Unit 1 · Linear Equations & Systems
Linear Equations
🔑
Memory Key
MOVE & SOLVE: variables → left, numbers → right
Subtract same term from both sides → isolate \(x\)
Quick Example
\(5x - 3 = 3x + 7\) → subtract \(3x\): \(2x - 3 = 7\) → add 3: \(2x = 10\) → \(x = 5\)
💡 Explanation
\(3x - 7 = 2x + 5\)
Subtract \(2x\) from both sides: \(x - 7 = 5\)
Add 7: \(x = 12\) ✓
Tricky part: Many students forget to move ALL terms. Always get variable alone on one side first.
Systems of Equations
🔑
Memory Key
INFINITE SOLUTIONS = SAME LINE:
One equation must be a multiple of the other
→ check ratio: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)
💡 Explanation
Divide the first equation by 2: \(x + 2y = 5\)
For infinitely many solutions, equations must be identical, so \(k = 5\).
Common mistake: Choosing \(k=10\) — that's the original RHS before dividing.
Linear Equations
🔑
Memory Key
WORD → EQUATION:
"flat fee" = constant (b), "per hour" = rate (m)
→ \(y = mx + b\) → solve for \(x\)
💡 Explanation
Equation: \(2h + 3 = 13\)
Subtract 3: \(2h = 10\)
Divide by 2: \(h = 5\) hours ✓
Unit 2 · Quadratic Equations
Quadratics
🔑
Memory Key
FACTOR TRICK — find two numbers that:
• Multiply to \(c\) (last number)
• Add to \(b\) (middle number)
→ \((x-p)(x-q)=0\) → \(x=p\) or \(x=q\)
Quick Example
\(x^2 - 7x + 12 = 0\): need × = 12, + = −7 → (−3)(−4) ✓ → \((x-3)(x-4)=0\) → \(x=3,4\)
💡 Explanation
Need two numbers: × = 6, + = −5 → that's −2 and −3
\((x-2)(x-3) = 0\) → \(x = 2\) or \(x = 3\) ✓
Tricky part: Don't confuse signs — the factors are \((x-2)\) not \((x+2)\).
Quadratics · Vertex
🔑
Memory Key
VERTEX x-coordinate: \(x = -\dfrac{b}{2a}\)
Then plug back in to find \(y\)
Or complete the square → \(y = (x-h)^2 + k\)
💡 Explanation
\(x = -\frac{-6}{2(1)} = \frac{6}{2} = 3\)
\(y = 3^2 - 6(3) + 8 = 9 - 18 + 8 = -1\)
Vertex: \((3, -1)\) ✓
Tricky part: Many students get \(y\) wrong — always substitute back in!
Quadratics · Discriminant
🔑
Memory Key
DISCRIMINANT \(\Delta = b^2 - 4ac\):
• \(\Delta > 0\) → 2 real solutions
• \(\Delta = 0\) → 1 real solution
• \(\Delta < 0\) → NO real solutions
💡 Explanation
\(\Delta = 3^2 - 4(2)(5) = 9 - 40 = -31\)
Since \(\Delta < 0\), no real solutions. ✓
Common mistake: Forgetting to multiply \(4ac\) fully — \(4 \times 2 \times 5 = 40\), not 20!
Functions
🔑
Memory Key
PLUG & CHUG: replace every \(x\) with the value
Negative input? Use parentheses: \((-1)^2 = 1\), NOT \(-1\)
💡 Explanation
\(f(-1) = 3(-1)^2 - 2(-1) + 1 = 3(1) + 2 + 1 = 6\) ✓
Tricky part: \((-1)^2 = +1\) and \(-2 \times (-1) = +2\). Both negatives flip to positive!
Functions · Transformations
🔑
Memory Key
TRANSFORMATION RULES (counterintuitive!):
\(f(x-h)\) → shift RIGHT \(h\) units (minus = right!)
\(f(x)+k\) → shift UP \(k\) units
💡 Explanation
\(f(x-3)\): shift RIGHT 3 (the sign inside is opposite to the direction!)
\(+2\) outside: shift UP 2
→ Right 3, Up 2 ✓
Most common mistake on SAT: Thinking \((x-3)\) moves LEFT. Remember: inside subtraction = move RIGHT.
Functions · Composition
🔑
Memory Key
COMPOSITION = INSIDE OUT:
Step 1: Compute the INNER function first
Step 2: Use that result as input for OUTER function
💡 Explanation
Step 1: \(g(3) = 3^2 = 9\)
Step 2: \(f(9) = 2(9) + 1 = 19\) ✓
Tricky part: \(f(g(3))\) means \(g\) first, then \(f\). Students often reverse the order!
Unit 4 · Exponents, Radicals & Polynomials
Exponents
🔑
Memory Key
NEGATIVE EXPONENT = FLIP:
\(x^{-n} = \dfrac{1}{x^n}\)
Never equals a negative number!
💡 Explanation
Negative exponent means reciprocal: \(x^{-3} = \frac{1}{x^3}\) ✓
Common mistake: Writing \(-x^3\) — a negative exponent does NOT make the base negative!
Polynomials
🔑
Memory Key
DISTRIBUTE EVERY TERM:
Multiply each term in first bracket by each term in second
Then collect LIKE TERMS (same power of \(x\))
💡 Explanation
\((2x^2)(x) = 2x^3\), \((2x^2)(2) = 4x^2\)
\((3x)(x) = 3x^2\), \((3x)(2) = 6x\)
\((-1)(x) = -x\), \((-1)(2) = -2\)
\(x^2\) terms: \(4x^2 + 3x^2 = 7x^2\) → coefficient = 7 ✓
Radicals
🔑
Memory Key
RADICAL DIVISION:
\(\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}\)
Simplify under one radical sign!
💡 Explanation
\(\frac{\sqrt{75}}{\sqrt{3}} = \sqrt{\frac{75}{3}} = \sqrt{25} = 5\) ✓
Note: \(\sqrt{25} = 5\) exactly (not approximate). Always check if answer simplifies to whole number!
Unit 5 · Statistics & Data Analysis
Statistics
🔑
Memory Key
MEDIAN = MIDDLE VALUE:
Step 1: Sort the data (smallest → largest)
Step 2: Cross off equal numbers from both ends
Step 3: Middle = median
💡 Explanation
Sorted: 4, 7, 7, 9, 13 → Middle (3rd of 5) = 7 ✓
Common mistake: Calculating the mean (average) instead of median.
Mean = (4+7+7+9+13)/5 = 8, but median = 7.
Statistics · Percent
🔑
Memory Key
% CHANGE ≠ CANCEL OUT:
Use multipliers: ×1.2 then ×0.8 = ×0.96
Result: 4% decrease (never 0%!)
💡 Explanation
Start: $100 → ×1.2 = $120 → ×0.8 = $96
Net change = $96 - $100 = −4% ✓
Classic SAT trap: +20% then −20% does NOT cancel out. Always use a starting value like 100!
Statistics · Probability
🔑
Memory Key
COMPLEMENT RULE:
\(P(\text{not A}) = 1 - P(A)\)
Total probability always = 1
💡 Explanation
P(red) = 3/10 → P(not red) = 1 − 3/10 = 7/10 ✓
Or directly: 7 blue out of 10 total = 7/10 ✓
Unit 6 · Geometry & Trigonometry
Geometry · Circles
🔑
Memory Key
CIRCLE FORMULAS:
Area: \(A = \pi r^2\) (r squared!)
Circumference: \(C = 2\pi r\)
Diameter: \(d = 2r\)
💡 Explanation
\(A = \pi r^2 = \pi(5)^2 = 25\pi\) ✓
Common mistake: Using diameter instead of radius, getting \(\pi(10)^2 = 100\pi\).
Always check: is the given value radius or diameter?
Geometry · Triangles
🔑
Memory Key
PYTHAGOREAN THEOREM: \(a^2 + b^2 = c^2\)
Memorize common triples: 3-4-5, 6-8-10, 5-12-13
Spot multiples: 6-8-10 = 2×(3-4-5)
💡 Explanation
\(c^2 = 6^2 + 8^2 = 36 + 64 = 100\) → \(c = \sqrt{100} = 10\) ✓
Recognize 6-8-10 = 2×(3-4-5) triple — saves time on SAT!
Trigonometry · SOHCAHTOA
🔑
Memory Key
SOH-CAH-TOA:
• \(\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
• \(\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
• \(\tan = \frac{\text{Opposite}}{\text{Adjacent}}\)
💡 Explanation
SOH: \(\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{4}{5}\) ✓
Adjacent = \(\sqrt{5^2 - 4^2} = \sqrt{9} = 3\)
So: \(\cos\theta = \frac{3}{5}\), \(\tan\theta = \frac{4}{3}\)
Geometry · Volume
🔑
Memory Key
CYLINDER VOLUME:
\(V = \pi r^2 h\) = (Circle area) × height
Think: stack of circles on top of each other!
💡 Explanation
\(V = \pi r^2 h = \pi(3)^2(4) = \pi(9)(4) = 36\pi\) ✓
Mistake to avoid: Squaring radius FIRST, then multiply by h. Never do \((3 \times 4)^2\)!
Mixed · Challenge
🔑
Memory Key
SYSTEM FROM POINTS:
Plug each point into equation → get 3 equations
Use (0,y) first: c = y₀ (easiest!)
Then solve remaining system for a and b
💡 Explanation
From (0,3): \(c = 3\)
From (1,6): \(a + b + 3 = 6\) → \(a + b = 3\)
From (2,11): \(4a + 2b + 3 = 11\) → \(4a + 2b = 8\) → \(2a + b = 4\)
Subtract: \((2a + b) - (a + b) = 4 - 3\) → \(a = 1\) ✓
Then \(b = 2\), so \(f(x) = x^2 + 2x + 3\).
0/20
Questions Correct
Keep going!