SAT Math — Grade 12 Study Notes

UNIT 1 · ALGEBRA

📘 Linear Equations & Systems

SLOPE = RISE/RUN

📌 Slope-intercept: y = mx + b | m = slope, b = y-intercept
📌 Standard form: Ax + By = C
📌 Two lines are parallel → same slope, different b
📌 Two lines are perpendicular → slopes are negative reciprocals: m₁ · m₂ = −1

📎 WARM-UP EXAMPLE

Line \( \ell \): \( y = 2x + 5 \). A line perpendicular to \(\ell\) passes through \((4, 1)\). Find its equation.
Perpendicular slope = \( -\frac{1}{2} \). Using point-slope: \( y - 1 = -\frac{1}{2}(x-4) \Rightarrow y = -\frac{1}{2}x + 3 \)

Q1 Linear Equations

⭐ Easy

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

💡 Distribute first, then isolate x. Watch the signs!

📖 EXPLANATION

\( 3x - 7 = 2x + 8 \)
\( 3x - 2x = 8 + 7 \)
\( x = 15 \) ✅

⚠️ Common mistake: forgetting to distribute the 2 to both terms inside the parentheses.

Q2 Systems of Equations

⭐⭐ Medium

The system below has no solution. What is the value of \( k \)?

\[ \begin{cases} 4x - 2y = 6 \\ 2x - y = k \end{cases} \]

💡 No solution = parallel lines = same slope, different y-intercept

📖 EXPLANATION

Divide eq.1 by 2: \( 2x - y = 3 \). This matches eq.2's left side exactly.
For no solution, the equations must be inconsistent: same left side, different right side.
So \( k \neq 3 \) ✅ (if k=3, infinitely many solutions instead!)

UNIT 2 · QUADRATICS

📗 Quadratic Equations & Parabolas

DISCRIMINANT = b² − 4ac

📌 \( b^2 - 4ac > 0 \): 2 real roots (crosses x-axis twice)
📌 \( b^2 - 4ac = 0 \): 1 real root (touches x-axis — "tangent")
📌 \( b^2 - 4ac < 0 \): No real roots (no x-intercepts)
📌 Vertex form: \( y = a(x-h)^2 + k \) → vertex = \((h, k)\)

Quadratic Formula: \(\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

📎 EXAMPLE

Find the vertex of \( f(x) = 2x^2 - 8x + 5 \).
Vertex x-coord: \( h = \frac{-(-8)}{2(2)} = \frac{8}{4} = 2 \)
\( f(2) = 2(4) - 16 + 5 = -3 \) → Vertex: \( (2, -3) \)

Q3 Quadratic Roots

⭐ Easy

How many real solutions does the equation \( x^2 + 6x + 9 = 0 \) have?

💡 Check the discriminant. Notice anything special about 6 and 9?

📖 EXPLANATION

\( x^2 + 6x + 9 = (x+3)^2 = 0 \)
This is a perfect square trinomial! \( x = -3 \) (one repeated root) ✅
Discriminant: \( 6^2 - 4(1)(9) = 36 - 36 = 0 \) → exactly 1 solution.

Q4 Vertex & Maximum

⭐⭐ Medium

A ball is thrown upward. Its height in feet after \( t \) seconds is:
\[ h(t) = -16t^2 + 64t + 5 \] What is the maximum height reached by the ball?

💡 For ax² + bx + c with a < 0, vertex gives maximum. Use h = −b/2a.

📖 EXPLANATION

\( t_{vertex} = \frac{-64}{2(-16)} = \frac{-64}{-32} = 2 \) seconds
\( h(2) = -16(4) + 64(2) + 5 = -64 + 128 + 5 = \mathbf{69} \) feet ✅
⚠️ Don't forget to add the initial height (+ 5)!

UNIT 3 · FUNCTIONS

📙 Functions & Transformations

SHIFT · FLIP · STRETCH

📌 \( f(x) + k \): shift UP k units
📌 \( f(x) - k \): shift DOWN k units
📌 \( f(x - h) \): shift RIGHT h units (opposite of sign!)
📌 \( -f(x) \): flip over x-axis | \( f(-x) \): flip over y-axis
📌 \( af(x) \): vertical stretch if \( a > 1 \), compression if \( 0 < a < 1 \)

Q5 Function Notation

⭐ Easy

Given \( f(x) = 3x^2 - x + 2 \), what is \( f(-2) \)?

💡 Substitute x = −2 carefully. Remember (−2)² = +4, not −4!

📖 EXPLANATION

\( f(-2) = 3(-2)^2 - (-2) + 2 = 3(4) + 2 + 2 = 12 + 4 = \mathbf{16} \) ✅
⚠️ Super common mistake: computing \( 3(-2^2) = 3(-4) \) instead of \( 3(-2)^2 = 3(4) \)

Q6 Transformations

⭐⭐ Medium

The graph of \( y = f(x) \) passes through \( (3, 7) \). Which point must be on the graph of \( y = f(x - 2) + 4 \)?

💡 (x − h) shifts RIGHT by h. Adding 4 shifts UP by 4. Apply both to the given point.

📖 EXPLANATION

\( f(x-2) \): x shifts +2 → x-coord: \( 3 + 2 = 5 \)
\( +4 \): y shifts +4 → y-coord: \( 7 + 4 = 11 \)
New point: \( \mathbf{(5, 11)} \) ✅
⚠️ The most common error: shifting LEFT instead of RIGHT for \( f(x-2) \).

UNIT 4 · EXPONENTS & LOGS

📕 Exponential Growth & Logarithms

LOG ↔ EXP (SWITCH)

📌 \( \log_b(x) = y \iff b^y = x \) ← Always convert between log and exp!
📌 \( \log(ab) = \log a + \log b \)
📌 \( \log(a/b) = \log a - \log b \)
📌 \( \log(a^n) = n\log a \)
📌 Growth: \( A = A_0 \cdot e^{rt} \) | Compound: \( A = P(1 + r/n)^{nt} \)

Q7 Exponential Equations

⭐ Easy

If \( 2^{x+3} = 32 \), what is the value of \( x \)?

💡 Write 32 as a power of 2. Then set exponents equal!

📖 EXPLANATION

\( 32 = 2^5 \), so \( 2^{x+3} = 2^5 \)
\( x + 3 = 5 \Rightarrow x = \mathbf{2} \) ✅

Q8 Exponential Growth

⭐⭐ Medium

A bacteria colony doubles every 3 hours. It starts with 500 bacteria. How many bacteria after 12 hours?

💡 Number of doublings = 12 ÷ 3 = 4. Each doubling multiplies by 2.

📖 EXPLANATION

\( N = 500 \cdot 2^{12/3} = 500 \cdot 2^4 = 500 \cdot 16 = \mathbf{8000} \) ✅
⚠️ Don't multiply 500 × 12 × 2. You must raise 2 to the power of doublings!

UNIT 5 · STATISTICS & DATA

📊 Statistics & Data Analysis

MEAN · MEDIAN · MODE · RANGE

📌 Mean = sum ÷ count (affected by outliers!)
📌 Median = middle value (NOT affected by outliers — use for skewed data)
📌 Standard Deviation = how spread out data is from the mean
📌 In a normal distribution: 68% within 1σ, 95% within 2σ, 99.7% within 3σ

Q9 Mean & Median

⭐ Easy

Five students scored: 72, 85, 90, 88, and 95 on a test. What is the mean score?

💡 Mean = sum of all values ÷ number of values

📖 EXPLANATION

Sum = \( 72 + 85 + 90 + 88 + 95 = 430 \)
Mean = \( 430 \div 5 = \mathbf{86} \) ✅

Q10 Outlier Effect

⭐⭐ Medium

A dataset has values: 10, 12, 11, 13, 12, 200. Which measure of center best represents the typical value, and why?

💡 Outlier = extreme value. Which measure ignores it?

📖 EXPLANATION

Mean = \( (10+12+11+13+12+200)/6 ≈ 43 \) — way too high, inflated by 200.
Median = sort: 10,11,12,12,13,200 → middle = \( (12+12)/2 = \mathbf{12} \) ✅
Median is resistant to outliers → best choice here.

UNIT 6 · GEOMETRY & TRIG

📐 Geometry & Trigonometry

SOH · CAH · TOA

📌 Sin = Opposite / Hypotenuse
📌 Cos = Adjacent / Hypotenuse
📌 Tan = Opposite / Adjacent
📌 Pythagorean identity: \( \sin^2\theta + \cos^2\theta = 1 \)
📌 Special angles: sin(30°) = 1/2, cos(60°) = 1/2, tan(45°) = 1

\( \sin^2\theta + \cos^2\theta = 1 \) | \( \tan\theta = \dfrac{\sin\theta}{\cos\theta} \)

Q11 Right Triangle Trig

⭐ Easy

In a right triangle, the leg opposite angle \( \theta \) is 5, and the hypotenuse is 13. What is \( \cos\theta \)?

💡 Find the adjacent leg using the Pythagorean theorem first!

📖 EXPLANATION

Adjacent leg: \( \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \)
\( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \mathbf{\dfrac{12}{13}} \) ✅

Q12 Pythagorean Identity

⭐⭐ Medium

If \( \sin\theta = \dfrac{3}{5} \) and \( \theta \) is in the first quadrant, what is \( \tan\theta \)?

💡 Use sin²θ + cos²θ = 1 to find cosθ first. Then tan = sin/cos.

📖 EXPLANATION

\( \cos^2\theta = 1 - (3/5)^2 = 1 - 9/25 = 16/25 \Rightarrow \cos\theta = 4/5 \)
\( \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{3/5}{4/5} = \mathbf{\dfrac{3}{4}} \) ✅

UNIT 7 · POLYNOMIALS

🔢 Polynomials & Rational Expressions

FACTOR = FIND ZEROS

📌 Factor Theorem: if \( f(a) = 0 \), then \( (x - a) \) is a factor
📌 Remainder Theorem: \( f(a) \) = remainder when divided by \( (x-a) \)
📌 FOIL: \((a+b)(a-b) = a^2 - b^2\) (difference of squares)
📌 \((a+b)^2 = a^2 + 2ab + b^2\) — DON'T forget the middle term!

Q13 Factoring

⭐ Easy

Which of the following is a factor of \( x^2 - 5x - 6 \)?

💡 Find two numbers that multiply to −6 and add to −5.

📖 EXPLANATION

Need two numbers: product = −6, sum = −5.
That's \( -6 \) and \( +1 \): \( (-6)(1)=-6 \), \( -6+1=-5 \) ✓
\( x^2 - 5x - 6 = (x-6)(x+1) \)
So \( (x+1) \) is a factor ✅

Q14 Remainder Theorem

⭐⭐ Medium

When \( p(x) = x^3 - 4x^2 + 5x - 2 \) is divided by \( (x - 2) \), what is the remainder?

💡 Remainder Theorem: just plug in x = 2 into p(x)! No long division needed.

📖 EXPLANATION

\( p(2) = 8 - 16 + 10 - 2 = \mathbf{0} \) ✅
Remainder = 0 means \( (x-2) \) is actually a factor of \( p(x) \)!

UNIT 8 · PROBABILITY & RATIOS

🎲 Probability, Ratios & Proportions

PROBABILITY = FAVORABLE / TOTAL

📌 \( P(\text{event}) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}} \)
📌 \( P(A \text{ and } B) = P(A) \cdot P(B) \) (if independent)
📌 \( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \)
📌 Complementary: \( P(\text{not } A) = 1 - P(A) \)

Q15 Basic Probability

⭐ Easy

A bag contains 4 red, 5 blue, and 3 green marbles. If one marble is picked at random, what is the probability it is NOT blue?

💡 Use complementary probability: P(not blue) = 1 − P(blue)

📖 EXPLANATION

Total = 12 marbles. P(blue) = 5/12.
P(not blue) = 1 − 5/12 = 7/12 ✅
(or directly: 4 red + 3 green = 7 non-blue out of 12)

Q16 Proportional Reasoning

⭐⭐ Medium

A recipe needs 2.5 cups of flour for 20 cookies. How many cups of flour are needed to make 56 cookies?

💡 Set up a proportion: cups/cookies = cups/cookies

📖 EXPLANATION

\( \dfrac{2.5}{20} = \dfrac{x}{56} \)
\( x = \dfrac{2.5 \times 56}{20} = \dfrac{140}{20} = \mathbf{7} \) cups ✅

UNIT 9 · ADVANCED TOPICS

🌀 Complex Numbers & Sequences

i² = −1 | CYCLE: i, −1, −i, 1

📌 \( i = \sqrt{-1} \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \) (then repeats!)
📌 Arithmetic sequence: \( a_n = a_1 + (n-1)d \) — common difference d
📌 Geometric sequence: \( a_n = a_1 \cdot r^{n-1} \) — common ratio r
📌 Sum of arithmetic: \( S_n = \frac{n}{2}(a_1 + a_n) \)

Q17 Complex Numbers

⭐⭐ Medium

Simplify: \( (3 + 2i)(1 - 4i) \)

💡 FOIL it out, then replace i² with −1. Combine real and imaginary parts.

📖 EXPLANATION

\( (3)(1) + (3)(-4i) + (2i)(1) + (2i)(-4i) \)
\( = 3 - 12i + 2i - 8i^2 \)
\( = 3 - 10i - 8(-1) \)
\( = 3 + 8 - 10i = \mathbf{11 - 10i} \) ✅

Q18 Arithmetic Sequences

⭐ Easy

The 3rd term of an arithmetic sequence is 11 and the 7th term is 27. What is the first term?

💡 Use a_n = a₁ + (n−1)d. Set up two equations to find d first.

📖 EXPLANATION

From 3rd to 7th: 4 steps, difference = \( (27-11)/4 = 4 \). So d = 4.
\( a_3 = a_1 + 2d \Rightarrow 11 = a_1 + 8 \Rightarrow a_1 = \mathbf{3} \) ✅

UNIT 10 · FINAL CHALLENGE

🏆 Mixed Review — Hard Traps!

WATCH THE TRAPS!

📌 Order of operations: PEMDAS — Parentheses, Exponents, Multiply/Divide, Add/Subtract
📌 Negative exponent: \( x^{-n} = \dfrac{1}{x^n} \)
📌 \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \) but \( \sqrt{a+b} \neq \sqrt{a} + \sqrt{b} \) ❌
📌 When multiplying inequalities by a negative: FLIP the sign!

Q19 Inequalities

⭐⭐ Medium (TRAP!)

Solve: \( -3x + 6 > 18 \)

💡 When you divide or multiply both sides by a NEGATIVE number → flip the inequality sign!

📖 EXPLANATION

\( -3x + 6 > 18 \)
\( -3x > 12 \)
Divide by −3 → FLIP sign!
\( x < -4 \) ✅

⚠️ Most students forget to flip and choose x > −4. This is the #1 inequality trap!

Q20 Word Problem Algebra

⭐⭐⭐ Hard (FINAL BOSS!)

Two trains start from cities 480 miles apart, heading toward each other. Train A travels at 60 mph and Train B at 80 mph. How many hours until they meet?

💡 Combined speed = speed A + speed B (they move toward each other!). Distance = Rate × Time.

📖 EXPLANATION

Combined speed = 60 + 80 = 140 mph
Time = Distance ÷ Speed = \( \dfrac{480}{140} = \dfrac{24}{7} = 3\tfrac{3}{7} \) hours ✅

⚠️ Trap: Some students divide by 60 or 80 alone, not the combined speed!

📋 FINAL SCORE
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