SAT Math Grade 12 — Study Notebook

Algebra

📘 Unit 1 · Algebra & Linear Functions

⚠️ Tricky! Watch signs & order of operations carefully

1

If 3x − 7 = 2(x + 4), what is the value of x? EASY

KEY WORD: DISTRIBUTE → COLLECT → ISOLATE
Step: expand → move variables left → move numbers right

Distribute first: 2(x + 4) = 2x + 8
Then: 3x − 7 = 2x + 8 → subtract 2x both sides → x − 7 = 8 → x = 15

Distribute: 3x − 7 = 2x + 8
Subtract 2x: x − 7 = 8
Add 7: x = 15 ✓
TRAP: Don't forget to distribute the 2!

2

A line passes through (2, 5) and (6, 13). What is the slope of the line? EASY

SLOPE = RISE over RUN = (y₂ − y₁) / (x₂ − x₁)
"Change in Y ÷ Change in X" — always Y first!

m = (13 − 5) / (6 − 2) = 8 / 4 = 2 ✓
TRAP: Never flip — it's (y₂−y₁)/(x₂−x₁), not (x₂−x₁)/(y₂−y₁)!

3

Which value of x satisfies |2x − 4| = 10? ⚠️ TRAP

ABSOLUTE VALUE = TWO CASES!
|A| = B means → A = B OR A = −B
Always solve both equations!

Case 1: 2x − 4 = 10 → 2x = 14 → x = 7
Case 2: 2x − 4 = −10 → 2x = −6 → x = −3
Both are valid answers!

|2x − 4| = 10 gives TWO cases:
① 2x − 4 = 10 → x = 7
② 2x − 4 = −10 → x = −3
Both x = 7 and x = −3 ✓
CLASSIC TRAP: Forgetting the negative case!

Quadratics

📗 Unit 2 · Quadratic Equations & Parabolas

4

What are the solutions of x² − 5x + 6 = 0? EASY

FACTOR: find two numbers that
MULTIPLY to +6 AND ADD to −5
→ think: −2 × −3 = 6 ✓ and −2 + (−3) = −5 ✓

Factor: (x − 2)(x − 3) = 0
x − 2 = 0 → x = 2
x − 3 = 0 → x = 3
x = 2 or x = 3 ✓
Check: 2² − 5(2) + 6 = 4 − 10 + 6 = 0 ✓

5

The quadratic y = x² − 6x + 5 has its vertex at which point? MEDIUM

VERTEX X-coordinate = −b / (2a)
Then plug back in to find Y
Formula: vertex = (−b/2a, f(−b/2a))

a = 1, b = −6, c = 5
x = −(−6) / (2×1) = 6/2 = 3
y = 3² − 6(3) + 5 = 9 − 18 + 5 = −4
Vertex = (3, −4)

x = −b/2a = 6/2 = 3
y = 9 − 18 + 5 = −4
Vertex = (3, −4) ✓
TRAP: Sign error — b = −6, so −b = +6!

6

The discriminant of 2x² + 3x + 5 = 0 is: ⚠️ TRAP

DISCRIMINANT = b² − 4ac
>0 → two real roots | =0 → one root | <0 → no real roots
"BIG-FOUR-AC"

a = 2, b = 3, c = 5
Δ = b² − 4ac = 9 − 4(2)(5) = 9 − 40 = −31 ✓
Since Δ < 0 → no real solutions!
TRAP: Many pick +31 by forgetting the minus sign in 4ac!

Functions

📙 Unit 3 · Functions & Graphs

7

If f(x) = 3x² − 2x + 1, what is f(−2)? EASY

SUBSTITUTION: replace every x with (−2)
Use parentheses! Write: 3(−2)² − 2(−2) + 1
"PLUG IN WITH PARENS"

f(−2) = 3(−2)² − 2(−2) + 1
= 3(4) − (−4) + 1
= 12 + 4 + 1 = 17 ✓
TRAP: (−2)² = +4, NOT −4! Squaring removes the negative!

8

If g(x) = 2x + 3, what is g⁻¹(x)? MEDIUM

INVERSE: SWAP x and y, then SOLVE for y
y = 2x+3 → swap: x = 2y+3 → solve: y = (x−3)/2
"SWITCH AND SOLVE"

Let y = 2x + 3. Swap x ↔ y: x = 2y + 3
Solve: 2y = x − 3 → y = (x−3)/2
g⁻¹(x) = (x − 3)/2 ✓
TRAP: ÷ 2 applies to ALL of (x−3), not just x!

9

Which transformation moves y = x² to y = (x − 3)² + 5? ⚠️ TRAP

TRANSFORMATION RULE:
y = (x − h)² + k → shifts RIGHT by h, UP by k
INSIDE parens: OPPOSITE direction! (x−3) → move RIGHT +3

(x − 3)² → horizontal shift RIGHT 3 (opposite of −3)
+ 5 → vertical shift UP 5
Right 3, Up 5 ✓
TRAP: (x−3) looks like left, but it's RIGHT! Inside = opposite!

Exponents

📕 Unit 4 · Exponents, Radicals & Exponential Growth

10

Simplify: x³ · x⁵ / x⁴ EASY

EXPONENT RULES:
Multiply → ADD exponents | Divide → SUBTRACT
"SAME BASE: add to multiply, subtract to divide"

x³ · x⁵ = x^(3+5) = x⁸
x⁸ / x⁴ = x^(8−4) = x⁴ ✓
Simple: (3+5−4) = 4

11

What is the value of 27^(2/3)? ⚠️ TRAP

FRACTIONAL EXPONENT: x^(m/n) = (ⁿ√x)^m
Denominator = ROOT, Numerator = POWER
"DENOMINATOR = ROOT, NUMERATOR = POWER"

27^(2/3) → cube root of 27, then square it
∛27 = 3 → 3² = 9

27^(2/3) = (∛27)² = 3² = 9 ✓
TRAP: Don't multiply 27 × (2/3)! This is ROOT then POWER!

12

A bacteria population doubles every 3 hours. Starting with 200, what is the population after 12 hours? MEDIUM

EXPONENTIAL GROWTH: P = P₀ · b^(t/period)
Count how many periods: 12 ÷ 3 = 4 periods
"INITIAL × BASE^(TIME ÷ PERIOD)"

Periods = 12/3 = 4
P = 200 × 2⁴ = 200 × 16 = 3200 ✓
Doubles: 200 → 400 → 800 → 1600 → 3200
TRAP: Don't multiply 200 × 12 × 2!

Geometry

📒 Unit 5 · Geometry & Trigonometry

13

In a right triangle, one leg = 5, hypotenuse = 13. What is sin(θ) for the angle opposite the unknown leg? MEDIUM

SOH-CAH-TOA:
Sin = Opposite / Hypotenuse
Cos = Adjacent / Hyp | Tan = Opp / Adj

Find missing leg: 5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = 12
The angle opposite the unknown leg (12): sin(θ) = 12/13

Missing leg: √(13² − 5²) = √(169−25) = √144 = 12
sin(θ) = opposite/hypotenuse = 12/13
sin(θ) = 12/13 ✓
TRAP: 5/13 is sin of the OTHER angle! Read which angle carefully!

14

What is the area of a circle with circumference 10π? ⚠️ TRAP

CIRCLE SHORTCUTS:
C = 2πr → find r first, THEN use A = πr²
"CIRCUMFERENCE GIVES r, r GIVES AREA"

C = 2πr = 10π → r = 5
A = πr² = π(5²) = 25π ✓
TRAP: Don't use 10 directly as radius! First solve for r!

Stats

📔 Unit 6 · Statistics & Data Analysis

15

The data set is: {3, 7, 7, 8, 10, 15}. What is the median? ⚠️ TRAP

MEDIAN with EVEN number of values:
→ AVERAGE the two MIDDLE numbers
"EVEN COUNT = AVERAGE THE MIDDLES"

Sorted: 3, 7, 7, 8, 10, 15 — 6 values (even!)
Two middle values: 7 and 8
Median = (7 + 8) / 2 = 7.5 ✓
TRAP: Most pick 7 (mode) or 8 — but even count needs averaging!

16

A bag has 4 red and 6 blue marbles. Two are drawn without replacement. P(both red) = ? ⚠️ TRAP

WITHOUT REPLACEMENT: denominator changes!
P(A then B) = P(A) × P(B | A already taken)
"DENOMINATOR SHRINKS BY 1 EACH DRAW"

1st red: 4/10. Now 3 red left, 9 total.
2nd red: 3/9
P(both) = 4/10 × 3/9 = 12/90 = 2/15

P = (4/10) × (3/9) = 12/90 = 2/15 ✓
TRAP: (4/10)×(4/10) = 4/25 is WITH replacement! Wrong here!

Systems

📘 Unit 7 · Systems of Equations & Inequalities

17

Solve the system: 2x + y = 10 and x − y = 2 EASY

ELIMINATION: Add or subtract equations to CANCEL one variable
Here: ADD both equations → y cancels → solve x → back-substitute
"ADD TO CANCEL, THEN SUBSTITUTE"

Add: (2x+y) + (x−y) = 10+2 → 3x = 12 → x = 4
Sub: 4 − y = 2 → y = 2
(x, y) = (4, 2) ✓
Verify: 2(4)+2=10 ✓ and 4−2=2 ✓

18

Which ordered pair satisfies 3x − 2y < 6? ⚠️ TRAP

INEQUALITY TEST: just PLUG IN the point and check!
If the inequality is true → it's in the solution region
"SUBSTITUTE AND CHECK THE SIGN"

Test (1,2): 3(1) − 2(2) = 3 − 4 = −1 < 6 ✓
Test (4,3): 12 − 6 = 6, NOT less than 6 ✗
Test (3,1): 9 − 2 = 7 > 6 ✗
(1, 2) ✓
TRAP: Boundary value (6=6) does NOT satisfy strict < !

Word Probs

📓 Unit 8 · Ratios, Rates & Word Problems

19

A car travels 240 miles in 4 hours. At the same rate, how far will it travel in 7 hours? EASY

UNIT RATE first, then SCALE UP
Speed = distance ÷ time → then multiply by new time
"RATE × TIME = DISTANCE"

Rate = 240 ÷ 4 = 60 mph
Distance = 60 × 7 = 420 miles ✓
Also: 240/4 = x/7 → x = 240×7/4 = 420

20

A store marks up a shirt by 25% to sell for $50. What was the original price? ⚠️ TRAP

PERCENT INCREASE: Final = Original × (1 + rate)
→ Solve: Original = Final ÷ (1 + rate)
NEVER subtract % from final price directly!

$50 = Original × 1.25
Original = 50 ÷ 1.25 = $40

Let p = original. Then p × 1.25 = 50
p = 50 / 1.25 = $40.00 ✓
TRAP: 50 × 0.75 = $37.50 is WRONG! You can't just subtract 25% from the final price!

🎓 ALL DONE!

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SAT MATH