Algebra I (Regents) Exam Flashcards

Q: How is the NY Regents Algebra I exam structured?

The Regents Examination in Algebra I has four parts and 35 questions. Part I has 24 multiple-choice questions (2 credits each). Part II has 6 constructed-response questions (2 credits each). Part III has 4 constructed-response questions (4 credits each). Part IV has 1 constructed-response question (6 credits). The maximum raw score is 82 credits.

Q: What score do you need to pass the Algebra I Regents?

A scale score of 65 is the Regents passing standard. Regents scores are scaled, not raw percentages, so NYSED requires schools to use the raw-to-scale conversion chart published for that specific administration. A scale score of 65 does not correspond to 65 percent of the raw credits.

Q: Is a reference sheet provided on the Algebra I Regents?

Yes. A detachable Next Generation Algebra I reference sheet is included at the end of the test booklet for administrations starting June 2024. It supplies common formulas, but you are still expected to know and apply core algebra methods such as factoring, function notation, and solving systems.

Q: What calculator can you use on the Algebra I Regents?

Schools must make a graphing calculator available for the exclusive use of each student, and students also need a straightedge. Calculators capable of symbolic algebraic manipulation (CAS) are prohibited.

Q: How long is the Algebra I Regents exam?

Students are permitted three hours to complete the Regents Examination in Algebra I, according to NYSED's Algebra I educator guide. Students should confirm report times and admission deadlines with their school.

Q: Which topics carry the most weight on the Algebra I Regents?

The Algebra domain carries 48 to 61 percent of credits and Functions carries 24 to 32 percent. Number and Quantity (4 to 10 percent) and Statistics and Probability (7 to 15 percent) make up smaller portions, with modeling embedded throughout the exam.

Question 1

How do you solve a one-variable linear equation?

Accepted Answer

Use inverse operations to isolate the variable: undo addition/subtraction first, then multiplication/division. Whatever you do to one side, do to the other. Example: 3x + 7 = 22 gives 3x = 15, so x = 5.

Question 2

What is the one special rule when solving inequalities?

Accepted Answer

Solve like an equation, but FLIP the inequality sign whenever you multiply or divide BOTH sides by a negative number. Dividing by a positive number keeps the direction the same.

Question 3

How do you solve a literal equation for one variable (e.g. solve A = lw for w)?

Accepted Answer

Treat all other letters as constants and isolate the target variable with inverse operations. For A = lw, divide both sides by l to get w = A/l. The method is identical to solving for a number.

Question 4

How do you solve a compound inequality like -3 < 2x + 1 <= 7?

Accepted Answer

Perform the same operation on all three parts. Subtract 1: -4 < 2x <= 6. Divide by 2: -2 < x <= 3. The solution is all x between -2 (open) and 3 (closed).

Question 5

What does the graph of a strict inequality (< or >) look like vs an inclusive one?

Accepted Answer

Strict (<, >): open circle on a number line, dashed boundary line on a coordinate plane (endpoint NOT included). Inclusive (<=, >=): closed/filled circle, solid line (endpoint included).

Question 6

What is the slope formula for two points?

Accepted Answer

Slope m = (y2 - y1) / (x2 - x1), the change in y over the change in x ('rise over run'). Example: through (2,5) and (6,13), m = (13-5)/(6-2) = 8/4 = 2.

Question 7

What is slope-intercept form, and what does each letter mean?

Accepted Answer

y = mx + b. Here m is the slope (rate of change) and b is the y-intercept (value of y when x = 0). It is the easiest form for graphing and for reading the starting value.

Question 8

What is point-slope form and when is it useful?

Accepted Answer

y - y1 = m(x - x1). Use it when you know the slope m and any one point (x1, y1) on the line. It lets you write a line's equation without first finding the y-intercept.

Question 9

How do slopes tell you if two lines are parallel or perpendicular?

Accepted Answer

Parallel lines have EQUAL slopes (m1 = m2). Perpendicular lines have slopes that are negative reciprocals (m1 * m2 = -1). Example: a line with slope 2 is perpendicular to one with slope -1/2.

Question 10

What do a horizontal line and a vertical line look like as equations, and their slopes?

Accepted Answer

Horizontal line: y = c, slope = 0 (no rise). Vertical line: x = c, slope is UNDEFINED (no run, division by zero). A vertical line is not a function.

Question 11

What does the slope represent in a real-world linear model?

Accepted Answer

Slope is the unit rate of change: how much the output changes per 1-unit increase in the input (e.g. dollars per hour, miles per gallon). The y-intercept is the starting/initial value.

Question 12

What are the three methods for solving a system of two linear equations?

Accepted Answer

Graphing (find the intersection point), Substitution (solve one equation for a variable and plug into the other), and Elimination (add/subtract equations to cancel a variable). The solution makes BOTH equations true.

Question 13

How many solutions can a linear system have, and what does each mean?

Accepted Answer

One solution: lines intersect once (different slopes). No solution: parallel lines, same slope but different intercepts. Infinitely many: the same line (identical equations).

Question 14

How do you solve a system by elimination?

Accepted Answer

Multiply equations so one variable has opposite coefficients, then add the equations to eliminate it. Solve for the remaining variable, then substitute back to find the other.

Question 15

How do you solve a system graphically, including a linear-quadratic system?

Accepted Answer

Graph both equations; the solution(s) are the point(s) of intersection. A line and a parabola can intersect at 0, 1, or 2 points, so a linear-quadratic system may have up to two solutions.

Question 16

How do you graph the solution region of a system of linear inequalities?

Accepted Answer

Graph each inequality (dashed for <, >; solid for <=, >=), shade each solution half-plane, and the OVERLAP of the shaded regions is the solution set. Test a point to confirm which side to shade.

Question 17

What is the quadratic formula?

Accepted Answer

For ax^2 + bx + c = 0 with a not 0: x = (-b ± sqrt(b^2 - 4ac)) / (2a). It solves ANY quadratic equation, even ones that do not factor nicely.

Question 18

What is the discriminant and what does it tell you?

Accepted Answer

The discriminant is b^2 - 4ac (inside the square root of the quadratic formula). Positive: 2 real roots. Zero: 1 real (double) root. Negative: no real roots (2 imaginary).

Question 19

How do you find the axis of symmetry of a parabola y = ax^2 + bx + c?

Accepted Answer

The axis of symmetry is the vertical line x = -b/(2a). The parabola is mirror-symmetric across this line, and the vertex lies on it.

Question 20

How do you find the vertex of a parabola from y = ax^2 + bx + c?

Accepted Answer

Find the x-coordinate with x = -b/(2a), then substitute it back into the equation to get the y-coordinate. The vertex is the minimum point if a > 0 (opens up) or the maximum if a < 0 (opens down).

Question 21

What is vertex form of a quadratic, and what does it reveal?

Accepted Answer

y = a(x - h)^2 + k, where (h, k) is the vertex. The sign of a tells the opening direction; |a| controls how wide/narrow. Vertex form makes the max/min and axis of symmetry x = h instant to read.

Question 22

How do you solve a quadratic by completing the square?

Accepted Answer

Move the constant aside, take half of the x-coefficient and square it, add it to both sides to form a perfect-square trinomial, write it as (x + p)^2 = q, then take the square root of both sides (keep ±).

Question 23

What does the zero-product property let you do?

Accepted Answer

If a product equals zero, at least one factor must be zero. So if (x - 3)(x + 5) = 0, then x - 3 = 0 or x + 5 = 0, giving roots x = 3 and x = -5. This is how factoring solves quadratics.

Question 24

What are the roots/zeros of a quadratic, and how do they relate to its graph?

Accepted Answer

Roots (zeros) are the x-values where the function equals 0. On the graph they are the x-intercepts where the parabola crosses the x-axis. A quadratic has 0, 1, or 2 real roots.

Question 25

How do you factor a difference of two squares?

Accepted Answer

a^2 - b^2 = (a + b)(a - b). Example: x^2 - 25 = (x + 5)(x - 5). It only works for a difference (subtraction) of two perfect squares, never a sum.

Question 26

How do you factor a simple trinomial x^2 + bx + c?

Accepted Answer

Find two numbers that MULTIPLY to c and ADD to b; those numbers become the constants in (x + m)(x + n). Example: x^2 + 7x + 12 = (x + 3)(x + 4) because 3*4 = 12 and 3 + 4 = 7.

Question 27

What is the first factoring step to always check for?

Accepted Answer

Factor out the greatest common factor (GCF) first. Example: 6x^2 + 9x = 3x(2x + 3). Removing the GCF often makes the remaining polynomial easier to factor further.

Question 28

How do you multiply two binomials (FOIL)?

Accepted Answer

Multiply First, Outer, Inner, Last terms, then combine like terms. Example: (x + 3)(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6.

Question 29

What are the perfect-square trinomial patterns?

Accepted Answer

(a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2. Recognizing these speeds up both expanding and factoring. Example: x^2 + 6x + 9 = (x + 3)^2.

Question 30

How do you add or subtract polynomials?

Accepted Answer

Combine like terms (terms with the same variable and exponent). For subtraction, distribute the negative to every term in the second polynomial first. Example: (3x^2 + 2x) - (x^2 - 5x) = 2x^2 + 7x.

Question 31

What do the laws of exponents say for multiplying and dividing powers?

Accepted Answer

Multiplying same base: ADD exponents (x^3 * x^4 = x^7). Dividing same base: SUBTRACT exponents (x^7 / x^2 = x^5). Power of a power: MULTIPLY exponents ((x^3)^2 = x^6).

Question 32

What is function notation f(x), and how do you evaluate f(4)?

Accepted Answer

f(x) names the output rule for input x. To evaluate f(4), substitute 4 for every x and simplify. Example: f(x) = 2x^2 - 3 gives f(4) = 2(16) - 3 = 29.

Question 33

What is the difference between domain and range?

Accepted Answer

Domain is the set of all valid INPUTS (x-values). Range is the set of all resulting OUTPUTS (y-values). For a real-world model, the domain is often limited by context (e.g. time cannot be negative).

Question 34

What is the vertical line test?

Accepted Answer

A graph represents a function only if no vertical line crosses it more than once. This guarantees each input x has exactly one output y. A circle or sideways parabola fails the test.

Question 35

How do the transformations f(x) + k, f(x + h), and a*f(x) change a graph?

Accepted Answer

f(x) + k shifts the graph UP k (or down if k is negative). f(x + h) shifts LEFT h (right if h is negative). a*f(x) stretches vertically; a negative a reflects over the x-axis.

Question 36

How do linear and exponential growth differ?

Accepted Answer

Linear functions change by a constant ADDED amount each step (common difference). Exponential functions change by a constant MULTIPLIED factor each step (common ratio), so they eventually grow much faster.

Question 37

What does the exponential model y = a*b^x mean?

Accepted Answer

a is the initial value (y when x = 0) and b is the growth/decay factor. If b > 1 it is growth; if 0 < b < 1 it is decay. A rate r gives growth b = 1 + r and decay b = 1 - r.

Question 38

What is the average rate of change of a function over [a, b]?

Accepted Answer

It is (f(b) - f(a)) / (b - a), the slope of the line connecting the two points on the graph. For a linear function it is constant; for a curve it varies by interval.

Question 39

How do you find the zeros of a function from its graph or equation?

Accepted Answer

Zeros are inputs where the output is 0 — the x-intercepts of the graph. From an equation, set f(x) = 0 and solve. Zeros are also called roots or solutions.

Question 40

What is an arithmetic sequence and its key feature?

Accepted Answer

A sequence with a constant COMMON DIFFERENCE d added to get the next term. Explicit form: a_n = a_1 + (n - 1)d. Arithmetic sequences model linear growth.

Question 41

What is a geometric sequence and its key feature?

Accepted Answer

A sequence with a constant COMMON RATIO r multiplied to get the next term. Explicit form: a_n = a_1 * r^(n-1). Geometric sequences model exponential growth or decay.

Question 42

What is the difference between a recursive and an explicit sequence rule?

Accepted Answer

Recursive defines each term using the PREVIOUS term (e.g. a_n = a_(n-1) + 3 with a_1 = 2). Explicit gives any term directly from its position n (e.g. a_n = 2 + 3(n-1)).

Question 43

What does the correlation coefficient r tell you?

Accepted Answer

r measures the strength and direction of a LINEAR relationship and ranges from -1 to 1. Near +1: strong positive; near -1: strong negative; near 0: little or no linear relationship.

Question 44

Why does correlation not imply causation?

Accepted Answer

A strong r shows two variables move together, but it does not prove one CAUSES the other. A hidden third factor (a lurking variable) or coincidence can create the correlation.

Question 45

What is a line of best fit (linear regression) used for?

Accepted Answer

It is the line that best models a linear trend in a scatter plot. Its slope and y-intercept let you interpret the rate of change in context and predict (interpolate) values within the data range.

Question 46

What is a residual in a regression model?

Accepted Answer

A residual is the actual y-value minus the predicted y-value (observed minus model). A residual plot with no clear pattern (random scatter) suggests a linear model is a good fit.

Question 47

How do you read a two-way frequency table?

Accepted Answer

It cross-classifies data by two categories. Joint frequency is a single cell; marginal frequency is a row/column total; relative frequency is a count divided by a total, used to compare proportions or check association.

Question 48

How is the NY Regents Algebra I exam organized?

Accepted Answer

Four parts, 35 questions: Part I has 24 multiple-choice (2 credits each); Part II has 6 short constructed-response (2 cr); Part III has 4 constructed-response (4 cr); Part IV has 1 constructed-response (6 cr). Max raw score is 82 credits.

Question 49

What is the passing score on the Algebra I Regents, and what tools are allowed?

Accepted Answer

A SCALE score of 65 is passing (set by each administration's conversion chart, not a raw 65 percent). A graphing calculator (no symbolic/CAS) and a straightedge are provided, plus a Next Generation Algebra I reference sheet.

Question 50

What is the smartest strategy for constructed-response (Parts II-IV) questions?

Accepted Answer

Show all work — define the variable, write the equation, and explain reasoning, since partial credit is awarded for correct steps even if the final answer is wrong. Always interpret answers in context.

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