — Number Types & Products

Question No. 02

The product of √25 and √2 will result in:

(1) an irrational number

(2) a rational number

(3) a natural number

(4) an integer

Step-by-Step Solution

Step 1 — Simplify √25.
Ask: what number times itself equals 25?
√25 = 5
✓ This is a perfect square — the result is a whole number.
Step 2 — Evaluate √2.
√2 is not a perfect square. Its decimal goes on forever without repeating:
√2 = 1.41421356…
✓ This is an irrational number (non-terminating, non-repeating decimal).
Step 3 — Multiply the two values.
√25 × √2 = 5 × √2 = 5√2
We get 5√2. This is 5 multiplied by an irrational number.
Step 4 — Classify the result.
Key rule: A non-zero rational number × an irrational number = always irrational.
Since 5 is rational and √2 is irrational:
5√2 ≈ 7.0710678…
The decimal never ends and never repeats → irrational. ✓

Step 5 — Eliminate the other choices.

Choice	Type	Why It's Wrong
(2) Rational	p/q form, terminating or repeating decimal	5√2 = 7.071… — non-terminating, non-repeating ✗
(3) Natural	Counting numbers: 1, 2, 3, …	5√2 ≈ 7.071, not a whole counting number ✗
(4) Integer	…−2, −1, 0, 1, 2, …	5√2 ≈ 7.071, not a whole integer ✗
(1) Irrational ✓	Non-terminating, non-repeating decimal	5√2 = 7.0710678… — fits perfectly ✓

✔ Quick Verification: We can also use the product rule: √25 × √2 = √(25 × 2) = √50 = 5√2 ≈ 7.071…
Since this decimal never terminates or repeats, the result is irrational. ✓

✅ Final Answer

Choice (1) — An Irrational Number
√25 × √2 = 5√2 ≈ 7.071…

📌 Regents Tip — Number Classification Rules:

Rational × Rational = always Rational (e.g., 3 × 4 = 12)
Irrational × Irrational = can be Rational or Irrational (e.g., √2 × √2 = 2 ✓ rational; √2 × √3 = √6 ✗ irrational)
Non-zero Rational × Irrational = always Irrational ← This question!
If you see √(perfect square) like √4, √9, √25, √36 — it simplifies to a whole number (rational).
If you see √(non-perfect square) like √2, √3, √7 — the result is always irrational.

Question No. 03 | Algebra 1 Regents — Absolute Value Functions

Question No. 03

When f(x) = |4x + 2| and g(x) = 3x + 5 are graphed on the same set of axes, for which value of x is f(x) = g(x)?

(1) 1

(2) 2

(3) 3

(4) 14

✅ Final Answer

Choice (3) — x = 3
Note: solving the equation gives x = −1 and x = 3, but only 3 appears in the choices shown.

📌 Regents Tip — Absolute Value Equations:

When you see |expression| = value, you usually solve two cases.
|A| = B means A = B or A = -B.
Always check both answers in the original equation.
If a multiple-choice question seems to miss a valid answer, solve carefully and compare with the listed options.

Question No. 04 | Algebra 1 Regents — Factoring Quadratics

Question No. 04

The expression x² − 26x − 120 is equivalent to

(1) (x + 4)(x − 30)

(2) (x − 4)(x + 30)

(3) (x − 20)(x + 6)

(4) (x + 20)(x − 6)

Step-by-Step Solution

Step 1 — Recognize that this is a factoring problem.
We want to factor:
x² − 26x − 120
into the form:
(x + m)(x + n)
Step 2 — Find two numbers that multiply to −120.
We need numbers whose product is:
m · n = -120

Step 3 — Find two numbers that add to −26.
We also need:
m + n = -26

Let’s test some factor pairs of 120:

Factor Pair	Product	Sum
4 and −30	−120	−26 ✓
−4 and 30	−120	26 ✗
−20 and 6	−120	−14 ✗
20 and −6	−120	14 ✗

Step 4 — Write the factored form.
Since 4 + (-30) = -26 and 4(-30) = -120, we get:
(x + 4)(x - 30)
Step 5 — Match to the correct choice.
That matches:
Choice (1)

✔ Quick Verification: Expand the answer:
(x + 4)(x - 30) = x² - 30x + 4x - 120 = x² - 26x - 120
It matches the original expression exactly. ✓

✅ Final Answer

Choice (1) — (x + 4)(x − 30)

📌 Regents Tip — Factoring x² + bx + c:

Find two numbers that multiply to c and add to b.
If the constant term is negative, the two numbers must have opposite signs.
Always expand your final answer to make sure it matches the original trinomial.
A fast check: the middle term comes from adding the outside and inside products.

Question No. 05 | Algebra 1 Regents — Simplifying Radical Expressions

Question No. 05

The expression 3 − 2√5 + 6√5 is equivalent to

(1) 7√5

(2) 7√10

(3) 3 + 4√5

(4) 3 + 4√10

Step-by-Step Solution

Step 1 — Identify like terms.
In the expression:
3 - 2√5 + 6√5
the terms -2√5 and 6√5 are like radicals because they both contain √5.
Step 2 — Combine the radical terms.
-2√5 + 6√5 = 4√5
because -2 + 6 = 4.
Step 3 — Rewrite the full expression.
Now put the constant term back:
3 - 2√5 + 6√5 = 3 + 4√5
Step 4 — Match the answer choice.
3 + 4√5 matches:
Choice (3)

Step 5 — Eliminate the wrong choices.

Choice	Why It's Wrong
(1) 7√5	This leaves out the constant 3. ✗
(2) 7√10	You do not add the numbers inside the radical like that. √5 + √5 = 2√5, not √10. ✗
(4) 3 + 4√10	The radical stays √5 because both radical terms are already like terms. ✗
(3) 3 + 4√5 ✓	This correctly combines the like radical terms. ✓

✔ Quick Verification: Group the radical terms first:
3 + (-2√5 + 6√5) = 3 + 4√5
So the simplified form is definitely 3 + 4√5. ✓

✅ Final Answer

Choice (3) — 3 + 4√5

📌 Regents Tip — Combining Radicals:

You can only combine radicals that have the same radicand (the same number under the radical).
√5 and √5 are like terms, so their coefficients can be added or subtracted.
2√5 + 3√5 = 5√5, just like 2x + 3x = 5x.
Do not add inside the radical unless the entire radical expression is being simplified correctly.

Question No. 06 | Algebra 1 Regents — Polynomial Vocabulary

Question No. 06

Students were asked to write a polynomial given the following conditions:

the degree of the expression is 3
the leading coefficient is 2
the constant term is −6

Which expression satisfies all three conditions?

(1) 4x − 6 + 3x²

(2) 3x² − 6x + 4

(3) 4 − 6x + 2x³

(4) 4x² + 2x³ − 6

✅ Final Answer

Choice (4) — 4x² + 2x³ − 6

📌 Regents Tip — Polynomial Vocabulary:

The degree is the greatest exponent in the polynomial.
The leading coefficient is the coefficient of the term with the highest power.
The constant term is the number by itself, with no variable attached.
Putting a polynomial in standard form makes it much easier to identify all of these features.

Question No. 07 | Algebra 1 Regents — Identifying a Function from a Graph

Question No. 07

Which graph below represents a function?

(1) Graph (1)

(2) Graph (2)

(3) Graph (3)

(4) Graph (4)

✅ Final Answer

Choice (1) — Graph (1)

📌 Regents Tip — Vertical Line Test:

A graph represents a function if every x-value has only one y-value.
Use the vertical line test: if a vertical line crosses the graph more than once, it is not a function.
Vertical segments always fail the vertical line test.
For point graphs, check whether any two points are stacked above each other at the same x-value.

Question No. 08 | Algebra 1 Regents — Exponential Functions

Question No. 08

The following function models the value of a diamond ring, in dollars, t years after it is purchased:

v(t) = 500(1.08)^t

What was the original price of the ring, in dollars?

(1) $108

(2) $460

(3) $500

(4) $540

✅ Final Answer

Choice (3) — $500

📌 Regents Tip — Initial Value in Exponential Functions:

In y = a(b)^x, the number a is the initial value.
The initial value is also found by substituting x = 0.
Remember: any nonzero number to the 0 power equals 1.
If the growth factor is greater than 1, the function represents exponential growth.

Question No. 09 | Algebra 1 Regents — Solving a Formula for a Variable

Question No. 09

The formula for the surface area of a cylinder can be expressed as

S = 2πr² + 2πrh, where r is the radius and h is the height of the cylinder. What is the height, h, expressed in terms of S, π, and r?

(1) h = (S − 2πr²) / (2πr)

(2) h = S − r

(3) h = (2πr² − S) / (2πr)

(4) h = r − S

✅ Final Answer

Choice (1) — h = (S − 2πr²) / (2πr)

📌 Regents Tip — Solving Formulas:

To solve for one variable, undo operations in the correct order.
First isolate the term containing the variable you want.
Then divide by the coefficient attached to that variable.
Be careful with subtraction order. a − b is not the same as b − a.

Question No. 10 | Algebra 1 Regents — Systems of Equations by Substitution

Question No. 10

When solving the following system of equations algebraically, Mason used the substitution method:

3x − y = 10
2x + 5y = 1

Which equation could he have used?

(1) 2(3x − 10) + 5x = 1

(2) 2(−3x + 10) + 5x = 1

(3) 2x + 5(3x − 10) = 1

(4) 2x + 5(−3x + 10) = 1

✅ Final Answer

Choice (3) — 2x + 5(3x − 10) = 1

📌 Regents Tip — Substitution Method:

First solve one equation for one variable, such as x = ... or y = ....
Then replace that variable in the other equation.
Be very careful with negative signs when isolating a variable.
If you solve for y, make sure you substitute that full expression everywhere y appears.

Question No. 11 | Algebra 1 Regents — Solving and Graphing an Inequality

Question No. 11

Which graph represents the solution to the inequality

4 + 3x > 9 − 7x?

(1) Graph (1)

(2) Graph (2)

(3) Graph (3)

(4) Graph (4)

✅ Final Answer

Choice (3) — x > 1/2

📌 Regents Tip — Graphing Inequalities:

An open circle means the endpoint is not included.
A closed circle means the endpoint is included.
x > a graphs to the right.
x < a graphs to the left.

Question No. 12 | Algebra 1 Regents — Properties of Equality

Question No. 12

When solving the equation 3(2x + 5) − 8 = 7x + 10, the first step could be

3(2x + 5) = 7x + 18

Which property justifies this step?

(1) addition property of equality

(2) commutative property of addition

(3) multiplication property of equality

(4) distributive property of multiplication over addition

Step-by-Step Solution

Step 1 — Compare the original equation and the new equation.
Original:
3(2x + 5) − 8 = 7x + 10

New first step:
3(2x + 5) = 7x + 18
Step 2 — Figure out what changed.
The −8 disappeared from the left side. To make that happen, we must add 8 to both sides.

Left side:
3(2x + 5) − 8 + 8 = 3(2x + 5)

Right side:
7x + 10 + 8 = 7x + 18
Step 3 — Identify the property used.
Since we added the same number, 8, to both sides of the equation, the property is the:
addition property of equality

Step 4 — Eliminate the wrong choices.

Choice	Why It's Wrong
(2) Commutative property of addition	This changes the order of terms, but here we are adding 8 to both sides. ✗
(3) Multiplication property of equality	Nothing was multiplied to both sides. ✗
(4) Distributive property	We did not expand 3(2x + 5) yet. ✗
(1) Addition property of equality ✓	We added 8 to both sides. ✓

✔ Quick Verification: Starting with 3(2x + 5) − 8 = 7x + 10, add 8 to both sides:
3(2x + 5) = 7x + 18
That matches the given first step exactly. ✓

✅ Final Answer

Choice (1) — Addition Property of Equality

📌 Regents Tip — Properties of Equality:

If you add the same number to both sides, that is the addition property of equality.
If you subtract the same number from both sides, that is the subtraction property of equality.
The distributive property is used only when multiplying across parentheses, like 3(x + 2) = 3x + 6.
Always ask yourself: What exact operation was done to both sides?

Question No. 13 | Algebra 1 Regents — Exponential Decay Tables

Question No. 13

Which table of values best models an exponential decay function?

(1) Table (1)

(2) Table (2)

(3) Table (3)

(4) Table (4)

✅ Final Answer

Choice (2) — Table (2)

📌 Regents Tip — Linear vs. Exponential:

Linear patterns have a constant difference.
Exponential patterns have a constant ratio.
If the ratio is between 0 and 1, the function shows decay.
If the ratio is greater than 1, the function shows growth.

Question No. 14 | Algebra 1 Regents — Evaluating a Function

Question No. 14

If f(x) = √(x + 1) + 5, then what is the value of f(3)?

(1) 9

(2) 7

(3) 3

(4) 10

✅ Final Answer

Choice (2) — 7

📌 Regents Tip — Evaluating Functions:

To find f(a), replace every x with a.
Use parentheses when substituting so signs stay correct.
Simplify inside the radical before taking the square root.
Then finish the rest of the arithmetic in order.

Question No. 15 | Algebra 1 Regents — Transformations of Functions

Question No. 15

Isabella wants to shift the graph of the function

f(x) = (x + 5)² − 2

left 3 units. Which function represents the shifted graph?

(1) g(x) = (x + 2)² − 2

(2) g(x) = (x + 8)² − 2

(3) g(x) = (x + 5)² − 5

(4) g(x) = (x + 5)² + 1

✅ Final Answer

Choice (2) — g(x) = (x + 8)² − 2

📌 Regents Tip — Function Shifts:

Inside the parentheses controls left-right movement.
Outside the parentheses controls up-down movement.
f(x + 3) shifts the graph left 3.
f(x − 3) shifts the graph right 3.

Question No. 16 | Algebra 1 Regents — Finding Zeros of a Polynomial

Question No. 16

What are the zeros of f(x) = x(x² − 36)?

(1) 0, only

(2) 6, only

(3) 6 and −6, only

(4) 0, 6, and −6

✅ Final Answer

Choice (4) — 0, 6, and −6

📌 Regents Tip — Finding Zeros:

To find zeros, set the function equal to 0.
Factor as much as possible before solving.
Use the zero product property: if ab = 0, then a = 0 or b = 0.
Watch for special factoring patterns like difference of squares.

Question No. 17 | Algebra 1 Regents — Points on a Parabola

Question No. 17

The point (x, −6) lies on the graph of a parabola whose equation is

y = −x² − x + 6. What is the value of x?

(1) −3 or 2

(2) −4 or 3

(3) 3, only

(4) −4, only

✅ Final Answer

Choice (2) — −4 or 3

📌 Regents Tip — Points on a Graph:

If a point lies on a graph, its coordinates must satisfy the equation.
Substitute the known value first, then solve for the unknown variable.
Quadratic equations can have two solutions, one solution, or no real solutions.
Always check your answers by plugging them back into the original equation.

Question No. 18 | Algebra 1 Regents — Relative Frequency from a Two-Way Table

Question No. 18

The two-way frequency table below is a summary of concession stand sales for a football game. Of the people making a purchase at the concession stand, what is the relative frequency of them buying pizza and a water?

(1) 0.58

(2) 0.35

(3) 0.455

(4) 0.145

✅ Final Answer

Choice (4) — 0.145

📌 Regents Tip — Relative Frequency:

Relative frequency is always part ÷ whole.
For a two-way table, read the question carefully to decide which cell is the part and which total is the whole.
The word “and” usually means use one specific interior cell of the table.
The phrase “of all people” usually means divide by the grand total.

Question No. 19 | Algebra 1 Regents — Unit Conversion

Question No. 19

When Theodore was driving in Canada, his speed was 104 kilometers per hour. Theodore was asked to convert his metric speed to a different rate using the following conversion:

104 km / 1 hr · 1 hr / 60 min · 1 min / 60 sec · 0.6214 mi / 1 km · 5280 ft / 1 mi

Assuming he did all the work correctly, what would the units be for Theodore’s rate?

(1) feet per second

(2) feet per minute

(3) seconds per foot

(4) minutes per foot

Step-by-Step Solution

Step 1 — Focus on the units, not the numbers.
We are asked what units remain after all the conversion factors are multiplied together. So we track only:
km / hr · hr / min · min / sec · mi / km · ft / mi

Step 2 — Cancel matching units.

Unit	What happens?
km	Cancels with km
hr	Cancels with hr
min	Cancels with min
mi	Cancels with mi

Step 3 — See what units are left.
After canceling, the only units that remain are:
ft / sec
which means feet per second.
Step 4 — Match the answer choice.
That is:
Choice (1)

✔ Quick Verification: The starting rate is a distance over time: km/hr.
The conversion changes kilometers into feet and hours into seconds, so the final rate must be feet/second. ✓

✅ Final Answer

Choice (1) — feet per second

📌 Regents Tip — Unit Conversions:

In conversion problems, write the units as fractions and cancel them carefully.
Matching units on top and bottom cancel out.
The units left at the end tell you the answer type.
If you start with a speed, your final answer should usually still be a distance per time unit.

Question No. 20 | Algebra 1 Regents — Powers of Monomials

Question No. 20

Which expression is equivalent to

(−2x²)³?

(1) −2x⁵

(2) −2x⁶

(3) −8x⁵

(4) −8x⁶

✅ Final Answer

Choice (4) — −8x⁶

📌 Regents Tip — Exponent Rules:

(ab)ⁿ = aⁿbⁿ
(x^m)ⁿ = x^mn
An odd exponent keeps a negative base negative.
Be careful not to add exponents when raising a power to another power. You multiply them.

Question No. 21 | Algebra 1 Regents — Average Rate of Change

Question No. 21

The table below shows the amount of a radioactive substance that remained for selected years. To the nearest tenth, what is the average rate of change, in grams per year, from 2000 to 2014?

(1) 39.1

(2) 51.8

(3) −39.1

(4) −51.8

✅ Final Answer

Choice (4) — −51.8

📌 Regents Tip — Average Rate of Change:

Use the slope formula: (y₂ − y₁) / (x₂ − x₁).
Read the two ordered pairs carefully from the table.
If the output decreases as the input increases, the average rate of change will be negative.
Always round only at the end unless the problem tells you otherwise.

Question No. 22 | Algebra 1 Regents — Subtracting Polynomial Expressions

Question No. 22

When 2x² − 3x + 4 is subtracted from x² + 2x − 5, the result is

(1) x² − 5x + 9

(2) x² − x + 1

(3) −x² + 5x − 9

(4) −x² − x − 1

✅ Final Answer

Choice (3) — −x² + 5x − 9

📌 Regents Tip — Subtracting Polynomials:

“A subtracted from B” means B − A, not A − B.
When subtracting a polynomial in parentheses, change every sign inside the second set of parentheses.
Then combine like terms carefully.
Write terms in standard form when possible.

Question No. 23 | Algebra 1 Regents — Equivalent Quadratic Equations

Question No. 23

Which equation has the same solution as

x² − 6x = 24?

(1) (x − 3)² = 24

(2) (x − 6)² = 24

(3) (x − 3)² = 33

(4) (x − 6)² = 60

✅ Final Answer

Choice (3) — (x − 3)² = 33

📌 Regents Tip — Completing the Square:

Take half of the x-coefficient, then square it.
Add that number to both sides to keep the equation balanced.
The goal is to turn the trinomial into a perfect square like (x − a)².
This method is especially useful when changing standard form into vertex form or solving quadratics.

Question No. 24 | Algebra 1 Regents — Geometric Sequence

Question No. 24

In a sequence, the first term is −2 and the common ratio is −3. The fourth term in this sequence is

(1) −162

(2) −11

(3) 24

(4) 54

✅ Final Answer

Choice (4) — 54

📌 Regents Tip — Geometric Sequences:

A geometric sequence multiplies by the same number each time.
That repeated multiplier is called the common ratio.
Use a_n = a₁rⁿ⁻¹ for any term.
If the ratio is negative, the signs of the terms will alternate.

Question No. 25 | Algebra 1 Regents — Solving a Linear Equation

Question No. 25

Solve the equation for x:

14x = 3(1 + 2x) − 4x

✅ Final Answer

x = 1/4

📌 Regents Tip — Solving Linear Equations:

Always distribute first if there are parentheses.
Then combine like terms on each side.
Move variable terms to one side and constants to the other.
Check your answer by plugging it back into the original equation.

Question No. 26 | Algebra 1 Regents — Graphing an Exponential Function

Question No. 26

Graph f(x) = 3(2)^x over the interval

−1 ≤ x ≤ 2

Step-by-Step Solution

Step 1 — Identify the x-values in the interval.
Since the graph is requested on −1 ≤ x ≤ 2, the most useful integer x-values are:
x = −1, 0, 1, 2
Step 2 — Evaluate the function at each x-value.
The function is:
f(x) = 3(2)^x

Step 3 — Find the ordered pairs.

x	f(x)	Ordered Pair
−1	3(2)⁻¹ = 3(1/2) = 3/2	(−1, 3/2)
0	3(2)⁰ = 3(1) = 3	(0, 3)
1	3(2)¹ = 6	(1, 6)
2	3(2)² = 12	(2, 12)

Step 4 — Plot the points.
Plot these four points on the coordinate plane:
(−1, 3/2), (0, 3), (1, 6), (2, 12)
Step 5 — Draw the curve correctly.
Connect the points with a smooth increasing exponential curve.
Since the interval is only from x = −1 to x = 2, only draw the graph on that part of the plane.

✔ Quick Verification: The outputs double each time x increases by 1:
3/2 → 3 → 6 → 12
That matches exponential growth, so the graph should rise more and more steeply from left to right. ✓

✅ Final Answer

Plot and connect the points:
(−1, 3/2), (0, 3), (1, 6), (2, 12)
with a smooth increasing exponential curve on −1 ≤ x ≤ 2

📌 Regents Tip — Graphing Exponential Functions:

Make a table of values first. This gives exact points to plot.
Remember: a⁰ = 1 and a⁻¹ = 1/a.
If the base is greater than 1, the graph shows exponential growth.
Draw a smooth curve, not straight line segments.

Question No. 27 | Algebra 1 Regents — Multiplying Polynomials

Question No. 27

Determine the product of (2x + 3) and (−6x² + 5x − 1).

Express the product in standard form.

✅ Final Answer

−12x³ − 8x² + 13x − 3

📌 Regents Tip — Multiplying Polynomials:

Distribute each term in the first polynomial to every term in the second polynomial.
Keep track of signs carefully, especially with negative terms.
After multiplying, always combine like terms.
Write the final answer in standard form, from highest power to lowest power.

Question No. 28 | Algebra 1 Regents — Constructing a Box Plot

Question No. 28

A student’s test scores for the semester are listed below:

83, 87, 90, 94, 94, 93, 95, 70, 72, 83, 85, 88, 98

Construct a box plot for this data set, using the number line below.

✅ Final Answer

Five-number summary:
Min = 70, Q1 = 83, Median = 88, Q3 = 94, Max = 98
Draw whiskers at 70 and 98, a box from 83 to 94, and a median line at 88.

📌 Regents Tip — Box Plots:

Always sort the data first.
The median is the middle value of the full data set.
Q1 is the median of the lower half, and Q3 is the median of the upper half.
A box plot uses the minimum, Q1, median, Q3, and maximum.

Question No. 29 | Algebra 1 Regents — Writing a Line in Slope-Intercept Form

Question No. 29

Write an equation, in slope-intercept form, of a line that passes through the point (6, 3) and has a slope of 2/3.

✅ Final Answer

y = (2/3)x − 1

📌 Regents Tip — Writing Linear Equations:

Use y = mx + b when the problem asks for slope-intercept form.
Substitute the slope first, then use the given point to solve for b.
Always check your answer by plugging the point back into the equation.
The slope tells you how steep the line is, and the y-intercept tells you where the line crosses the y-axis.

Question No. 30 | Algebra 1 Regents — Writing and Solving an Inequality

Question No. 30

Abby has $20 to spend at a community festival. She uses $8.50 to purchase food coupons for popcorn, a hot dog, and a soda.

She can buy individual ride tickets for $2.25 each. Determine algebraically the maximum number of ride tickets Abby can buy.

✅ Final Answer

Abby can buy a maximum of 5 ride tickets.

📌 Regents Tip — Word Problems with Inequalities:

Use an inequality when the total can be “at most,” “no more than,” or “within” a budget.
Define your variable clearly before writing the inequality.
After solving, check whether the answer must be a whole number.
Always test the next whole number to make sure you truly found the maximum.

Question No. 31 | Algebra 1 Regents — Graphing a Quadratic Function

Question No. 31

A rocket was launched from the ground into the air at an initial velocity of 80 feet per second. The path of the rocket can be modeled by

h(t) = −16t² + 80t, where t represents the time after the rocket has been launched, and h(t) represents the height of the rocket.

Sketch the function on the set of axes below.

State how many seconds it will take for the rocket to reach its maximum height.

State the maximum height, in feet, of the rocket.

Step-by-Step Solution

Step 1 — Recognize the shape of the graph.
The function is:
h(t) = −16t² + 80t
Since the coefficient of t² is negative, the graph is a parabola opening downward.
That means the rocket rises, reaches a maximum height, and then falls back down.
Step 2 — Find when the rocket reaches maximum height.
For a quadratic in the form at² + bt + c, the x-coordinate of the vertex is:
t = −b / 2a
Here, a = −16 and b = 80, so:
t = −80 / (2 · −16) = −80 / −32 = 2.5
Step 3 — Find the maximum height.
Substitute t = 2.5 into the function:
h(2.5) = −16(2.5)² + 80(2.5)
h(2.5) = −16(6.25) + 200
h(2.5) = −100 + 200 = 100
So the maximum height is:
100 feet

Step 4 — Find important points for the sketch.
Start with the y-intercept:
h(0) = 0 → point (0, 0)

Now make a table:

t	h(t)	Point
0	0	(0, 0)
1	−16(1)² + 80(1) = 64	(1, 64)
2	−16(4) + 160 = 96	(2, 96)
2.5	100	(2.5, 100)
3	−16(9) + 240 = 96	(3, 96)
4	−16(16) + 320 = 64	(4, 64)
5	−16(25) + 400 = 0	(5, 0)

Step 5 — Describe the sketch.
Plot the points (0,0), (1,64), (2,96), (2.5,100), (3,96), (4,64), (5,0) and draw a smooth downward-opening parabola through them.
Step 6 — State the two requested answers.
Time to maximum height:
2.5 seconds

Maximum height:
100 feet

✔ Quick Verification: The graph starts at (0,0) because the rocket launches from the ground, reaches its highest point at the vertex (2.5,100), and lands back on the ground at (5,0). ✓

✅ Final Answer

Sketch a downward-opening parabola through
(0,0), (1,64), (2,96), (2.5,100), (3,96), (4,64), (5,0)

Time to maximum height = 2.5 seconds
Maximum height = 100 feet

📌 Regents Tip — Graphing Quadratics in Context:

The vertex gives the maximum or minimum point of a parabola.
Use x = −b / 2a to find the x-coordinate of the vertex quickly.
If the coefficient of the squared term is negative, the parabola opens downward.
In projectile problems, the graph often starts at ground level, rises to a maximum, then returns to ground level.

Question No. 32 | Algebra 1 Regents — Solving a Quadratic with the Quadratic Formula

Question No. 32

Use the quadratic formula to solve

2x² − 4x − 3 = 0, and express the answer in simplest radical form.

✅ Final Answer

x = (2 + √10)/2 or x = (2 − √10)/2

📌 Regents Tip — Quadratic Formula:

Always identify a, b, and c carefully before substituting.
Use parentheses around negative numbers when squaring or multiplying.
Simplify the discriminant first, then simplify the radical if possible.
Leave irrational answers in simplest radical form unless the question asks for a decimal approximation.

Question No. 33 | Algebra 1 Regents — Linear Regression and Correlation

Question No. 33

The table below shows the ages of drivers and the annual cost of their car insurance.

Write the linear regression equation for this set of data. Round all values to the nearest hundredth.

State the correlation coefficient of this line of best fit, to the nearest hundredth.

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

✅ Final Answer

Linear regression equation:
y = −56.97x + 2352.22

Correlation coefficient:
r = −0.98

This indicates a strong negative linear relationship, so the line of best fit is a very good fit for the data.

📌 Regents Tip — Regression and Correlation:

The regression equation has the form y = mx + b.
The correlation coefficient r tells how strong the linear relationship is.
If r is close to 1, the relationship is strongly positive.
If r is close to −1, the relationship is strongly negative.
If r is close to 0, there is little or no linear relationship.