Algebra 1 Regents Review Sheet

Q: How should I study for the Algebra 1 Regents?

Work real past-exam questions by topic, review this one-page summary of the must-know concepts, and read the worked solution every time you miss one. The exam reuses the same ideas year after year.

Everything is a function. Solve = find the input that makes the output true.

Practice Algebra 1 free Reference & formulas

Big ideas

A function turns each input x into exactly one output y. Test on a graph: no vertical line hits it twice.
Linear = adds the same amount every step. Exponential = multiplies by the same factor every step. Quadratic = the rate of change itself grows linearly.
'Solve' means find the x that makes both sides equal. 'Graph the solution' means show every x that works on a number line or plane.

Linear equations & slope

Straight lines. Constant rate of change. Read them like a story: 'starts at b, grows by m each step.'

Slope-intercept form

y = mx + b

m is slope (rise/run), b is y-intercept (where line hits the y-axis).

Example: Line: y = 2x + 3 → Starts at (0,3), goes up 2 for every 1 right.

Slope from two points

m = (y_2 - y_1)/(x_2 - x_1)

Rise over run — how much y changes per unit x.

Example: Points (1,4) and (3,10) → m = (10−4)/(3−1) = 6/2 = 3

Point-slope form

y - y_1 = m(x - x_1)

Use when you know a point and a slope.

Example: Slope 4 through (2, 5) → y − 5 = 4(x − 2)

Parallel vs perpendicular

Parallel lines share the same slope. Perpendicular slopes multiply to −1 (negative reciprocals).

Example: Line with slope 2/3, perpendicular? → Slope = −3/2

Trap: Sign errors on slope. Always subtract y's and x's in the same order — mixing them up flips the sign.

Tip: If a word problem says 'per', 'each', or 'every', that's your slope. If it says 'starting amount', that's b.

Systems of equations

Two lines, one puzzle: the solution is where they meet.

Substitution

Solve one equation for a variable, then plug into the other.

Example: y = x + 1, 2x + y = 7 → Sub: 2x + (x+1) = 7 → 3x = 6 → x = 2, y = 3

Elimination

Add or subtract equations so a variable cancels.

Example: x + y = 5, x − y = 1 → Add → 2x = 6 → x = 3, y = 2

No solution vs infinite

Parallel lines: no solution. Same line: infinitely many.

Example: y = 2x + 1 and y = 2x + 4 → Parallel — no solution

Trap: Distribute the negative when you subtract equations. Losing a sign is the #1 miss on Regents systems.

Tip: If both variables have coefficient 1, elimination is faster. If one equation is already 'y = something', substitution is faster.

Quadratics

y = ax² + bx + c. Graph is a parabola. Opens up if a > 0, down if a < 0.

Factor a trinomial

Find two numbers that multiply to c and add to b.

Example: x² + 5x + 6 → 2 × 3 = 6, 2 + 3 = 5 → (x + 2)(x + 3)

Quadratic formula

x = (-b ± \sqrt{b^2 - 4ac})/(2a)

Works on any ax² + bx + c = 0, even when factoring won't.

Example: x² − 4x + 3 = 0 → x = (4 ± √4)/2 = 3 or 1

Vertex form

y = a(x - h)^2 + k

Vertex at (h, k). h is the x of vertex, k is the y.

Example: y = (x − 3)² + 2 → Vertex at (3, 2), opens up

Discriminant

b^2 - 4ac

> 0: two real roots. = 0: one repeated. < 0: no real roots.

Example: x² − 6x + 9 → b²−4ac = 36 − 36 = 0 → one repeated root (x = 3)

Trap: Vertex form's h has a FLIP: (x − 3)² has h = +3, not −3. The formula subtracts h, so the sign flips.

Tip: If asked for 'zeros', factor first — cleaner numbers. If the numbers are ugly, jump to the quadratic formula immediately.

Exponential functions

y = a·b^x. The output MULTIPLIES by the same factor each step (not adds).

Growth

y = a(1 + r)^t

Starts at a, grows by rate r each period.

Example: $500 growing 4% per year for 3 years → 500 × 1.04³ ≈ $562.43

Decay

y = a(1 - r)^t

Same shape, replace + with −.

Example: 800 mg drug losing 20% per hour, 4 hours → 800 × 0.8⁴ = 327.68 mg

Doubling / half-life

Doubling every T means base = 2^(t/T). Halving every T means (1/2)^(t/T).

Example: Bacteria double every 3 hours, 12 hours later → ×2^(12/3) = ×16

Trap: Linear vs exponential: if the difference is constant, linear. If the ratio is constant, exponential. Regents loves that table question.

Tip: Compound interest is exponential growth. Half-life is exponential decay with r = 0.5.

Functions & notation

f(x) is just 'the output of the function f at input x.' Nothing more, nothing less.

Evaluate

Substitute the number for x, do the arithmetic.

Example: f(x) = 2x + 3, find f(4) → f(4) = 8 + 3 = 11

Domain & range

Domain = legal inputs (x). Range = outputs (y). Watch for square roots (must be ≥ 0) and denominators (can't be 0).

Example: f(x) = 1/(x − 5) → Domain: all reals except x = 5

Average rate of change

(f(b) - f(a))/(b - a)

Just the slope between two points on the curve.

Example: f(x) = x², a = 1, b = 3 → (9 − 1)/(3 − 1) = 4

Trap: f(x + 2) is NOT f(x) + 2. It means 'plug in (x+2) wherever x appears'.

Tip: Domain 'in interval notation': parentheses exclude, brackets include. Infinity always gets a parenthesis.

Statistics

Center and spread. Then decide what the number actually means.

Center

Mean = average. Median = middle. Mode = most common. Skewed data: median is more honest.

Example: Salaries 20, 25, 30, 35, 1000 → Median 30 (mean 222 — misleading)

Spread

Range = max − min. IQR = Q3 − Q1 (middle 50%). Standard deviation = typical distance from mean.

Example: Data 1, 2, 3, 4, 5 → Range 4, mean 3, sd ≈ 1.58

Correlation vs causation

r near +1 or −1: strong linear relationship. r near 0: no linear pattern. Correlation ≠ causation.

Example: Ice cream sales and drownings go up together → Both caused by heat — not each other

Trap: Standard deviation on the calculator: know which button. The sample sd (Sx) is what the Regents wants, not σx.

Tip: For box plots: Q1, median, Q3 = quarters of the data. The IQR box holds the MIDDLE half.

Frequently asked questions

How should I study for the Algebra 1 Regents?