- Slope formula
- Slope m = (y₂ − y₁) ÷ (x₂ − x₁) — the rise over the run between two points.
- Words in Context question
- Asks you to choose the word or phrase that best completes a passage based on its surrounding meaning. Predict your own word first, then match.
- Text Structure and Purpose question
- Asks why the author wrote a text or how a part functions within it (e.g., introduces a counterargument, gives an example).
- Cross-Text Connections question
- Gives two short passages on one topic and asks how the authors' viewpoints relate — support, challenge, or extend.
- Best strategy for Words in Context
- Predict your own word for the blank before reading the choices; a correct synonym can still be wrong if the tone or connotation clashes.
- Connotation vs denotation
- Denotation is a word's literal dictionary meaning; connotation is its emotional or tonal association. The SAT tests both.
- What 'main purpose' means
- The author's overall goal for the text — to inform, persuade, describe, or analyze — not a single detail in it.
- Tone in a passage
- The author's attitude toward the subject (e.g., critical, admiring, skeptical), revealed through word choice.
- How to handle a paired-passage question
- Pin down each author's claim separately, then describe how Text 2 responds to Text 1 (agrees, disagrees, qualifies, adds).
- Rule for evidence in R&W
- Every correct answer must be supported by the passage itself — never by outside knowledge or what is merely true in the real world.
- Function of a transitional sentence
- It may shift focus, introduce a contrast, or signal a conclusion — identify the role it plays, not just its content.
- Author's point of view
- The perspective from which the text is written; it shapes which details the author emphasizes.
- Reading the part after the blank
- Always read the whole sentence, including text after the blank — it often determines the right word in context.
- Central idea
- The main point a passage makes — the claim the whole text supports, broader than any single detail.
- Command of evidence (textual)
- Asks which quotation from the passage best supports a given claim or conclusion.
- Command of evidence (quantitative)
- Gives a graph or table and asks which choice the data supports. Read the axis labels and units first.
- Inference question
- Asks for a conclusion the passage implies but does not state. Pick the best-supported choice, not the most extreme.
- Detail question
- Asks for a specific fact the passage states. The answer is directly in the text.
- How to avoid inference traps
- Eliminate any choice that needs information the passage never provides, and any that overstate or distort the text.
- Reading a graph for evidence
- Check the title, axis labels, and units before the answer choices; the right choice must match both the data and the claim.
- Main idea vs supporting detail
- The main idea is the overall point; a supporting detail is one piece of evidence that backs it up.
- Summarizing a passage
- Capture the central claim and how the author supports it — leave out minor examples.
- Implicit vs explicit
- Explicit information is stated outright; implicit information must be inferred from what the text implies.
- Why 'most strongly supported' matters
- The credited inference is the one the text most directly supports — not merely plausible or interesting.
- Rhetorical Synthesis question
- Gives bulleted notes and a stated goal, and asks for the sentence that uses the notes to meet that goal.
- Transitions question
- Tests the logical relationship between ideas — choose the transition that matches (contrast, cause, addition, example).
- Transition for contrast
- However, nevertheless, on the other hand, conversely, yet — signals the second idea opposes the first.
- Transition for cause/effect
- Therefore, thus, consequently, as a result — signals the second idea follows from the first.
- Transition for addition
- Furthermore, moreover, in addition, also — signals the second idea adds to the first.
- Transition for example
- For example, for instance, specifically — signals an illustration of the prior idea follows.
- Transition for sequence
- First, next, then, finally, subsequently — signals order in time or steps.
- How to solve a transition question
- Cover the choices, name the relationship between the two ideas yourself, then pick the matching transition.
- Common transition trap
- A contrast word (however) where the ideas actually agree, or 'for example' where no example follows.
- How to handle synthesis goals
- Read the goal first (e.g., 'emphasize a difference'), then choose the sentence that actually accomplishes it.
- Concise writing on the SAT
- When choices say the same thing, the shortest grammatically correct, unambiguous one is usually right.
- Independent clause
- A group of words with a subject and verb that can stand alone as a complete sentence.
- Dependent (subordinate) clause
- Has a subject and verb but cannot stand alone (e.g., 'because it rained'); it depends on a main clause.
- Comma splice
- Two independent clauses joined by only a comma — an error. Fix with a period, semicolon, or comma + conjunction.
- Joining two independent clauses
- Use a period, a semicolon, or a comma plus a FANBOYS conjunction (and, but, or, nor, for, so, yet).
- FANBOYS
- The seven coordinating conjunctions: for, and, nor, but, or, yet, so.
- When to use a semicolon
- To join two closely related independent clauses without a conjunction.
- When to use a colon
- After a complete sentence, to introduce a list, explanation, or example.
- Subject-verb agreement
- A verb must match its subject in number; ignore words between them and match the true subject.
- Pronoun-antecedent agreement
- A pronoun must match its antecedent in number (a company = 'it,' not 'they').
- Its vs it's
- 'Its' is possessive (its color); 'it's' means 'it is' (it's raining).
- Their vs there vs they're
- 'Their' = possessive; 'there' = place; 'they're' = 'they are.'
- Who vs whom
- 'Who' is the subject (who called?); 'whom' is the object (to whom did you speak?).
- Dangling modifier
- An opening modifier that doesn't logically describe the noun right after the comma. Fix by putting the right noun there.
- Parallel structure
- Items in a list or comparison must share the same grammatical form (running, swimming, biking).
- Verb tense consistency
- Keep verb tense consistent with the timeline the passage establishes.
- Apostrophe for possession
- Singular: add 's (the dog's bone); plural ending in s: add only ' (the dogs' bones).
- Nonessential clause punctuation
- Set off nonessential information with a pair of commas, dashes, or parentheses.
- Restrictive vs nonrestrictive
- Restrictive info is essential (no commas); nonrestrictive info is extra (use commas).
- Comma after an introductory phrase
- Place a comma after an introductory word, phrase, or dependent clause before the main clause.
- Pronoun case
- Use subject pronouns (I, he, she, they) as subjects and object pronouns (me, him, her, them) as objects.
- Slope-intercept form
- y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis).
- Standard form of a line
- Ax + By = C. Convert to y = mx + b to read the slope and intercept.
- Point-slope form
- y − y₁ = m(x − x₁), using a known point (x₁, y₁) and slope m.
- Slopes of parallel lines
- Parallel lines have equal slopes.
- Slopes of perpendicular lines
- Perpendicular slopes are negative reciprocals; their product is −1.
- Slope of a horizontal line
- Zero (no rise).
- Slope of a vertical line
- Undefined (no run).
- Solving a system by substitution
- Solve one equation for a variable, substitute into the other, then solve.
- Solving a system by elimination
- Add or subtract the equations to cancel one variable, then solve for the other.
- System with no solution
- The lines are parallel — same slope, different y-intercept.
- System with infinite solutions
- The two equations are multiples of each other (the same line).
- Solving a linear inequality
- Solve like an equation, but flip the inequality sign when multiplying or dividing by a negative.
- Graphing a linear inequality
- Solid line for ≤ or ≥, dashed for < or >; shade the side that satisfies the inequality.
- Solution to an equation
- The value(s) of the variable that make the equation true.
- x-intercept
- Where a graph crosses the x-axis (y = 0).
- y-intercept
- Where a graph crosses the y-axis (x = 0); the b in y = mx + b.
- Absolute value equation
- |x| = a (a ≥ 0) has two solutions: x = a and x = −a.
- Direct proportion
- y = kx: y changes by a constant factor k as x changes; the graph passes through the origin.
- Distributing
- a(b + c) = ab + ac.
- Combining like terms
- Add or subtract terms with the same variable and exponent (3x + 5x = 8x).
- Quadratic equation
- An equation of the form ax² + bx + c = 0; its graph is a parabola.
- Quadratic formula
- x = (−b ± √(b² − 4ac)) ÷ (2a), which solves any quadratic ax² + bx + c = 0.
- Discriminant
- b² − 4ac. Positive → two real solutions; zero → one; negative → none.
- Difference of squares
- a² − b² = (a + b)(a − b).
- Perfect-square trinomial
- (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b².
- Factoring a quadratic
- Find two numbers that multiply to ac and add to b, then split the middle term.
- Vertex form of a parabola
- y = a(x − h)² + k, with vertex (h, k).
- Vertex of a parabola
- The maximum (a < 0) or minimum (a > 0) point; x-coordinate is −b ÷ (2a).
- Sum of the roots
- For ax² + bx + c = 0, the roots add to −b ÷ a.
- Product of the roots
- For ax² + bx + c = 0, the roots multiply to c ÷ a.
- Exponent product rule
- xᵐ · xⁿ = xᵐ⁺ⁿ.
- Exponent quotient rule
- xᵐ ÷ xⁿ = xᵐ⁻ⁿ.
- Power of a power rule
- (xᵐ)ⁿ = xᵐⁿ.
- Negative exponent
- x⁻ⁿ = 1 ÷ xⁿ.
- Zero exponent
- x⁰ = 1 (for x ≠ 0).
- Fractional exponent
- A fractional exponent means a root: x to the power 1/n is the nth root of x, and x to the power m/n is the nth root of x raised to m.
- Function notation
- f(x) is the output for input x; f(3) means evaluate the function at x = 3.
- Exponential function
- y = a · bˣ: grows by a constant percent if b > 1, decays if 0 < b < 1.
- Linear vs exponential growth
- Linear adds a constant amount each step; exponential multiplies by a constant factor each step.
- Growth/decay factor
- A 5% increase makes b = 1.05; a 5% decrease makes b = 0.95 in y = a·bˣ.
- Polynomial
- A sum of terms with whole-number exponents (e.g., 2x³ − x + 4).
- Zero of a function
- An input x where f(x) = 0; it is an x-intercept of the graph.
- Percent change
- Change ÷ original value × 100. Always divide by the starting amount.
- Increase a number by a percent
- Multiply by (1 + the percent as a decimal); +25% means × 1.25.
- Decrease a number by a percent
- Multiply by (1 − the percent as a decimal); −25% means × 0.75.
- Percent of a number
- Convert the percent to a decimal and multiply (20% of 50 = 0.20 × 50 = 10).
- Setting up a proportion
- Write two equal ratios and cross-multiply to solve (a/b = c/d → ad = bc).
- Unit rate
- A rate with a denominator of 1 (miles per 1 hour); divide to find it.
- Mean (average)
- Sum of the values ÷ how many there are; sensitive to outliers.
- Median
- The middle value of an ordered data set; resists outliers.
- Mode
- The most frequently occurring value in a data set.
- Range (statistics)
- The largest value minus the smallest value.
- Standard deviation
- A measure of spread around the mean; larger means more spread out.
- Effect of an outlier
- An outlier pulls the mean toward it but barely affects the median.
- Skew and the mean
- Right-skewed: mean > median; left-skewed: mean < median.
- Probability
- Favorable outcomes ÷ total outcomes; always between 0 and 1.
- Reading a two-way table
- Find the value at the right row and column; for 'given that,' divide within that subgroup.
- Conditional probability
- P(A given B) uses only the cases where B happens as the denominator.
- Margin of error / sampling
- A larger random sample narrows the margin of error and improves an estimate's reliability.
- Line of best fit
- A line that models the trend in scattered data; its slope shows the rate of change.
- Converting units
- Multiply by a conversion factor equal to 1 (e.g., 60 min / 1 hr) so unwanted units cancel.
- Probability of 'not' an event
- P(not A) = 1 − P(A).
- Pythagorean theorem
- For a right triangle with legs a, b and hypotenuse c: a² + b² = c².
- Area of a triangle
- ½ × base × height.
- Area of a rectangle
- length × width.
- Area of a circle
- πr², where r is the radius.
- Circumference of a circle
- 2πr (or πd, where d is the diameter).
- SOH-CAH-TOA
- sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent.
- sin θ
- Opposite ÷ hypotenuse in a right triangle.
- cos θ
- Adjacent ÷ hypotenuse in a right triangle.
- tan θ
- Opposite ÷ adjacent in a right triangle.
- 45-45-90 triangle
- Side ratio 1 : 1 : √2 (legs equal, hypotenuse = leg × √2).
- 30-60-90 triangle
- Side ratio 1 : √3 : 2 (short leg : long leg : hypotenuse).
- Complementary angle trig identity
- sin θ = cos(90° − θ): the sine of an angle equals the cosine of its complement.
- Sum of a triangle's angles
- 180°.
- Angles on a straight line
- Sum to 180° (supplementary).
- Angles around a point
- Sum to 360°.
- Complementary vs supplementary angles
- Complementary angles sum to 90°; supplementary angles sum to 180°.
- Parallel lines cut by a transversal
- Corresponding and alternate angles are equal; co-interior angles sum to 180°.
- Similar triangles
- Same shape, proportional sides, equal angles; set up a ratio to find a missing side.
- Equation of a circle
- (x − h)² + (y − k)² = r², center (h, k), radius r.
- Arc length
- (central angle ÷ 360°) × circumference.
- Sector area
- (central angle ÷ 360°) × πr².
- Volume of a rectangular prism
- length × width × height.
- Volume of a cylinder
- πr²h (base area × height).
- Radians and degrees
- 180° = π radians; multiply degrees by π/180 to convert to radians.
- Inscribed angle
- An inscribed angle is half the central angle that subtends the same arc.
- Connotation in answer choices
- Two synonyms can differ in feel; pick the one whose connotation matches the passage's tone.
- Identifying the claim
- Before answering, restate the author's main claim in your own words.
- Quantitative claim support
- The right data choice must be accurate to the graph AND relevant to the claim being made.
- Eliminating extreme answers
- Reject inferences using 'always,' 'never,' or 'only' unless the text fully supports them.
- Choosing the best transition
- Match the transition to the actual logic — don't be swayed by a familiar-sounding word.
- Sentence fragment
- A group of words missing a subject, verb, or complete thought; not a complete sentence.
- Run-on sentence
- Two independent clauses joined with no punctuation or conjunction.
- Colon vs semicolon
- A colon introduces (list/explanation after a full sentence); a semicolon joins two full sentences.
- Cross-multiplying
- For a/b = c/d, ad = bc — used to solve proportions.
- Literal equation
- An equation solved for one variable in terms of others (solve A = lw for w → w = A/l).
- Completing the square
- Rewrite ax² + bx + c into a(x − h)² + k form to find the vertex.
- Rational expression
- A ratio of polynomials; simplify by factoring and canceling common factors.
- Expected value idea
- A long-run average outcome — weight each outcome by its probability and add.
- Reading a scatterplot
- A positive trend rises left to right; a negative trend falls; strength is how tightly points cluster.
- Surface area concept
- The total area of all faces of a 3-D solid; add the areas of each face.
- Diameter vs radius
- The diameter is twice the radius (d = 2r).
- Vocabulary: 'ambivalent'
- Having mixed or contradictory feelings about something.
- Vocabulary: 'arbitrary'
- Based on random choice or personal whim, not reason or system.
- Vocabulary: 'candid'
- Truthful and straightforward; frank.
- Vocabulary: 'cogent'
- Clear, logical, and convincing.
- Vocabulary: 'didactic'
- Intended to teach, often with a moral lesson.
- Vocabulary: 'empirical'
- Based on observation or experiment rather than theory.
- Vocabulary: 'nuanced'
- Characterized by subtle distinctions or shades of meaning.
- Vocabulary: 'pragmatic'
- Dealing with things sensibly and practically.
- Vocabulary: 'skeptical'
- Not easily convinced; having doubts or reservations.
- Vocabulary: 'ubiquitous'
- Present, appearing, or found everywhere.
- Vocabulary: 'undermine'
- To weaken or damage something, often gradually.
- Vocabulary: 'novel' (adjective)
- New or original; not previously known.
- Vocabulary: 'comprehensive'
- Complete; covering all or nearly all elements.
- Vocabulary: 'substantiate'
- To provide evidence to support or prove a claim.
- Vocabulary: 'mitigate'
- To make something less severe or serious.
- Synthesizing across a passage
- Combine multiple details into the single conclusion the whole text supports.
- Counterargument
- A point that opposes the author's claim; authors raise it to address or refute it.
- Hypothesis in a science passage
- A proposed, testable explanation; questions may ask which result would support or weaken it.
- Identifying the strongest evidence
- The best evidence directly and specifically backs the exact claim, not a related idea.
- What weakens a claim
- Evidence that contradicts the claim or shows its reasoning fails.
- Transition 'in fact'
- Emphasizes or intensifies the previous statement.
- Transition 'likewise/similarly'
- Signals that the second idea is comparable to the first.
- Transition 'in contrast'
- Signals opposition between two ideas.
- Transition 'as a result'
- Signals an effect or consequence.
- Transition 'meanwhile'
- Signals events happening at the same time.
- Possessive plural noun
- For a plural noun ending in s, add only an apostrophe (the students' books).
- Singular 'they' vs antecedent
- On the SAT, match a pronoun to its antecedent's number; a singular antecedent usually takes 'it' or 'he/she.'
- Comparative vs superlative
- Comparative compares two (taller); superlative compares three or more (tallest).
- Fewer vs less
- 'Fewer' for countable nouns (fewer cars); 'less' for uncountable (less water).
- Affect vs effect
- 'Affect' is usually a verb (to influence); 'effect' is usually a noun (a result).
- Then vs than
- 'Then' refers to time; 'than' is used in comparisons.
- Modifier must touch its noun
- Place a descriptive phrase next to the word it modifies to avoid ambiguity.
- Verb agreement with collective nouns
- Treat a group (team, jury) as singular when acting as one unit.
- Apostrophe is never for plurals
- Don't add an apostrophe to make a noun plural (cats, not cat's).
- Em dash uses
- A dash can set off a nonessential element or introduce an emphatic addition.
- Linear function rate of change
- The slope is the constant rate of change of y with respect to x.
- Interpreting b in context
- In y = mx + b, b is the starting value (the y-value when x = 0).
- Interpreting m in context
- The slope m is the amount y changes for each one-unit increase in x.
- Solving for a variable
- Use inverse operations to isolate the variable on one side.
- Graph of x = a
- A vertical line through x = a.
- Graph of y = a
- A horizontal line through y = a.
- Number of solutions (one line)
- A single linear equation in one variable has exactly one solution unless it's contradictory or an identity.
- Identity equation
- True for all values (e.g., 2x + 2 = 2(x + 1)) → infinitely many solutions.
- Roots / x-intercepts
- The solutions of f(x) = 0; where the graph crosses the x-axis.
- Axis of symmetry
- The vertical line x = −b ÷ (2a) through a parabola's vertex.
- Maximum vs minimum (parabola)
- Opens up (a > 0) → minimum at the vertex; opens down (a < 0) → maximum.
- Multiplying binomials (FOIL)
- (a + b)(c + d) = ac + ad + bc + bd.
- Sum/difference of cubes
- a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²).
- Domain of a function
- The set of all allowable input (x) values.
- Range of a function
- The set of all output (y) values the function produces.
- Extraneous solution
- A solution that appears algebraically but fails the original equation; check radical/rational answers.
- Average speed
- Total distance ÷ total time (not the average of the speeds).
- Compound interest idea
- Interest earned on both principal and accumulated interest; modeled by an exponential function.
- Percent greater than
- If A is 30% greater than B, A = 1.30 × B.
- Ratio to total
- If a ratio is 3:2, the parts are 3/5 and 2/5 of the total.
- Frequency table mean
- Multiply each value by its frequency, sum, then divide by the total frequency.
- Interpreting slope of a model
- The slope is the predicted change in the output per one-unit change in the input.
- Random sampling validity
- A conclusion can be generalized only to the population the sample was randomly drawn from.
- Correlation vs causation
- A relationship in data does not by itself prove one variable causes the other.
- Perimeter
- The total distance around a 2-D shape (sum of the side lengths).
- Triangle inequality
- The sum of any two sides of a triangle is greater than the third side.
- Isosceles triangle
- Two equal sides and two equal base angles.
- Equilateral triangle
- All three sides equal and all angles 60°.
- Exterior angle of a triangle
- Equals the sum of the two non-adjacent interior angles.
- Congruent vs similar
- Congruent = same size and shape; similar = same shape, proportional sizes.
- Central angle
- An angle whose vertex is the center of a circle; equals its intercepted arc's measure.
- Tangent to a circle
- A line touching the circle at one point; it is perpendicular to the radius at that point.
- Volume of a cone
- (1/3)πr²h.
- Volume of a sphere
- (4/3)πr³.