What is the slope-intercept form of a line?
Answer: y = mx + b, where m = slope and b = y-intercept.
Explanation
Example: y = 2x + 3 has slope 2 and y-intercept 3.
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SAT Math: Algebra, Advanced Math and Data Formulas (High School)
Linear EquationsSystems of EquationsQuadraticsExponentsGrowth & PercentStatisticsCirclesTrigonometrySAT Traps
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Linear Equations
What is the point-slope form of a line?
Answer: y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a known point.
Explanation
Example: line through (2,5) with slope 3: y − 5 = 3(x − 2).
Formula for slope given two points (x₁,y₁) and (x₂,y₂)?
Answer: m = (y₂ − y₁)/(x₂ − x₁)
Explanation
Example: points (1,2) and (3,8): m = (8−2)/(3−1) = 3.
Systems of Equations
When does a system of two linear equations have exactly one solution?
Answer: When the two lines have different slopes, so they cross at exactly one point.
Explanation
Example: y=2x+1 and y=−x+4 intersect once.
When does a system of two linear equations have no solution?
Answer: When the lines have the same slope but different y-intercepts — they're parallel and never meet.
Explanation
Example: y=2x+1 and y=2x+5.
When does a system of two linear equations have infinitely many solutions?
Answer: When both equations describe the exact same line (same slope and same y-intercept).
Explanation
Example: y=2x+1 and 2y=4x+2 simplify to the same line.
For ax+by=c and dx+ey=f, what coefficient ratio means no solution?
Answer: a/d = b/e ≠ c/f (same slope/direction, different intercept — parallel lines).
Explanation
Spotting this ratio quickly tells you the system is inconsistent without graphing.
For ax+by=c and dx+ey=f, what coefficient ratio means infinitely many solutions?
Answer: a/d = b/e = c/f (the second equation is just a multiple of the first).
Quadratics
What is the vertex form of a quadratic?
Answer: y = a(x − h)² + k, where (h, k) is the vertex of the parabola.
Explanation
Example: y = 2(x − 3)² + 1 has vertex (3, 1).
In standard form y = ax² + bx + c, what's the formula for the vertex's x-coordinate?
Answer: x = −b/(2a)
Explanation
Example: y = x² − 4x + 3 → vertex x = 4/2 = 2.
How do you factor x² + bx + c (a = 1)?
Answer: Find two numbers that multiply to c and add to b, then write (x + p)(x + q).
Explanation
Example: x² + 5x + 6 = (x + 2)(x + 3) since 2·3=6 and 2+3=5.
What is the discriminant and its formula?
Answer: Δ = b² − 4ac; it determines the number and type of roots of ax² + bx + c = 0.
Explanation
It's the part of the quadratic formula under the square root.
What do discriminant values Δ>0, Δ=0, Δ<0 tell you?
Answer: Δ>0: two real roots; Δ=0: one repeated real root; Δ<0: no real roots (complex pair).
Explanation
Example: x² + 2x + 5 → Δ = 4 − 20 = −16 < 0, no real roots.
What is the quadratic formula?
Answer: x = (−b ± √(b² − 4ac)) / (2a)
Explanation
Example: x² − 3x + 2 = 0 → x = (3 ± 1)/2 = 2 or 1.
Formula for the sum of the roots of ax² + bx + c = 0?
Answer: Sum of roots = −b/a
Explanation
Example: x² − 5x + 6 = 0 → sum = 5.
Formula for the product of the roots of ax² + bx + c = 0?
Answer: Product of roots = c/a
Explanation
Example: x² − 5x + 6 = 0 → product = 6.
Exponents
Exponent rule: a^m × a^n = ?
Answer: a^(m+n) — add exponents when multiplying same base.
Explanation
Example: 2³ × 2⁴ = 2⁷.
Exponent rule: a^m ÷ a^n = ?
Answer: a^(m−n) — subtract exponents when dividing same base.
Explanation
Example: 5⁶ ÷ 5² = 5⁴.
Exponent rule: (a^m)^n = ?
Answer: a^(mn) — multiply exponents for a power raised to a power.
Explanation
Example: (3²)³ = 3⁶.
What does a negative exponent mean? a^(−n) = ?
Answer: a^(−n) = 1/a^n
Explanation
Example: 2⁻³ = 1/8.
What is a^0 for a ≠ 0?
Answer: a^0 = 1
Explanation
Example: 7^0 = 1.
Growth & Percent
Formula for exponential growth or decay?
Answer: y = a(1 ± r)^t, where a = initial amount, r = rate (decimal), t = time; use + for growth, − for decay.
Explanation
Example: $1000 growing 5%/year for 3 years: 1000(1.05)³ ≈ $1157.63.
Formula for percent change?
Answer: Percent change = (New − Old)/Old × 100%
Explanation
Example: price goes from $50 to $60: (60−50)/50 = 20% increase.
How do you apply a percent increase or decrease as a multiplier?
Answer: Increase by r%: multiply by (1 + r/100). Decrease by r%: multiply by (1 − r/100).
Explanation
Example: decreasing $80 by 25%: 80 × 0.75 = $60.
Statistics
How does an outlier affect the mean of a data set?
Answer: It pulls the mean strongly toward its own value, since the mean uses every data point.
Explanation
Example: adding 100 to {2,3,4} changes the mean from 3 to 27.25.
How does an outlier affect the median of a data set?
Answer: The median barely changes — it's resistant to outliers since it depends only on the middle value(s).
Explanation
Example: {2,3,4} has median 3; {2,3,100} still has median 3.
Circles
Standard equation of a circle?
Answer: (x − h)² + (y − k)² = r², with center (h, k) and radius r.
Explanation
Example: (x−2)² + (y+1)² = 9 has center (2, −1) and radius 3.
Formula for arc length given a central angle θ (degrees)?
Answer: Arc length = (θ/360°) × 2πr
Explanation
Example: 90° arc of radius 4: (90/360) × 2π(4) = 2π.
Trigonometry
What does SOH-CAH-TOA stand for?
Answer: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
Explanation
Example: right triangle with opposite=3, hypotenuse=5 → sin = 3/5.
Pythagorean theorem formula?
Answer: a² + b² = c², where c is the hypotenuse of a right triangle and a, b are the legs.
Explanation
Example: legs 3 and 4 give hypotenuse 5.
Name three common Pythagorean triples.
Answer: 3-4-5, 5-12-13, 8-15-17 (plus their multiples, e.g. 6-8-10).
Explanation
Memorizing these speeds up right-triangle problems without a calculator.
SAT Traps
SAT trap: solving for the wrong variable — how do you avoid it?
Answer: Reread the question after solving; the SAT often asks for an expression (like 2x+1) or a different variable, not just x.
Explanation
Example: you solve x=5, but the question asks for the value of x+3.
SAT trap: unit mismatches — how do you avoid it?
Answer: Confirm all quantities use the same units (minutes vs. hours, feet vs. inches) before combining them, converting first if needed.
Explanation
Example: speed given in mph but time given in minutes — convert minutes to hours first.
SAT trap: extraneous roots — when do they appear and how do you catch them?
Answer: They arise from squaring both sides or clearing denominators; always plug each solution back into the original equation to check it's valid.
Explanation
Example: √(x+2) = x−4 may produce a root that fails when substituted back.
SAT trap: dropped negative signs — how do you avoid it?
Answer: Carefully distribute negatives across every term in parentheses, especially when squaring or expanding.
Explanation
Example: −(x−3) = −x+3, not −x−3.
SAT trap: misreading "must be true" questions — how do you approach them?
Answer: Test multiple valid values/cases; a statement is only correct if it holds for ALL allowed cases, not just one example you tried.
Explanation
Example: for x>0, "x² > x" is false when 0<x<1, so it's not always true.
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