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NY Regents · Algebra I

Algebra I Study Guide Expanded Edition

11 Units · 220 Quiz Questions · 62 Flashcards · Diagnostic · Worked Examples · Practice Exam · Modeled on the NY Regents

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Solving linear equations
Every equation is a balance — whatever you do to one side, you must do to the other. The goal is always to isolate the variable.
  • Combine like terms on each side first, then use inverse operations (undo addition/subtraction, then multiplication/division) to isolate the variable.
  • Distribute before combining like terms: 3(x + 4) = 3x + 12, not 3x + 4.
  • An equation with the SAME variable term on both sides that cancels out to a TRUE statement (like 5 = 5) has infinite solutions.
  • An equation that cancels out to a FALSE statement (like 5 = 8) has no solution.
Literal equations
A literal equation has more than one variable — you solve for ONE of them in terms of the others, using the exact same steps as any other equation.
  • Treat every other variable as if it were a number — isolate the target variable using inverse operations.
  • Example: solve A = ½bh for h → multiply both sides by 2, then divide by b → h = 2A/b.
  • Common formulas to practice on: perimeter, simple interest I = Prt, temperature conversion.
Solving & graphing inequalities
Inequalities solve almost exactly like equations, with one critical rule that trips people up constantly.
  • Flip the inequality symbol whenever you multiply or divide both sides by a NEGATIVE number.
  • Graph on a number line: open circle for < or >, closed (filled) circle for ≤ or ≥. Shade the direction the inequality points.
  • Compound inequalities like −3 < x ≤ 5 mean x is between −3 (exclusive) and 5 (inclusive) — graph both boundaries.
Absolute value equations & inequalities
Absolute value measures distance from zero, so it always splits into two cases.
  • |x| = a (a > 0) means x = a OR x = −a. Solve both.
  • |x| < a means −a < x < a (a compound 'between' inequality).
  • |x| > a means x < −a OR x > a (two separate rays going outward).
Forgetting to flip the inequality sign when multiplying/dividing by a negative is the #1 Regents error on this unit.
Distributing a negative: −2(x − 5) = −2x + 10, NOT −2x − 10. Watch the sign on the second term.
|x| < a and |x| > a graph completely differently — one is a segment between two points, the other is two separate rays.
Solve: isolate the variable using inverse operations, in reverse PEMDAS order
Flip inequality sign when multiplying/dividing by a negative
|x| = a → x = a or x = −a (a > 0)
Diagram
-3 5 open closed

Compound inequality −3 < x ≤ 5 on a number line: open circle at −3 (not included), closed circle at 5 (included).

Slope
Slope measures steepness — how much y changes for every step in x.
  • Slope formula: m = (y₂ − y₁) / (x₂ − x₁), often remembered as 'rise over run.'
  • Positive slope rises left to right; negative slope falls left to right.
  • A horizontal line has slope 0 (y = constant). A vertical line has UNDEFINED slope (x = constant) — you cannot divide by zero.
Forms of a linear equation
The same line can be written three different ways — pick whichever form fits the information you're given.
  • Slope-intercept form: y = mx + b (m = slope, b = y-intercept). Best for graphing quickly.
  • Point-slope form: y − y₁ = m(x − x₁). Best when you have a point and a slope.
  • Standard form: Ax + By = C. Useful for finding intercepts quickly (set x = 0 for the y-intercept, y = 0 for the x-intercept).
Writing equations of lines
Most Regents problems give you two pieces of information and ask for the equation.
  • Given two points: find slope first with the slope formula, then plug one point into point-slope form.
  • Parallel lines have the SAME slope.
  • Perpendicular lines have slopes that are NEGATIVE RECIPROCALS (they multiply to −1). Example: 2 and −½.
Intercepts & graphing
Intercepts are where a line crosses each axis — fast, reliable points for graphing.
  • x-intercept: set y = 0 and solve for x (where the line crosses the x-axis).
  • y-intercept: set x = 0 and solve for y (where the line crosses the y-axis) — this is the 'b' in y = mx + b.
  • Two points (like the two intercepts) are all you need to graph any line.
Slope formula order matters for the SIGN of individual differences, but as long as you keep y's and x's in the same point-order in numerator and denominator, the slope comes out the same either way.
Vertical lines (x = #) have UNDEFINED slope; horizontal lines (y = #) have ZERO slope — these are opposite and commonly swapped.
Perpendicular slope is the negative reciprocal, not just the negative. For 2/3 it's −3/2, not −2/3.
Slope: m = (y₂ − y₁) / (x₂ − x₁)
Slope-intercept: y = mx + b · Point-slope: y − y₁ = m(x − x₁)
Parallel: same slope · Perpendicular: negative reciprocal slopes
Diagram
(x₁,y₁) (x₂,y₂) run (Δx) rise (Δy)

Slope = rise/run between two points on a line. This line rises 3 for every 2 it runs, so m = 3/2.

Solving systems by substitution
Substitution works best when one equation is already solved for a variable (or easy to solve).
  • Solve one equation for one variable, then substitute that expression into the OTHER equation.
  • Solve the resulting one-variable equation, then plug that value back in to find the second variable.
  • Always write your answer as an ordered pair (x, y).
Solving systems by elimination
Elimination works by adding or subtracting the equations to cancel out one variable.
  • Line up the equations, then multiply one or both by a constant so one variable's coefficients are opposites.
  • Add the equations to eliminate that variable, solve for what's left, then back-substitute.
  • Elimination is fastest when both equations are already in standard form (Ax + By = C).
Number of solutions
Two lines can relate to each other in exactly three ways.
  • Different slopes → lines cross once → ONE solution (the normal case).
  • Same slope, different y-intercept → parallel lines → NO solution.
  • Same slope AND same y-intercept → the same line → INFINITE solutions.
Systems of linear inequalities
Graphing a system of inequalities means finding the region where BOTH shaded areas overlap.
  • Graph each inequality's boundary line (dashed for < or >, solid for ≤ or ≥) and shade its solution side.
  • The solution to the SYSTEM is only the region where both shadings overlap.
  • Test a point (often the origin, if not on a line) in the ORIGINAL inequality to check which side to shade.
Solving a system algebraically but forgetting to give BOTH coordinates of the answer — a system's solution is a point (x, y), not just a single number.
Dashed boundary lines mean the line itself is NOT included (< or >); solid lines mean it IS included (≤ or ≥).
When a system has no solution, students often think they made an error — but 0 = 8 (a false, variable-free statement) correctly means 'no solution,' not a mistake.
Substitution: solve for one variable, plug into the other equation
Elimination: add/subtract equations to cancel a variable
Same slope + different intercept = no solution · Same slope + same intercept = infinite solutions
Diagram
solution

Two lines with different slopes cross at exactly one point — that point (x, y) is the system's one solution.

Laws of exponents
These rules let you simplify expressions with exponents without ever expanding them out.
  • Product rule: xᵃ · xᵇ = x^(a+b) — multiplying same bases, ADD the exponents.
  • Quotient rule: xᵃ / xᵇ = x^(a−b) — dividing same bases, SUBTRACT the exponents.
  • Power rule: (xᵃ)ᵇ = x^(ab) — a power raised to a power, MULTIPLY the exponents.
  • Zero exponent: x⁰ = 1 (any nonzero base). Negative exponent: x⁻ⁿ = 1/xⁿ (flip to the denominator).
Polynomial vocabulary
A polynomial is a sum of terms with whole-number exponents on the variable.
  • Monomial: one term (5x²). Binomial: two terms (x + 3). Trinomial: three terms (x² + 2x − 1).
  • The degree of a polynomial is the HIGHEST exponent present. A degree-2 polynomial is 'quadratic'; degree-1 is 'linear.'
  • Like terms have the exact same variable AND exponent (3x² and −5x² are like terms; 3x² and 3x are not).
Adding & subtracting polynomials
Combine only like terms — everything else stays as it is.
  • Line up like terms and add/subtract their coefficients.
  • When subtracting a polynomial, distribute the negative sign to EVERY term inside the parentheses first.
  • Example: (3x² + 2x) − (x² − 5x) = 3x² + 2x − x² + 5x = 2x² + 7x.
Multiplying polynomials
Every term in the first polynomial must multiply every term in the second.
  • Monomial × polynomial: distribute the monomial to every term.
  • Binomial × binomial: use FOIL (First, Outer, Inner, Last), then combine like terms.
  • Binomial × trinomial: distribute each term of the binomial across all three terms of the trinomial, then combine like terms.
x⁰ = 1, NOT 0 — a very common Regents trap answer.
When subtracting polynomials, forgetting to distribute the negative to EVERY term in the second polynomial, not just the first.
(x + 3)² is NOT x² + 9 — you must FOIL it out: (x+3)(x+3) = x² + 6x + 9.
xᵃ·xᵇ = x^(a+b) · xᵃ/xᵇ = x^(a−b) · (xᵃ)ᵇ = x^(ab)
x⁰ = 1 · x⁻ⁿ = 1/xⁿ
FOIL: First, Outer, Inner, Last
Greatest Common Factor (GCF)
Always check for a GCF FIRST, before trying any other factoring method.
  • Find the largest number and variable power common to every term, then pull it out front.
  • Example: 6x² + 9x = 3x(2x + 3) — 3x is the GCF of 6x² and 9x.
  • Factoring is the reverse of distributing — you can always check by multiplying back out.
Factoring trinomials (a = 1)
For x² + bx + c, find two numbers that multiply to c and add to b.
  • x² + 7x + 12: need two numbers that multiply to 12 and add to 7 → 3 and 4 → (x + 3)(x + 4).
  • If c is positive and b is positive, both factors are positive. If c is positive and b is negative, both factors are negative.
  • If c is negative, the two factors have opposite signs (one positive, one negative).
Difference of squares & special patterns
Two patterns show up constantly and can be factored instantly once recognized.
  • Difference of squares: a² − b² = (a + b)(a − b). There is NO middle term.
  • There is no such pattern for a SUM of squares (a² + b²) — it does not factor over the reals.
  • Perfect square trinomial: a² + 2ab + b² = (a + b)². Recognize by checking if the first and last terms are perfect squares.
Factoring by grouping
Used for four-term polynomials or trinomials where a ≠ 1.
  • Split the middle term(s) so you can group the polynomial into two pairs, each with its own GCF.
  • Factor the GCF out of each pair — if done correctly, both pairs share the same leftover binomial.
  • Factor that common binomial out one final time.
Always check for a GCF first — trying to factor 4x² + 8x + 4 as a trinomial directly is much harder than pulling out the GCF of 4 first: 4(x² + 2x + 1) = 4(x+1)².
a² − b² factors, but a² + b² does NOT factor (over the real numbers) — a very common Regents trap.
Signs matter: c positive & b positive → both factors positive; c negative → factors have opposite signs.
GCF: pull out the largest shared factor first
a² − b² = (a+b)(a−b)
x² + bx + c: find two numbers that multiply to c, add to b
Solving by factoring
The Zero Product Property is the engine behind solving quadratics by factoring.
  • Set the equation equal to 0 first (move everything to one side).
  • Factor completely, then set EACH factor equal to zero and solve — this is the Zero Product Property.
  • A quadratic can have 0, 1, or 2 real solutions.
The quadratic formula
Works for ANY quadratic equation, even ones that don't factor nicely.
  • For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / 2a.
  • Identify a, b, and c carefully from the standard form BEFORE plugging in — a common source of errors.
  • The ± means you'll generally get two answers: one using + and one using −.
Graphing parabolas
Every quadratic function graphs as a U-shaped curve called a parabola.
  • If a > 0, the parabola opens UP (has a minimum). If a < 0, it opens DOWN (has a maximum).
  • Axis of symmetry: x = −b/2a — a vertical line through the vertex that splits the parabola into two mirror halves.
  • The vertex is the highest or lowest point; find its y-coordinate by plugging the axis-of-symmetry x-value back into the function.
The discriminant
The expression under the square root in the quadratic formula tells you how many real solutions exist, without solving.
  • Discriminant = b² − 4ac.
  • If b² − 4ac > 0: two different real solutions (the parabola crosses the x-axis twice).
  • If b² − 4ac = 0: exactly one real solution (the parabola just touches the x-axis at its vertex).
  • If b² − 4ac < 0: no real solutions (the parabola never touches the x-axis).
Forgetting to set the quadratic equal to ZERO before factoring — the Zero Product Property only works when one side is 0.
Sign errors plugging into the quadratic formula, especially when b or c is already negative — use parentheses around each substituted value.
Confusing the discriminant's sign meaning: POSITIVE discriminant means TWO solutions, not one.
Zero Product Property: if AB = 0, then A = 0 or B = 0
Quadratic formula: x = (−b ± √(b²−4ac)) / 2a
Axis of symmetry: x = −b/2a · Discriminant: b² − 4ac
Diagram
vertex axis of symmetry

Parabola y = a(x−h)² + k opening upward: vertex at the minimum, axis of symmetry through the vertex.

Simplifying radicals
Simplify a radical by pulling out the largest perfect-square factor.
  • Find the largest perfect square that divides evenly into the number under the radical.
  • √50 = √(25 · 2) = √25 · √2 = 5√2.
  • A radical is fully simplified when no perfect-square factor remains inside it.
Operating with radicals
Radicals combine like variables — you can only add/subtract 'like radicals.'
  • Like radicals have the SAME number under the root: 3√2 + 5√2 = 8√2 (add the coefficients, keep the radical).
  • √2 + √3 CANNOT be combined — they are not like radicals.
  • Multiplying radicals: √a · √b = √(ab). Multiply what's under the roots together, then simplify.
Rational exponents
Rational (fraction) exponents are just another way to write roots.
  • x^(1/n) = ⁿ√x. Example: x^(1/2) = √x, x^(1/3) = ∛x.
  • x^(m/n) = ⁿ√(x^m) = (ⁿ√x)^m — the denominator is the root, the numerator is the power.
  • All the normal exponent laws (product, quotient, power rules) still apply to rational exponents.
Solving radical equations
To undo a square root, square both sides — but this can introduce a fake answer you must check for.
  • Isolate the radical on one side first, then square both sides to eliminate it.
  • Solve the resulting equation normally.
  • ALWAYS check your answer(s) in the ORIGINAL equation — squaring can create an 'extraneous' (fake) solution that doesn't actually work.
√2 + √3 does NOT equal √5 — you cannot add unlike radicals by adding what's under the root.
Forgetting to check for extraneous solutions after squaring both sides of a radical equation — this is tested constantly on the Regents.
x^(m/n): the DENOMINATOR n is the root, the NUMERATOR m is the power — easy to flip by accident.
√(ab) = √a · √b · Combine only LIKE radicals
x^(1/n) = ⁿ√x · x^(m/n) = ⁿ√(xᵐ)
Radical equations: isolate radical, square both sides, CHECK for extraneous solutions
Exponential growth & decay
Exponential functions model quantities that grow or shrink by a constant PERCENT each time period, not a constant amount.
  • General form: y = a(1 + r)ᵗ for growth, y = a(1 − r)ᵗ for decay, where a is the starting amount and r is the rate (as a decimal).
  • The base (1 + r) or (1 − r) is called the growth/decay factor — if it's greater than 1, it's growth; if it's between 0 and 1, it's decay.
  • Unlike linear growth (constant amount added each time), exponential growth compounds — it multiplies by the same factor each period.
Arithmetic sequences
A sequence where you ADD the same number (the common difference) to get the next term.
  • Common difference d = any term minus the term before it.
  • Explicit formula: aₙ = a₁ + (n − 1)d, where a₁ is the first term.
  • Arithmetic sequences correspond to LINEAR functions — the graph of term number vs. value is a straight line.
Geometric sequences
A sequence where you MULTIPLY by the same number (the common ratio) to get the next term.
  • Common ratio r = any term divided by the term before it.
  • Explicit formula: aₙ = a₁ · r^(n−1), where a₁ is the first term.
  • Geometric sequences correspond to EXPONENTIAL functions.
Comparing linear and exponential models
Knowing which model fits a situation is a core Regents skill.
  • Linear/arithmetic: same amount ADDED each period (+3 every year).
  • Exponential/geometric: same PERCENT or FACTOR multiplied each period (×1.05 every year, or ×2 every hour).
  • Over a long enough time, exponential growth will always eventually overtake linear growth, no matter the starting values.
Mixing up common difference (arithmetic, addition) with common ratio (geometric, multiplication) — check whether you're adding or multiplying between terms.
In y = a(1+r)ᵗ, r is written as a DECIMAL (5% is 0.05, not 5), and you add/subtract it from 1 before raising to the t power.
aₙ = a₁ + (n−1)d uses (n − 1), not n — a common off-by-one error.
Exponential: y = a(1 ± r)ᵗ
Arithmetic: aₙ = a₁ + (n−1)d (common difference d)
Geometric: aₙ = a₁ · r^(n−1) (common ratio r)
Diagram
linear exponential

Exponential growth (curving upward, compounding) eventually overtakes linear growth (straight line, constant amount added).

Measures of center
Mean, median, and mode each describe the 'typical' value of a data set differently.
  • Mean: sum of all values divided by the count. Sensitive to outliers (extreme values pull it strongly).
  • Median: the middle value when data is ordered (average the two middle values if there's an even count). Resistant to outliers.
  • Mode: the most frequently occurring value(s). A data set can have no mode, one mode, or multiple modes.
Measures of spread
Spread describes how varied or consistent the data is.
  • Range = maximum − minimum. The simplest measure of spread, but very sensitive to outliers.
  • Interquartile Range (IQR) = Q3 − Q1 (the range of the middle 50% of the data) — resistant to outliers.
  • Standard deviation measures the typical distance each data point is from the mean; a bigger standard deviation means more spread-out data.
Five-number summary & box plots
A box plot visually displays five key statistics at once.
  • The five numbers, in order: minimum, Q1 (first quartile), median (Q2), Q3 (third quartile), maximum.
  • The 'box' spans from Q1 to Q3 (the IQR); the line inside the box is the median; the 'whiskers' extend to the min and max.
  • An outlier is often flagged if it falls more than 1.5 × IQR beyond Q1 or Q3.
Shape of distributions
The overall shape of a histogram tells you a lot about the data before you compute a single statistic.
  • Symmetric (roughly bell-shaped/normal): mean ≈ median.
  • Skewed right (tail stretches toward higher values): mean is pulled HIGHER than the median.
  • Skewed left (tail stretches toward lower values): mean is pulled LOWER than the median.
Mean is pulled toward the tail in a skewed distribution — 'skewed right' means the mean is greater than the median, which feels backwards to many students.
Forgetting to order the data before finding the median — an unordered 'middle value' is meaningless.
IQR uses Q3 − Q1, NOT max − min (that's the range).
Mean = sum ÷ count · Median = middle value (ordered data)
Range = max − min · IQR = Q3 − Q1
Skewed right: mean > median · Skewed left: mean < median
Diagram
min Q1 median Q3 max

Box plot (five-number summary): whiskers reach the min and max, the box spans Q1 to Q3, and the line inside is the median.

Scatter plots & association
A scatter plot shows the relationship between two numerical variables.
  • Positive association: as x increases, y tends to increase (points trend upward left to right).
  • Negative association: as x increases, y tends to decrease (points trend downward left to right).
  • No association: no clear pattern — the points look scattered randomly.
Linear regression
A line of best fit summarizes the overall trend in a scatter plot with linear association.
  • The line of best fit (regression line) minimizes the overall distance between itself and all the data points.
  • Its equation has the same form as any line, ŷ = mx + b, and can be used to PREDICT y-values for given x-values.
  • Predicting within the range of the given data is called interpolation (generally reliable); predicting far outside the given range is extrapolation (much less reliable).
Correlation coefficient (r)
r is a single number from −1 to 1 that measures how strong and what direction a LINEAR relationship is.
  • r close to +1: strong positive linear correlation. r close to −1: strong negative linear correlation.
  • r close to 0: little to no linear correlation (the data may still have a pattern, just not a linear one).
  • The closer |r| is to 1, the more tightly the points cluster around the regression line.
Correlation vs. causation
A strong correlation does NOT prove that one variable causes the other to change.
  • Two variables can be strongly correlated because of a third, unseen variable — or by pure coincidence.
  • Only a controlled experiment (not just observed data) can establish causation.
  • A residual is the difference between an actual data point and the value the regression line predicts for it — residuals scattered randomly around 0 suggest a linear model is a good fit.
A strong correlation (r near ±1) does NOT mean one variable CAUSES the other — this is tested directly and often.
r = 0 doesn't mean 'no relationship at all' — it means no LINEAR relationship; the data could still follow a strong curved pattern.
Extrapolating far beyond the collected data range is unreliable, even with a strong r value.
−1 ≤ r ≤ 1 · closer to ±1 = stronger linear correlation
Positive association: y increases with x · Negative: y decreases as x increases
Correlation ≠ Causation
Function notation
f(x) is just a labeled way of writing 'y' that tells you exactly which function produced the value.
  • f(x) is read 'f of x' — it means the OUTPUT of function f when the input is x. It does NOT mean f times x.
  • To evaluate f(3), substitute 3 everywhere you see x in the function's rule.
  • If f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7.
Domain & range
Domain is every valid input; range is every possible output.
  • Domain: the set of all possible x-values (inputs) a function can accept.
  • Range: the set of all possible y-values (outputs) a function can produce.
  • Common domain restrictions: you cannot divide by zero, and you cannot take the square root of a negative number (in the real numbers).
Identifying functions
A relation is a function only if every input maps to exactly ONE output.
  • Vertical Line Test: if any vertical line crosses the graph more than once, it is NOT a function.
  • In a table or set of ordered pairs, a function cannot repeat an x-value with two different y-values.
  • A relation CAN repeat a y-value for different x-values and still be a function — only repeated x-values (with different outputs) break it.
Transformations & rate of change
Simple changes to a function's equation shift, flip, or stretch its graph in predictable ways.
  • f(x) + k shifts the graph UP k units (k > 0) or down (k < 0). f(x + k) shifts LEFT k units (k > 0) or right (k < 0) — horizontal shifts feel backwards.
  • −f(x) reflects the graph over the x-axis; f(−x) reflects it over the y-axis.
  • Average rate of change between two points = (change in y) ÷ (change in x) — for a linear function this is just the slope; for other functions it changes depending on which interval you pick.
f(x) does NOT mean f multiplied by x — it means the output of function f for input x.
Horizontal shifts feel backwards: f(x − 3) shifts RIGHT, not left. Vertical shifts feel normal: f(x) − 3 shifts DOWN.
A relation with a repeated y-value is still fine — only a repeated x-value with two different outputs makes it NOT a function.
f(x) = output of function f at input x
Vertical Line Test: crosses more than once → not a function
Avg. rate of change = (change in y) / (change in x)
📝 Practice Question Bank — 220 questions
Unit 1: Solving Equations & Inequalities (20)

  1. Solve for x: 3x + 5 = 20

    • 3
    • 5
    • 7
    • 25

    3x = 15, so x = 5.

  2. Solve for x: 2(x − 4) = 10

    • 3
    • 5
    • 7
    • 9

    2x − 8 = 10 → 2x = 18 → x = 9.

  3. Solve for x: 5x − 3 = 2x + 12

    • 3
    • 4
    • 5
    • 6

    3x = 15 → x = 5.

  4. Solve the literal equation A = ½bh for h

    • h = 2A/b
    • h = A/2b
    • h = 2Ab
    • h = Ab/2

    Multiply both sides by 2, then divide by b: h = 2A/b.

  5. Solve the literal equation I = Prt for r

    • r = I/(Pt)
    • r = IPt
    • r = Pt/I
    • r = I − Pt

    Divide both sides by Pt: r = I/(Pt).

  6. Solve the inequality: −2x + 6 > 10

    • x > −2
    • x < −2
    • x > 2
    • x < 2

    −2x > 4 → x < −2 (flip the sign when dividing by a negative).

  7. Solve the inequality: 4x − 7 ≤ 9

    • x ≤ 4
    • x ≥ 4
    • x ≤ 3
    • x ≥ 3

    4x ≤ 16 → x ≤ 4.

  8. Which equation has infinite solutions?

    • 2x + 3 = 2x + 5
    • 2x + 3 = 2x + 3
    • 2x + 3 = 3x + 2
    • 2(x + 1) = 2x + 3

    2x + 3 = 2x + 3 simplifies to a true statement (3 = 3) for every x, so it has infinite solutions.

  9. Which equation has no solution?

    • 3x + 1 = 3x + 1
    • 3x + 1 = 3x + 4
    • 3x + 1 = 2x + 4
    • x = x

    3x + 1 = 3x + 4 simplifies to 1 = 4, a false statement — no value of x works.

  10. Solve |x − 2| = 7

    • x = 9 or x = −5
    • x = 5 or x = −9
    • x = 9 only
    • x = −9 or x = 5

    x − 2 = 7 → x = 9, or x − 2 = −7 → x = −5.

  11. Which interval represents |x| < 4?

    • −4 < x < 4
    • x < −4 or x > 4
    • x > 4
    • x = 4 or x = −4

    |x| < a means −a < x < a, so −4 < x < 4.

  12. Which interval represents |x| > 3?

    • −3 < x < 3
    • x < −3 or x > 3
    • x > 3 only
    • x = 3 or x = −3

    |x| > a means x < −a or x > a.

  13. Solve for x: −3(x + 2) = 15

    • −7
    • 7
    • −3
    • 3

    −3x − 6 = 15 → −3x = 21 → x = −7.

  14. Solve for x: 6x − 4 = 2x + 12

    • 4
    • 2
    • 6
    • 8

    4x = 16, so x = 4.

  15. Solve for x: −5x + 3 = 18

    • −3
    • 3
    • 5
    • −5

    −5x = 15, so x = −3.

  16. Solve the literal equation for w: P = 2l + 2w

    • w = (P − 2l)/2
    • w = P − 2l
    • w = (P + 2l)/2
    • w = 2l − P

    P − 2l = 2w, so w = (P − 2l)/2.

  17. Solve the inequality: 5 − 2x ≤ 11

    • x ≥ −3
    • x ≤ −3
    • x ≥ 3
    • x ≤ 3

    −2x ≤ 6 → x ≥ −3 (flip the sign when dividing by a negative).

  18. Which value of x satisfies |2x − 4| = 10?

    • x = 7 or x = −3
    • x = 7 only
    • x = 3 or x = −7
    • x = −7 only

    2x−4=10 → x=7, or 2x−4=−10 → x=−3.

  19. Solve: 3(2x − 1) = 4x + 7

    • 5
    • 3
    • 2
    • −5

    6x−3=4x+7 → 2x=10 → x=5.

  20. Graph the solution to x > 5 on a number line. What kind of circle is used at 5?

    • Open circle
    • Closed circle
    • No circle
    • A square

    Strict inequality (>) uses an open circle, since 5 itself is not included.

Unit 2: Linear Functions & Graphing (20)

  1. Find the slope between (2, 3) and (6, 11)

    • 2
    • 4
    • 1/2
    • 8

    m = (11 − 3)/(6 − 2) = 8/4 = 2.

  2. Find the slope between (−1, 5) and (3, −3)

    • −2
    • 2
    • 4
    • −4

    m = (−3 − 5)/(3 − (−1)) = −8/4 = −2.

  3. What is the slope of the line y = −3x + 7?

    • −3
    • 7
    • 3
    • 1/7

    In y = mx + b form, m = −3.

  4. What is the y-intercept of y = 5x − 2?

    • 5
    • −2
    • 2
    • −5

    In y = mx + b form, b = −2.

  5. A line has y-intercept 4 and slope −2. What is its equation?

    • y = −2x + 4
    • y = 2x + 4
    • y = −2x − 4
    • y = 4x − 2

    y = mx + b with m = −2, b = 4.

  6. Find the equation (in y = mx + b form) of the line with slope 3 through the point (2, 1)

    • y = 3x − 5
    • y = 3x + 5
    • y = 3x − 1
    • y = −3x − 5

    y − 1 = 3(x − 2) → y − 1 = 3x − 6 → y = 3x − 5.

  7. A line is parallel to y = 4x − 1. What is its slope?

    • 4
    • −1/4
    • −4
    • 1/4

    Parallel lines have the same slope.

  8. A line is perpendicular to y = 2x + 3. What is its slope?

    • 2
    • −2
    • 1/2
    • −1/2

    Perpendicular slopes are negative reciprocals: the negative reciprocal of 2 is −1/2.

  9. Which of these lines is horizontal?

    • y = 5
    • x = 5
    • y = 5x
    • y = x + 5

    y = 5 is a constant y-value for every x — a horizontal line.

  10. What is the slope of a vertical line?

    • 0
    • Undefined
    • 1
    • −1

    A vertical line's slope is undefined (division by zero in the slope formula).

  11. Find the x-intercept of 2x + 3y = 12

    • 6
    • 4
    • 12
    • 3

    Set y = 0: 2x = 12 → x = 6.

  12. Find the y-intercept of 2x + 3y = 12

    • 6
    • 4
    • 12
    • 2

    Set x = 0: 3y = 12 → y = 4.

  13. Two lines have slopes 2/3 and −3/2. What is their relationship?

    • Parallel
    • Perpendicular
    • The same line
    • Neither

    (2/3)(−3/2) = −1, so the lines are perpendicular.

  14. Find the slope between (0, 0) and (4, −8)

    • −2
    • 2
    • 4
    • −4

    m = (−8−0)/(4−0) = −2.

  15. What is the y-intercept of y = −4x + 9?

    • 9
    • −4
    • −9
    • 4

    In y=mx+b form, b=9.

  16. Write the equation of the line through (0, −3) with slope 5

    • y = 5x − 3
    • y = 5x + 3
    • y = −3x + 5
    • y = 3x − 5

    y = mx+b with m=5, b=−3.

  17. A line is parallel to x = 7. Which of these could be its equation?

    • x = −2
    • y = 7
    • y = −2x
    • x = y

    Vertical lines (x=constant) are parallel to other vertical lines.

  18. Find the x-intercept of y = 3x − 12

    • 4
    • −12
    • 12
    • −4

    Set y=0: 0=3x−12 → x=4.

  19. Two lines have slopes −4 and 1/4. What is their relationship?

    • Perpendicular
    • Parallel
    • The same line
    • Neither

    (−4)(1/4) = −1, so they are perpendicular.

  20. Write the equation, in slope-intercept form, of the line through (4, 5) and (4, −1)

    • x = 4 (undefined slope, no slope-intercept form)
    • y = 4
    • y = 5x − 15
    • x = 5

    Both points share x=4, making this a vertical line — it has no slope-intercept form.

Unit 3: Systems of Equations & Inequalities (20)

  1. Solve by substitution: y = x + 2 and 2x + y = 11

    • (3, 5)
    • (5, 3)
    • (2, 4)
    • (4, 2)

    2x + (x + 2) = 11 → 3x = 9 → x = 3, y = 5.

  2. Solve by elimination: x + y = 10 and x − y = 2

    • (6, 4)
    • (4, 6)
    • (5, 5)
    • (8, 2)

    Adding the equations: 2x = 12 → x = 6, then y = 4.

  3. Solve by substitution: y = 2x and x + y = 9

    • (3, 6)
    • (6, 3)
    • (9, 0)
    • (0, 9)

    x + 2x = 9 → 3x = 9 → x = 3, y = 6.

  4. Solve: 2x + 3y = 12 and 2x − y = 4

    • (3, 2)
    • (2, 3)
    • (4, 1)
    • (1, 4)

    Subtracting: (2x+3y)−(2x−y) = 12−4 → 4y = 8 → y = 2. Then 2x − 2 = 4 → x = 3.

  5. How many solutions does the system y = 3x + 1 and y = 3x − 4 have?

    • One
    • None
    • Infinite
    • Two

    Same slope (3), different y-intercepts — the lines are parallel and never meet.

  6. How many solutions does the system y = 2x + 5 and 2y = 4x + 10 have?

    • One
    • None
    • Infinite
    • Two

    Dividing the second equation by 2 gives y = 2x + 5 — the exact same line, so infinite solutions.

  7. How many solutions does the system x + y = 7 and x − y = 1 have?

    • None
    • One
    • Infinite
    • Two

    The lines have different slopes (−1 and 1), so they cross exactly once.

  8. A system has two equations with the same slope but different y-intercepts. What do the graphs look like?

    • Intersecting lines
    • Parallel lines
    • The same line
    • Perpendicular lines

    Same slope, different intercept — always parallel, never intersecting.

  9. Solve: x + 2y = 8 and 3x − 2y = 8

    • (4, 2)
    • (2, 4)
    • (6, 1)
    • (1, 6)

    Adding the equations: 4x = 16 → x = 4. Then 4 + 2y = 8 → y = 2.

  10. When graphing y > 2x − 1, how should the boundary line be drawn?

    • Solid, shaded above
    • Dashed, shaded above
    • Solid, shaded below
    • Dashed, shaded below

    Strict inequality (>) means a dashed line; shade above since y is greater than the line.

  11. How do you determine which side of a boundary line to shade?

    • Always shade above the line
    • Test a point not on the line in the original inequality
    • Shade toward the y-axis
    • Shade the larger-looking region

    Plug a test point (often the origin) into the original inequality — if true, shade that side.

  12. The solution region for a system of two linear inequalities is:

    • The union of both shaded regions
    • The region where both shaded areas overlap
    • The boundary lines only
    • Always the entire plane

    A system's solution must satisfy BOTH inequalities — only the overlap works.

  13. Solve: 4x − y = 9 and x + y = 6

    • (3, 3)
    • (2, 4)
    • (4, 2)
    • (1, 5)

    Adding the equations: 5x = 15 → x = 3. Then 3 + y = 6 → y = 3.

  14. Solve by substitution: x = 3y and 2x + y = 21

    • (9, 3)
    • (3, 9)
    • (7, 7)
    • (6, 2)

    2(3y)+y=21 → 7y=21 → y=3, x=9.

  15. Solve by elimination: 2x + y = 9 and x − y = 3

    • (4, 1)
    • (1, 4)
    • (3, 3)
    • (5, −1)

    Adding: 3x=12 → x=4, then y=9−8=1.

  16. How many solutions does x − y = 4 and 2x − 2y = 8 have?

    • Infinite
    • None
    • One
    • Two

    Dividing the second equation by 2 gives x−y=4, the same line — infinite solutions.

  17. Solve: 4x − 3y = 1 and 2x + 3y = 11

    • (2, 7/3)
    • (7/3, 2)
    • (1, 1)
    • (3, 11/3)

    Adding: 6x=12 → x=2. Then 8−3y=1 → y=7/3.

  18. A system has one solution. What must be true about the two lines' slopes?

    • They are different
    • They are the same
    • They are both zero
    • They are both undefined

    Different slopes guarantee the lines cross at exactly one point.

  19. Graphing y ≤ −x + 2, which type of boundary line and shading is used?

    • Solid line, shaded below
    • Dashed line, shaded below
    • Solid line, shaded above
    • Dashed line, shaded above

    ≤ uses a solid line (included); shade below since y is less than or equal to the line.

  20. Solve: y = 4 − x and x + y = 4

    • Infinite solutions
    • No solution
    • (0, 4) only
    • (4, 0) only

    Substituting: x + (4−x) = 4 → 4=4, always true — the equations describe the same line.

Unit 4: Exponents & Polynomials (20)

  1. Simplify: x⁴ · x³

    • x⁷
    • x¹²
    • x¹
    • 2x⁷

    Product rule: add the exponents, 4 + 3 = 7.

  2. Simplify: x⁹ / x⁴

    • x⁵
    • x¹³
    • x²·²⁵
    • x³⁶

    Quotient rule: subtract the exponents, 9 − 4 = 5.

  3. Simplify: (x³)⁴

    • x⁷
    • x¹²
    • x⁶⁴
    • 3x⁴

    Power rule: multiply the exponents, 3 × 4 = 12.

  4. Simplify: x⁰ (x ≠ 0)

    • 0
    • 1
    • x
    • Undefined

    Any nonzero number raised to the 0 power equals 1.

  5. Simplify: x⁻³

    • 1/x³
    • x³
    • −1/x³
    • 3x

    A negative exponent flips the base to the denominator: x⁻³ = 1/x³.

  6. Simplify: (2x²)³

    • 8x⁶
    • 6x⁶
    • 2x⁶
    • 8x⁵

    Cube both factors: 2³ = 8 and (x²)³ = x⁶, giving 8x⁶.

  7. What is the degree of 5x³ − 2x + 7?

    • 3
    • 5
    • 7
    • 2

    The degree is the highest exponent present, which is 3.

  8. Classify x² + 3x − 1 by its number of terms

    • Monomial
    • Binomial
    • Trinomial
    • Polynomial of degree 4

    It has three terms: x², 3x, and −1.

  9. Add: (3x² + 2x − 5) + (x² − 4x + 1)

    • 4x² − 2x − 4
    • 4x² + 2x − 4
    • 4x² − 2x + 4
    • 2x² − 2x − 4

    Combine like terms: (3x²+x²)=4x², (2x−4x)=−2x, (−5+1)=−4.

  10. Subtract: (5x² + 3x) − (2x² − x)

    • 3x² + 4x
    • 3x² + 2x
    • 7x² + 4x
    • 3x² − 4x

    Distribute the negative: 5x²+3x−2x²+x = 3x²+4x.

  11. Multiply: (x + 3)(x + 5)

    • x² + 8x + 15
    • x² + 15x + 8
    • x² + 8x + 8
    • x² + 2x + 15

    FOIL: x² + 5x + 3x + 15 = x² + 8x + 15.

  12. Multiply: (x − 4)(x + 4)

    • x² − 16
    • x² + 16
    • x² − 8x − 16
    • x² − 8

    This is a difference-of-squares pattern: x² − 16.

  13. Multiply: 3x(2x² − 5x + 1)

    • 6x³ − 15x² + 3x
    • 6x³ − 5x² + x
    • 6x² − 15x + 3
    • 5x³ − 15x² + 3x

    Distribute 3x to every term: 6x³ − 15x² + 3x.

  14. Simplify: (3x)² · x

    • 9x³
    • 3x³
    • 9x²
    • 6x³

    (3x)²=9x², then times x gives 9x³.

  15. Simplify: x¹² / x³

    • x⁹
    • x⁴
    • x¹⁵
    • 9x

    Subtract exponents: 12−3=9.

  16. Simplify: (x⁻²)³

    • x⁻⁶
    • x⁻⁵
    • x⁶
    • x⁻¹

    Multiply exponents: −2×3=−6.

  17. What is the degree of the polynomial 7x⁴y² + 3xy?

    • 6
    • 4
    • 2
    • 3

    Degree of a term is the sum of its exponents; 7x⁴y² has degree 4+2=6, the highest in the polynomial.

  18. Add: (2x² − 3x + 4) + (−x² + 5x − 6)

    • x² + 2x − 2
    • x² − 2x + 2
    • 3x² + 2x − 2
    • x² + 2x + 2

    (2−1)x² + (−3+5)x + (4−6) = x² + 2x − 2.

  19. Multiply: (2x − 1)(x + 5)

    • 2x² + 9x − 5
    • 2x² − 9x − 5
    • 2x² + 4x − 5
    • 2x² + 9x + 5

    FOIL: 2x²+10x−x−5 = 2x²+9x−5.

  20. Simplify: (x²)⁰

    • 1
    • 0
    • x²
    • Undefined

    Anything nonzero to the 0 power is 1.

Unit 5: Factoring (20)

  1. Factor completely: 6x² + 9x

    • 3x(2x + 3)
    • 3(2x² + 3x)
    • x(6x + 9)
    • 3x(2x + 9)

    The GCF of 6x² and 9x is 3x: 3x(2x + 3).

  2. Factor: x² + 7x + 12

    • (x + 3)(x + 4)
    • (x + 2)(x + 6)
    • (x + 1)(x + 12)
    • (x − 3)(x − 4)

    Need two numbers that multiply to 12 and add to 7: 3 and 4.

  3. Factor: x² − 2x − 15

    • (x − 5)(x + 3)
    • (x + 5)(x − 3)
    • (x − 15)(x + 1)
    • (x − 3)(x − 5)

    Need two numbers that multiply to −15 and add to −2: −5 and 3.

  4. Factor: x² − 9x + 20

    • (x − 4)(x − 5)
    • (x + 4)(x + 5)
    • (x − 2)(x − 10)
    • (x − 20)(x − 1)

    Need two numbers that multiply to 20 and add to −9: −4 and −5.

  5. Factor: x² − 16

    • (x − 4)(x + 4)
    • (x − 8)(x + 8)
    • (x − 4)²
    • (x − 16)(x + 1)

    Difference of squares: a² − b² = (a+b)(a−b), with a=x, b=4.

  6. Factor: 4x² − 25

    • (2x − 5)(2x + 5)
    • (4x − 5)(x + 5)
    • (2x − 25)(2x + 1)
    • (4x − 25)(x + 1)

    Difference of squares: (2x)² − 5² = (2x−5)(2x+5).

  7. Factor: x² + 10x + 25

    • (x + 5)²
    • (x − 5)²
    • (x + 25)
    • (x + 5)(x + 2)

    This is a perfect square trinomial: (x+5)(x+5) = (x+5)².

  8. Which expression does NOT factor over the real numbers?

    • x² − 9
    • x² + 9
    • x² − 4
    • x² − 1

    A sum of squares like x² + 9 has no real-number factoring pattern.

  9. Factor completely: 2x² + 8x + 8

    • 2(x + 2)²
    • 2(x + 4)(x + 2)
    • (2x + 4)(x + 2)
    • 2(x + 2)(x − 2)

    Pull out the GCF of 2 first: 2(x² + 4x + 4) = 2(x+2)².

  10. Factor by grouping: x³ + 3x² + 2x + 6

    • (x + 3)(x² + 2)
    • (x + 2)(x² + 3)
    • (x − 3)(x² + 2)
    • (x + 3)(x² − 2)

    Group (x³+3x²)+(2x+6) = x²(x+3) + 2(x+3) = (x+3)(x²+2).

  11. What should you always check for first when factoring any polynomial?

    • A greatest common factor
    • A perfect square trinomial
    • A difference of squares
    • The quadratic formula

    Always pull out a GCF first — it makes every later step simpler.

  12. Factor: x² − x − 6

    • (x − 3)(x + 2)
    • (x + 3)(x − 2)
    • (x − 6)(x + 1)
    • (x + 6)(x − 1)

    Need two numbers that multiply to −6 and add to −1: −3 and 2.

  13. Factor completely: 3x² + 9x − 12

    • 3(x + 4)(x − 1)
    • 3(x − 4)(x + 1)
    • (3x + 4)(x − 1)
    • 3(x + 2)(x − 2)

    Pull out the GCF of 3: 3(x² + 3x − 4) = 3(x+4)(x−1).

  14. Factor: x² − 25

    • (x − 5)(x + 5)
    • (x − 5)²
    • (x + 5)²
    • (x − 25)(x + 1)

    Difference of squares: x² − 5² = (x−5)(x+5).

  15. Factor: x² + 12x + 36

    • (x + 6)²
    • (x + 6)(x − 6)
    • (x + 12)(x + 3)
    • (x + 4)(x + 9)

    Perfect square trinomial: (x+6)(x+6) = (x+6)².

  16. Factor: x² + x − 20

    • (x + 5)(x − 4)
    • (x − 5)(x + 4)
    • (x + 10)(x − 2)
    • (x − 10)(x + 2)

    Need two numbers that multiply to −20 and add to 1: 5 and −4.

  17. Factor completely: 5x² − 20

    • 5(x − 2)(x + 2)
    • 5(x − 4)(x + 1)
    • (5x − 10)(x + 2)
    • 5(x² − 4)

    GCF of 5 first: 5(x²−4) = 5(x−2)(x+2).

  18. Factor: x² − 3x − 18

    • (x − 6)(x + 3)
    • (x + 6)(x − 3)
    • (x − 9)(x + 2)
    • (x + 9)(x − 2)

    Need two numbers that multiply to −18 and add to −3: −6 and 3.

  19. Which of these is a difference of squares?

    • x² − 49
    • x² − 7x
    • x² + 49
    • x² − 7

    x² − 49 = x² − 7² fits the a²−b² pattern.

  20. Factor by grouping: x³ + 5x² + 2x + 10

    • (x + 5)(x² + 2)
    • (x + 2)(x² + 5)
    • (x − 5)(x² + 2)
    • (x + 5)(x² − 2)

    Group (x³+5x²)+(2x+10) = x²(x+5)+2(x+5) = (x+5)(x²+2).

Unit 6: Quadratic Equations & Functions (21)

  1. Solve by factoring: x² − 5x + 6 = 0

    • x = 2 or x = 3
    • x = −2 or x = −3
    • x = 1 or x = 6
    • x = 2 or x = −3

    (x−2)(x−3)=0, so x = 2 or x = 3.

  2. Solve: x² − 9 = 0

    • x = 3 or x = −3
    • x = 9 or x = −9
    • x = 3 only
    • x = −3 only

    (x−3)(x+3)=0, so x = 3 or x = −3.

  3. Solve: x² + 4x = 0

    • x = 0 or x = −4
    • x = 0 or x = 4
    • x = 4 or x = −4
    • x = 0 only

    x(x+4)=0, so x = 0 or x = −4.

  4. Use the quadratic formula to solve x² + 2x − 8 = 0

    • x = 2 or x = −4
    • x = −2 or x = 4
    • x = 4 or x = −2
    • x = −2 or x = −4

    a=1,b=2,c=−8. Discriminant = 4+32=36, √36=6. x=(−2±6)/2 → x=2 or x=−4.

  5. Use the quadratic formula to solve x² − 4x + 4 = 0

    • x = 2 only
    • x = 2 or x = −2
    • x = 4 only
    • No real solution

    Discriminant = 16−16=0, so there's exactly one solution: x=(4±0)/2=2.

  6. For ax² + bx + c = 0, the discriminant is:

    • b² − 4ac
    • b² + 4ac
    • −b/2a
    • 4ac − b²

    The discriminant is the expression under the square root: b² − 4ac.

  7. A quadratic has discriminant = −12. How many real solutions does it have?

    • 0
    • 1
    • 2
    • 3

    A negative discriminant means no real solutions (the parabola never touches the x-axis).

  8. A quadratic has discriminant = 25. How many real solutions does it have?

    • 0
    • 1
    • 2
    • 3

    A positive discriminant means two distinct real solutions.

  9. A quadratic has discriminant = 0. How many real solutions does it have?

    • 0
    • 1
    • 2
    • Infinite

    A discriminant of exactly 0 means one real solution (the vertex touches the x-axis).

  10. For y = 2x² − 8x + 3, does the parabola open up or down?

    • Up
    • Down
    • Left
    • Right

    Since a = 2 > 0, the parabola opens upward.

  11. Find the axis of symmetry for y = x² − 6x + 5

    • x = 3
    • x = −3
    • x = 6
    • x = −6

    Axis of symmetry: x = −b/2a = −(−6)/2(1) = 3.

  12. Find the axis of symmetry for y = 2x² + 8x + 1

    • x = −2
    • x = 2
    • x = −4
    • x = 4

    Axis of symmetry: x = −b/2a = −8/(2·2) = −2.

  13. The vertex of a parabola is:

    • Where it crosses the x-axis twice
    • The highest or lowest point on the curve
    • Always at the origin
    • The y-intercept

    The vertex is the maximum point (if it opens down) or minimum point (if it opens up).

  14. Solve: x² = 49

    • x = 7 or x = −7
    • x = 7 only
    • x = 49
    • x = −7 only

    Taking the square root of both sides gives x = ±7.

  15. Solve by factoring: x² + x − 12 = 0

    • x = 3 or x = −4
    • x = −3 or x = 4
    • x = 12 or x = −1
    • x = 4 or x = −3

    (x−3)(x+4)=0, so x=3 or x=−4.

  16. Solve: x² − 25 = 0

    • x = 5 or x = −5
    • x = 25 or x = −25
    • x = 5 only
    • x = −5 only

    (x−5)(x+5)=0, so x=±5.

  17. Use the quadratic formula to solve x² − 6x + 5 = 0

    • x = 5 or x = 1
    • x = −5 or x = −1
    • x = 6 or x = 1
    • x = 5 or x = −1

    a=1,b=−6,c=5. disc=36−20=16, √16=4. x=(6±4)/2 → x=5 or x=1.

  18. A quadratic has discriminant = 0. Describe its graph.

    • It touches the x-axis at exactly one point (the vertex)
    • It crosses the x-axis twice
    • It never touches the x-axis
    • It is a straight line

    A zero discriminant means one repeated real solution — the vertex sits exactly on the x-axis.

  19. For y = −3x² + 12x − 5, does the parabola open up or down?

    • Down
    • Up
    • Left
    • Right

    Since a=−3 < 0, the parabola opens downward.

  20. Find the axis of symmetry for y = x² + 4x − 1

    • x = −2
    • x = 2
    • x = 4
    • x = −4

    x = −b/2a = −4/2 = −2.

  21. Solve: 2x² = 32

    • x = 4 or x = −4
    • x = 16 or x = −16
    • x = 4 only
    • x = 8

    x² = 16, so x = ±4.

Unit 7: Radicals & Rational Exponents (20)

  1. Simplify √48

    • 4√3
    • 16√3
    • 2√12
    • 4√12

    48 = 16 · 3, so √48 = √16 · √3 = 4√3.

  2. Simplify √72

    • 6√2
    • 36√2
    • 2√36
    • 8√9

    72 = 36 · 2, so √72 = √36 · √2 = 6√2.

  3. Simplify √20

    • 2√5
    • 4√5
    • 5√4
    • 2√20

    20 = 4 · 5, so √20 = √4 · √5 = 2√5.

  4. Simplify: 3√2 + 5√2

    • 8√2
    • 8√4
    • 15√2
    • 8

    Like radicals combine by adding coefficients: (3+5)√2 = 8√2.

  5. Simplify: 7√3 − 2√3

    • 5√3
    • 5√0
    • 9√3
    • 5

    Like radicals: (7−2)√3 = 5√3.

  6. Which expression CANNOT be simplified by combining like radicals?

    • 2√5 + 3√5
    • √2 + √3
    • 4√7 − √7
    • 6√2 + 2√2

    √2 and √3 have different numbers under the root — they are not like radicals.

  7. Multiply: √3 · √12

    • 6
    • 36
    • √15
    • 15

    √3 · √12 = √36 = 6.

  8. Which expression equals x^(1/2)?

    • √x
    • 2x
    • x/2
    • x²

    A power of 1/n equals the nth root: x^(1/2) = √x.

  9. Which expression equals x^(2/3)?

    • ∛(x²)
    • √(x³)
    • x²/3
    • 3√x

    x^(m/n) = ⁿ√(xᵐ), so x^(2/3) = ∛(x²).

  10. Solve: √(x + 3) = 5

    • 22
    • 2
    • 25
    • −22

    Square both sides: x+3=25 → x=22. Check: √25=5 ✓.

  11. Solve: √(2x − 1) = 3

    • 5
    • 4
    • 10
    • −5

    Square both sides: 2x−1=9 → x=5. Check: √9=3 ✓.

  12. Solve: √x = −4

    • No real solution
    • x = 16
    • x = −16
    • x = 4

    Squaring gives x=16, but √16=4, not −4 — this is extraneous, so there is no real solution.

  13. Why must you check solutions to radical equations in the original equation?

    • Squaring both sides can introduce extraneous solutions
    • Radicals are always negative
    • You might divide by zero
    • It isn't necessary if you show work

    Squaring both sides can create solutions that don't actually satisfy the original equation.

  14. Simplify √75

    • 5√3
    • 3√5
    • 25√3
    • 5√15

    75 = 25·3, so √75 = 5√3.

  15. Simplify √32

    • 4√2
    • 2√8
    • 8√2
    • 16√2

    32 = 16·2, so √32 = 4√2.

  16. Simplify: 4√5 + 3√5

    • 7√5
    • 7√10
    • 12√5
    • 7

    Like radicals: (4+3)√5 = 7√5.

  17. Simplify: √6 · √3

    • 3√2
    • √18
    • 9√2
    • 18

    √6·√3 = √18 = √(9·2) = 3√2.

  18. Simplify x^(3/2) using radical notation

    • √(x³)
    • 3√x
    • √x/3
    • x√3

    x^(3/2) = √(x³) (the denominator 2 is the root, numerator 3 is the power).

  19. Solve √(3x) = 9

    • x = 27
    • x = 3
    • x = 9
    • x = 81

    Square both sides: 3x=81 → x=27.

  20. Solve √(x − 1) = −3

    • No real solution
    • x = 10
    • x = 8
    • x = −8

    Squaring gives x=10, but √9=3≠−3 — extraneous, so there's no real solution.

Unit 8: Exponential Functions & Sequences (20)

  1. A population of 500 grows 6% per year. Which equation models this?

    • y = 500(1.06)ᵗ
    • y = 500(0.06)ᵗ
    • y = 500(1.6)ᵗ
    • y = 500 + 1.06t

    Growth: y = a(1+r)ᵗ with a=500, r=0.06.

  2. A car worth $20,000 depreciates 10% per year. Which equation models its value?

    • y = 20000(0.9)ᵗ
    • y = 20000(1.1)ᵗ
    • y = 20000(0.1)ᵗ
    • y = 20000 − 0.1t

    Decay: y = a(1−r)ᵗ with a=20000, r=0.10, so the factor is 0.9.

  3. In y = a(1 + r)ᵗ, if r = 0.08, this represents:

    • 8% growth
    • 8% decay
    • 80% growth
    • 0.8% growth

    r as a decimal of 0.08 means an 8% growth rate.

  4. In y = a(1 − r)ᵗ, if r = 0.15, this represents:

    • 15% growth
    • 15% decay
    • 85% decay
    • 1.5% decay

    Subtracting r means decay — a 15% decay rate.

  5. Find the common difference: 4, 9, 14, 19, ...

    • 5
    • 4
    • 9
    • 13

    Each term increases by 5 (9−4=5, 14−9=5).

  6. Find the common ratio: 3, 6, 12, 24, ...

    • 2
    • 3
    • 6
    • 9

    Each term is multiplied by 2 (6/3=2, 12/6=2).

  7. Find the 10th term of the arithmetic sequence with a₁ = 3, d = 4

    • 39
    • 43
    • 36
    • 35

    aₙ = a₁+(n−1)d = 3+(9)(4) = 3+36 = 39.

  8. Find the 6th term of the geometric sequence with a₁ = 2, r = 3

    • 486
    • 243
    • 729
    • 162

    aₙ = a₁·r^(n−1) = 2·3⁵ = 2·243 = 486.

  9. Find the 5th term of the arithmetic sequence 7, 11, 15, 19, ...

    • 23
    • 19
    • 27
    • 25

    d=4, a₅ = 7+(4)(4) = 23.

  10. Is the sequence 2, 4, 8, 16, ... arithmetic or geometric?

    • Arithmetic
    • Geometric
    • Neither
    • Both

    Each term is multiplied by 2 (a constant ratio), so it's geometric.

  11. Is the sequence 5, 8, 11, 14, ... arithmetic or geometric?

    • Arithmetic
    • Geometric
    • Neither
    • Both

    Each term increases by 3 (a constant difference), so it's arithmetic.

  12. Which grows faster in the long run: linear growth or exponential growth?

    • Linear always
    • Exponential eventually overtakes linear
    • They grow at the same rate
    • It depends only on the y-intercept

    No matter the starting values, exponential growth eventually outpaces linear growth.

  13. The explicit formula for an arithmetic sequence is:

    • aₙ = a₁ + (n−1)d
    • aₙ = a₁ · r^(n−1)
    • aₙ = a₁ + nd
    • aₙ = a₁ · d^n

    Arithmetic sequences add the common difference d, using (n−1) steps from the first term.

  14. A population of 800 declines 4% per year. Which equation models this?

    • y = 800(0.96)ᵗ
    • y = 800(1.04)ᵗ
    • y = 800(0.04)ᵗ
    • y = 800 − 0.04t

    Decay factor is 1−0.04=0.96.

  15. Find the common difference: 20, 15, 10, 5, ...

    • −5
    • 5
    • −10
    • 10

    Each term decreases by 5.

  16. Find the common ratio: 1, 1/3, 1/9, 1/27, ...

    • 1/3
    • 3
    • 1/9
    • 9

    Each term is multiplied by 1/3.

  17. Find the 12th term of the arithmetic sequence with a₁ = 6, d = 2

    • 28
    • 26
    • 30
    • 24

    a₁₂ = 6+(11)(2) = 28.

  18. Find the 4th term of the geometric sequence with a₁ = 5, r = 4

    • 320
    • 80
    • 1280
    • 20

    a₄ = 5·4³ = 5·64 = 320.

  19. Which type of sequence corresponds to a LINEAR function?

    • Arithmetic
    • Geometric
    • Neither
    • Both

    Arithmetic sequences (constant difference) match linear functions.

  20. In y = a(1+r)ᵗ, what does 'a' represent?

    • The initial (starting) value
    • The growth rate
    • The number of years
    • The final value

    'a' is the starting amount before any growth or decay is applied.

Unit 9: Statistics: Data & Distributions (20)

  1. Find the mean of 4, 7, 9, 12, 13

    • 9
    • 7
    • 45
    • 10

    Sum = 45, divided by 5 values = 9.

  2. Find the median of 3, 8, 5, 12, 9

    • 8
    • 5
    • 9
    • 7.4

    Ordered: 3, 5, 8, 9, 12 — the middle value is 8.

  3. Find the median of 2, 4, 6, 8

    • 5
    • 4
    • 6
    • 4.5

    With an even count, average the two middle values: (4+6)/2 = 5.

  4. Find the mode of 2, 3, 3, 5, 7, 3, 8

    • 3
    • 5
    • 7
    • 2

    3 appears three times, more than any other value.

  5. Find the range of 12, 5, 19, 8, 3

    • 16
    • 19
    • 14
    • 22

    Range = max − min = 19 − 3 = 16.

  6. A data set has Q1 = 10 and Q3 = 22. Find the IQR.

    • 12
    • 32
    • 22
    • 10

    IQR = Q3 − Q1 = 22 − 10 = 12.

  7. Which measure of center is LEAST affected by an outlier?

    • Mean
    • Median
    • Range
    • Sum

    The median only depends on the middle position, not the actual size of extreme values.

  8. In a box plot, the line inside the box represents the:

    • Mean
    • Median
    • Mode
    • Range

    The line inside the box marks the median (Q2).

  9. In a box plot, the box itself spans from:

    • Min to max
    • Q1 to Q3
    • Mean to median
    • 0 to the median

    The box represents the interquartile range, from Q1 to Q3.

  10. A distribution is skewed right. Which is true?

    • Mean > median
    • Mean < median
    • Mean = median
    • There is no mean

    A right-skewed tail pulls the mean higher than the median.

  11. A distribution is skewed left. Which is true?

    • Mean > median
    • Mean < median
    • Mean = median
    • The median doesn't exist

    A left-skewed tail pulls the mean lower than the median.

  12. Which measure of spread is MOST resistant to outliers?

    • Range
    • IQR
    • Maximum minus mean
    • Sum of all values

    IQR only uses the middle 50% of data, so extreme outliers don't affect it.

  13. For the data set 1, 2, 3, 4, 100 — which measure of center is most affected by the outlier 100?

    • Mean
    • Median
    • Mode
    • IQR

    The mean is pulled dramatically higher by the outlier; the median stays at 3.

  14. Find the mean of 6, 10, 14, 18, 22

    • 14
    • 12
    • 18
    • 16

    Sum=70, divided by 5 = 14.

  15. Find the median of 9, 2, 7, 4, 11, 6

    • 6.5
    • 7
    • 6
    • 4

    Ordered: 2,4,6,7,9,11 — average the two middle values (6,7): 6.5.

  16. Find the range of 45, 12, 38, 7, 29

    • 38
    • 45
    • 33
    • 7

    Range = max − min = 45 − 7 = 38.

  17. A data set has Q1=15 and Q3=27. Find the IQR

    • 12
    • 21
    • 42
    • 15

    IQR = Q3−Q1 = 27−15 = 12.

  18. In a box plot, what does a longer whisker on one side suggest?

    • More spread out data on that side
    • An error in the data
    • A smaller sample size
    • The data is symmetric

    A longer whisker indicates the data extends further and is more spread out on that side.

  19. Which measure of center would be most useful for a strongly skewed data set?

    • Median
    • Mean
    • Range
    • Mode only

    The median resists the pull of extreme values, making it more representative in a skewed distribution.

  20. A histogram has a long tail extending to the left. How does this describe the distribution?

    • Skewed left
    • Skewed right
    • Symmetric
    • Uniform

    A distribution is named for the direction its tail stretches — a left-stretching tail means skewed left.

Unit 10: Statistics: Regression & Correlation (19)

  1. A scatter plot shows points trending upward from left to right. This shows:

    • Positive association
    • Negative association
    • No association
    • Causation

    An upward trend means as x increases, y tends to increase — positive association.

  2. A scatter plot shows points trending downward from left to right. This shows:

    • Positive association
    • Negative association
    • No association
    • Causation

    A downward trend means as x increases, y tends to decrease — negative association.

  3. A correlation coefficient of r = 0.95 indicates:

    • A strong positive linear correlation
    • A strong negative linear correlation
    • Almost no correlation
    • Causation

    r close to +1 means a strong positive linear correlation.

  4. A correlation coefficient of r = −0.88 indicates:

    • A strong positive linear correlation
    • A strong negative linear correlation
    • No correlation
    • A weak correlation

    r close to −1 means a strong negative linear correlation.

  5. A correlation coefficient close to r = 0.05 indicates:

    • A strong linear correlation
    • Almost no linear correlation
    • A perfect correlation
    • Strong causation

    r near 0 means little to no LINEAR relationship.

  6. Which value of r shows the STRONGEST linear relationship?

    • r = 0.3
    • r = −0.91
    • r = 0.05
    • r = −0.4

    Strength depends on |r| — |−0.91| = 0.91 is closest to 1.

  7. The line that minimizes overall distance to all points on a scatter plot is called the:

    • Correlation line
    • Line of best fit
    • Axis of symmetry
    • Median line

    This is the regression line, or line of best fit.

  8. A residual is:

    • The slope of the regression line
    • The difference between an actual value and the predicted value
    • The correlation coefficient
    • The y-intercept of the line

    A residual = actual y − predicted y for a given data point.

  9. Predicting a y-value using an x-value far outside the range of the collected data is called:

    • Interpolation
    • Extrapolation
    • Regression
    • Correlation

    Extrapolation goes beyond the collected data range and is less reliable.

  10. A strong correlation between ice cream sales and drowning incidents does NOT mean ice cream causes drowning because:

    • Correlation never happens by chance
    • A third variable (like hot weather) likely explains both
    • The correlation coefficient must be negative
    • Causation only applies to negative correlations

    A lurking (third) variable — like summer heat — likely drives both.

  11. If r = 1 exactly, the data points:

    • Lie exactly on a line with positive slope
    • Lie exactly on a line with negative slope
    • Are randomly scattered
    • Form a curve, not a line

    r = 1 is a perfect positive linear correlation — every point lies exactly on the line.

  12. Interpolation means predicting a value:

    • Within the range of the collected data
    • Far outside the range of the collected data
    • Using only the mean
    • Using only the mode

    Interpolation stays within the range of the data already collected, so it's more reliable.

  13. A scatter plot shows no clear pattern between x and y. What kind of association is this?

    • No association
    • Strong positive association
    • Strong negative association
    • Perfect correlation

    Randomly scattered points with no trend indicate no association.

  14. Which correlation coefficient indicates the weakest linear relationship?

    • r = 0.02
    • r = 0.85
    • r = −0.79
    • r = 0.99

    |0.02| is closest to 0, indicating almost no linear relationship.

  15. A line of best fit is ŷ = −2x + 50. What does the slope tell you?

    • y decreases by 2 for every 1 unit increase in x
    • y increases by 2 for every 1 unit increase in x
    • x decreases by 2 for every unit of y
    • The starting value is −2

    The slope −2 means y decreases by 2 units for each 1-unit increase in x.

  16. Using ŷ = 3x + 10, predict y when x = 5

    • 25
    • 15
    • 30
    • 23

    ŷ = 3(5)+10 = 25.

  17. Which best describes a residual?

    • Actual y-value minus predicted y-value
    • The slope of the regression line
    • The correlation coefficient squared
    • The x-intercept of the line

    A residual measures how far off a prediction was from the actual data point.

  18. Two variables have r = 0.91. What can you conclude?

    • They have a strong positive linear relationship, but not necessarily causation
    • One variable definitely causes the other
    • There is no relationship
    • The relationship is negative

    A strong r shows correlation, not proof of causation.

  19. Predicting a value using x far beyond the range of the collected data is risky because:

    • It is extrapolation, which is less reliable
    • It always produces negative results
    • Correlation coefficients don't apply outside the data
    • It violates the slope formula

    Extrapolating beyond the observed data range reduces prediction reliability.

Unit 11: Functions: Notation, Domain/Range & Transformations (20)

  1. If f(x) = 2x + 3, find f(4)

    • 11
    • 9
    • 8
    • 14

    f(4) = 2(4) + 3 = 11.

  2. If f(x) = x² − 1, find f(3)

    • 8
    • 9
    • 5
    • 2

    f(3) = 3² − 1 = 9 − 1 = 8.

  3. If f(x) = 3x − 5, find f(−2)

    • −11
    • 1
    • −6
    • 11

    f(−2) = 3(−2) − 5 = −6 − 5 = −11.

  4. If g(x) = x² + 2x, find g(−1)

    • −1
    • 1
    • 3
    • −3

    g(−1) = (−1)² + 2(−1) = 1 − 2 = −1.

  5. What does f(x) mean?

    • f multiplied by x
    • The output of function f for input x
    • The slope of f
    • The x-intercept of f

    f(x) is notation for 'the output of f when the input is x' — not multiplication.

  6. What is the domain of f(x) = 1/(x − 3)?

    • All real numbers except x = 3
    • All real numbers
    • x ≥ 3
    • x ≤ 3

    The denominator cannot equal 0, so x ≠ 3.

  7. What is the domain of f(x) = √(x − 5)?

    • x ≥ 5
    • x ≤ 5
    • x ≥ −5
    • All real numbers

    The expression under the root cannot be negative: x − 5 ≥ 0 → x ≥ 5.

  8. Which test determines if a graph represents a function?

    • Horizontal Line Test
    • Vertical Line Test
    • Slope Test
    • Intercept Test

    The Vertical Line Test: if any vertical line crosses the graph more than once, it's not a function.

  9. A relation contains the points (2, 3), (2, 5), (4, 7). Is this a function?

    • Yes, it's a function
    • No, x = 2 has two different outputs
    • No, because 7 repeats
    • Yes, because the y-values differ

    x = 2 maps to both 3 and 5 — a function cannot have one input with two different outputs.

  10. A relation contains the points (1, 4), (2, 4), (3, 6). Is this a function?

    • Yes, every x has exactly one output
    • No, y = 4 repeats so it's not a function
    • No, because 1 ≠ 2
    • Yes, but only if x-values are consecutive

    Repeated y-values are fine — a function is only broken by a repeated x-value with different outputs.

  11. The graph of f(x) + 3 compared to f(x) is shifted:

    • Up 3 units
    • Down 3 units
    • Left 3 units
    • Right 3 units

    Adding outside the function shifts the graph vertically — up 3 units.

  12. The graph of f(x − 2) compared to f(x) is shifted:

    • Left 2 units
    • Right 2 units
    • Up 2 units
    • Down 2 units

    Subtracting inside the function shifts the graph right — horizontal shifts feel backwards.

  13. Find the average rate of change of f(x) = x² between x = 1 and x = 3

    • 4
    • 8
    • 2
    • 9

    f(1)=1, f(3)=9. Average rate of change = (9−1)/(3−1) = 8/2 = 4.

  14. If f(x) = 5x − 2, find f(0)

    • −2
    • 5
    • 0
    • 2

    f(0) = 5(0)−2 = −2.

  15. If f(x) = x² + 4, find f(−3)

    • 13
    • 1
    • −5
    • 7

    f(−3) = 9+4 = 13.

  16. What is the domain of f(x) = 3/(x+5)?

    • All real numbers except x = −5
    • All real numbers except x = 5
    • x ≥ −5
    • x ≤ 5

    The denominator cannot be zero, so x ≠ −5.

  17. A relation contains (1,2), (2,4), (3,6), (1,8). Is this a function?

    • No, x=1 has two different outputs
    • Yes, it's a function
    • No, because 8 is too large
    • Yes, because y-values differ

    x=1 maps to both 2 and 8, which violates the definition of a function.

  18. The graph of g(x) = f(x) − 4 compared to f(x) is:

    • Shifted down 4 units
    • Shifted up 4 units
    • Shifted left 4 units
    • Shifted right 4 units

    Subtracting outside the function shifts the graph down.

  19. Which relation passes the Vertical Line Test?

    • y = x²
    • x = y²
    • A circle x²+y²=9
    • x = 4 (a vertical line)

    y=x² is a function — every vertical line crosses it exactly once. The others fail the test.

  20. Find the average rate of change of f(x) = 2x + 3 between x=1 and x=4

    • 2
    • 3
    • 6
    • 9

    For a linear function, average rate of change always equals the slope: 2.

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Unit 1: Solving Equations & Inequalities

Inverse operations
Operations that undo each other: addition/subtraction, multiplication/division. Used to isolate a variable.
Literal equation
An equation with more than one variable, solved for one variable in terms of the others.
Compound inequality
Two inequalities joined together, like −3 < x ≤ 5, describing a range of values.
No solution (equation)
When solving leads to a false statement (like 5 = 8) — no value of the variable works.
Infinite solutions
When solving leads to a true statement (like 5 = 5) — every value of the variable works.
|x| = a
Absolute value equation meaning x = a or x = −a (for a > 0).

Unit 2: Linear Functions & Graphing

Slope
Steepness of a line: m = (y₂−y₁)/(x₂−x₁), or 'rise over run.'
Slope-intercept form
y = mx + b, where m is the slope and b is the y-intercept.
Point-slope form
y − y₁ = m(x − x₁), used to write an equation from one point and a slope.
Undefined slope
The slope of a vertical line (x = constant) — division by zero, so it has no numeric value.
Zero slope
The slope of a horizontal line (y = constant).
Perpendicular slopes
Negative reciprocals of each other (multiply to −1). Example: 2 and −½.
x-intercept
The point where a line crosses the x-axis; found by setting y = 0.

Unit 3: Systems of Equations & Inequalities

System of equations
Two or more equations considered together; the solution is the point (x, y) that satisfies all of them.
Substitution method
Solving a system by solving one equation for a variable and plugging that expression into the other equation.
Elimination method
Solving a system by adding or subtracting the equations to cancel out one variable.
No solution (system)
When two lines are parallel (same slope, different intercept) — they never intersect.
Infinite solutions (system)
When two equations describe the exact same line (same slope AND intercept).

Unit 4: Exponents & Polynomials

Product rule (exponents)
xᵃ · xᵇ = x^(a+b) — multiplying same bases adds the exponents.
Power rule (exponents)
(xᵃ)ᵇ = x^(ab) — a power raised to a power multiplies the exponents.
Zero exponent
Any nonzero base raised to the 0 power equals 1: x⁰ = 1.
Negative exponent
x⁻ⁿ = 1/xⁿ — flips the base to the denominator and makes the exponent positive.
Monomial / Binomial / Trinomial
Polynomials with one, two, and three terms, respectively.
Degree of a polynomial
The highest exponent on the variable in the polynomial.
FOIL
First, Outer, Inner, Last — the order for multiplying two binomials together.

Unit 5: Factoring

Greatest Common Factor (GCF)
The largest number/variable expression that divides evenly into every term — factor this out first.
Difference of squares
a² − b² = (a + b)(a − b). There is no similar pattern for a sum of squares.
Factoring by grouping
Splitting a polynomial into pairs of terms, factoring each pair's GCF, then factoring out the shared binomial.
Factor completely
Keep factoring until no factor can be broken down any further (always check for a GCF first).

Unit 6: Quadratic Equations & Functions

Zero Product Property
If A · B = 0, then A = 0 or B = 0. The basis for solving factored quadratics.
Quadratic formula
x = (−b ± √(b² − 4ac)) / 2a, used to solve any quadratic equation ax² + bx + c = 0.
Vertex of a parabola
The highest or lowest point of a parabola; occurs on the axis of symmetry.
Axis of symmetry
The vertical line x = −b/2a that splits a parabola into two mirror-image halves.
Discriminant
b² − 4ac. Positive → 2 real solutions; zero → 1 real solution; negative → 0 real solutions.

Unit 7: Radicals & Rational Exponents

Simplifying a radical
Pulling the largest perfect-square factor out from under the root sign.
Like radicals
Radicals with the same number under the root — only like radicals can be added or subtracted.
Rational exponent
x^(m/n) = ⁿ√(xᵐ) — the denominator is the root, the numerator is the power.
Extraneous solution
A solution produced by squaring both sides of a radical equation that does not actually work in the original equation.

Unit 8: Exponential Functions & Sequences

Exponential growth
y = a(1 + r)ᵗ — a quantity that grows by the same PERCENT each time period.
Exponential decay
y = a(1 − r)ᵗ — a quantity that shrinks by the same PERCENT each time period.
Arithmetic sequence
A sequence where the same number (common difference) is added to get each next term.
Common difference
The constant amount added between consecutive terms of an arithmetic sequence.
Geometric sequence
A sequence where each term is multiplied by the same number (common ratio) to get the next term.
Common ratio
The constant factor multiplied between consecutive terms of a geometric sequence.

Unit 9: Statistics: Data & Distributions

Mean
The sum of all data values divided by the number of values; sensitive to outliers.
Median
The middle value of ordered data; resistant to outliers.
Mode
The most frequently occurring value(s) in a data set.
Range
Maximum value minus minimum value.
Interquartile Range (IQR)
Q3 − Q1; the range of the middle 50% of the data, resistant to outliers.
Skewed right
A distribution with a tail stretching toward higher values; mean is greater than median.
Skewed left
A distribution with a tail stretching toward lower values; mean is less than median.

Unit 10: Statistics: Regression & Correlation

Positive association
As one variable increases, the other tends to increase too (upward trend on a scatter plot).
Negative association
As one variable increases, the other tends to decrease (downward trend on a scatter plot).
Line of best fit
A line that summarizes the trend in a scatter plot and can be used to make predictions.
Correlation coefficient (r)
A number from −1 to 1 measuring the strength and direction of a LINEAR relationship.
Residual
The difference between an actual data value and the value predicted by the regression line.
Correlation ≠ Causation
A strong correlation between two variables does not prove that one causes the other.

Unit 11: Functions: Notation, Domain/Range & Transformations

Function notation f(x)
Read 'f of x' — the output of function f when the input is x. Not f multiplied by x.
Domain
The set of all valid input (x) values for a function.
Range (of a function)
The set of all possible output (y) values a function can produce.
Vertical Line Test
If any vertical line crosses a graph more than once, the graph does NOT represent a function.
Average rate of change
(change in y) ÷ (change in x) between two points on a function.

Core Formulas

Slope
m = (y₂−y₁)/(x₂−x₁)
Slope-Intercept
y = mx + b
Point-Slope
y − y₁ = m(x − x₁)
Standard Form
Ax + By = C
Perpendicular Slopes
Negative reciprocals (multiply to −1)
Quadratic Formula
x = (−b ± √(b²−4ac)) / 2a
Discriminant
b² − 4ac (sign tells # of real solutions)
Axis of Symmetry
x = −b / 2a
Zero Product Property
If AB = 0, then A = 0 or B = 0
Product Rule
xᵃ · xᵇ = x^(a+b)
Quotient Rule
xᵃ / xᵇ = x^(a−b)
Power Rule
(xᵃ)ᵇ = x^(ab)
Zero / Negative Exponent
x⁰ = 1 · x⁻ⁿ = 1/xⁿ
Difference of Squares
a² − b² = (a+b)(a−b)
Rational Exponent
x^(m/n) = ⁿ√(xᵐ)
Arithmetic Sequence
aₙ = a₁ + (n−1)d
Geometric Sequence
aₙ = a₁ · r^(n−1)
Exponential Model
y = a(1 ± r)ᵗ
Mean
sum of values ÷ count
IQR
Q3 − Q1
Correlation Coefficient
−1 ≤ r ≤ 1 (closer to ±1 = stronger)
Average Rate of Change
(change in y) / (change in x)
Vertical Line Test
Crosses graph twice → not a function

Factoring Patterns

PatternFormExample
Greatest Common Factorab + ac = a(b + c)6x² + 9x = 3x(2x + 3)
Trinomial (a = 1)x² + bx + c = (x + p)(x + q), pq=c, p+q=bx² + 7x + 12 = (x+3)(x+4)
Difference of Squaresa² − b² = (a+b)(a−b)x² − 16 = (x+4)(x−4)
Perfect Square Trinomiala² + 2ab + b² = (a+b)²x² + 10x + 25 = (x+5)²
Factoring by Groupingax³+bx²+cx+d → pair and factorx³+3x²+2x+6 = (x+3)(x²+2)

Transformations of Functions

ChangeEffect on Graph
f(x) + kShifts UP k units (k > 0)
f(x) − kShifts DOWN k units
f(x − k)Shifts RIGHT k units (feels backwards)
f(x + k)Shifts LEFT k units
−f(x)Reflects over the x-axis
f(−x)Reflects over the y-axis

Fast Facts

  • Flip the inequality sign whenever multiplying or dividing by a negative number.
  • Same slope + different intercept = no solution. Same slope + same intercept = infinite solutions.
  • A sum of squares (a² + b²) does NOT factor over the real numbers — only a difference does.
  • Always check radical-equation solutions in the ORIGINAL equation for extraneous answers.
  • Correlation does not imply causation — a lurking third variable can explain both.
  • Check units in your final answer before moving on.
Quick ways to lock in the facts you keep forgetting. Read the big trick, then the small note tells you what it unlocks. Say them out loud — silly is memorable.

Equations & Inequalities

Flip it for negatives
Multiplying or dividing an inequality by a negative number flips the sign. Nothing else in algebra flips a sign like this — it's the #1 inequality trap.
True = infinite, false = none
If solving an equation leaves a TRUE statement (5=5), every x works (infinite solutions). If it leaves a FALSE statement (5=8), no x works (no solution).
Absolute value splits in two
|x| = a always means x = a OR x = −a. Distance from zero can go either direction.

Linear Functions

Rise over run, not run over rise
Slope = (y₂−y₁)/(x₂−x₁). The y's go on top. Mixing up the order flips the sign of the slope.
Parallel = same, perpendicular = flip-and-negate
Parallel lines share the exact same slope. Perpendicular lines have slopes that are negative reciprocals of each other (2 and −½).
Vertical is UNdefined, horizontal is ZERO
A vertical line (x = #) has undefined slope. A horizontal line (y = #) has zero slope. Easy to swap under pressure — say them together to lock the pairing in.

Systems

Same slope decides everything
Different slopes → one solution. Same slope, different intercept → parallel, no solution. Same slope AND intercept → the same line, infinite solutions.
Test a point to know which way to shade
When graphing an inequality, plug a test point (often the origin) into the ORIGINAL inequality. If it's true, shade that side.

Exponents & Polynomials

Same base: add to multiply, subtract to divide
xᵃ·xᵇ = x^(a+b) when multiplying; xᵃ/xᵇ = x^(a−b) when dividing. A power of a power MULTIPLIES the exponents instead.
Anything to the zero is one
x⁰ = 1 for any nonzero base — a constant Regents trap answer people second-guess.
FOIL: First, Outer, Inner, Last
The order for multiplying two binomials, so you never miss a term.

Factoring

GCF first, always
Before trying any other factoring method, check for a greatest common factor — it makes every later step smaller and easier.
Squares subtract, they don't add
a² − b² factors neatly into (a+b)(a−b). a² + b² does NOT factor over the real numbers — there's no pattern for a sum of squares.

Quadratics

Zero first, then factor
The Zero Product Property only works when one side of the equation is 0 — always move everything over before factoring and solving.
Discriminant sign tells the story
Positive discriminant → two real solutions. Zero → exactly one. Negative → none. No need to finish solving just to count solutions.

Radicals

Only like radicals combine
3√2 + 5√2 = 8√2, but √2 + √3 cannot be combined — the number under the root has to match, just like combining like terms.
Squaring can lie to you
Always plug your answer back into the ORIGINAL radical equation. Squaring both sides can create a fake (extraneous) solution that doesn't actually work.

Exponential Functions & Sequences

Add for arithmetic, multiply for geometric
Arithmetic sequences ADD a common difference between terms. Geometric sequences MULTIPLY by a common ratio. Check which operation connects the terms.
Exponential always wins eventually
No matter how big a linear function's starting advantage is, exponential growth eventually overtakes it given enough time.

Statistics

Mean chases the tail
In a right-skewed distribution, the mean is pulled higher than the median (toward the tail). In a left-skewed distribution, it's pulled lower.
Correlation isn't causation
A strong r value just means two variables move together — it never proves one causes the other. Look for a hidden third variable.

Functions

f(x) is a label, not multiplication
f(x) means "the output of f when the input is x," not f times x. Read it as a whole phrase, not two separate pieces.
Horizontal shifts feel backwards
f(x − 3) shifts the graph RIGHT, not left. Vertical shifts feel normal: f(x) − 3 shifts DOWN. When in doubt, plug in a test value and check.
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