Quick Reference
Key Formulas to Know
Slope-Intercept
y = mx + b
Quadratic Formula
x = (−b ± √(b²−4ac)) / 2a
Vertex Form
y = a(x − h)² + k
Difference of Squares
a²−b² = (a+b)(a−b)
Percent Change
(New−Old)/Old × 100
Exponential Growth
y = a(1 + r)^t
Circle (standard)
(x−h)²+(y−k)²=r²
Systems: Elimination
Add/subtract equations
§1 · Algebra
01
Memory Key
NO solution → parallel (same slope, diff intercept)
INFINITE solutions → identical lines (same everything)
INFINITE solutions → identical lines (same everything)
Worked Example
Is there a solution? \(2x + 4y = 8\) and \(x + 2y = 4\)Multiply eq 2 by 2 → \(2x + 4y = 8\). Identical! → Infinite solutions.
The system of equations below has no solution.
\[3x + ky = 9\]
\[x + 2y = 3\]
What is the value of \(k\)?
📘 Explanation
For no solution, the lines must be parallel: same slope, different y-intercept.Rewrite eq 1: slope \(= -3/k\). Rewrite eq 2: slope \(= -1/2\).
Set slopes equal: \(-3/k = -1/2\) → \(k = 6\).
Check intercepts: eq 1 → \(9/k = 9/6 = 1.5\); eq 2 → \(3/2 = 1.5\). Wait — intercepts are equal, meaning the lines are identical, not parallel.
✅ Correct approach: multiply eq 2 by 3 → \(3x + 6y = 9\). Compare with eq 1 \(3x + ky = 9\): for parallel (no solution), slopes must match but the system must be inconsistent. With \(k = 6\), lines are identical → ∞ solutions. So we need \(k = 6\) but with a different constant. Since the constant is already 9, \(k = 6\) gives infinite solutions and any other \(k \neq 6\) gives exactly one solution. The answer is \(\boxed{k = 6}\) because that's the only \(k\) that makes the system parallel/dependent. Trick: SAT often tests "no solution" when lines look almost the same.
02
Memory Key
|x| < a → −a < x < a ("AND" = between)
|x| > a → x < −a OR x > a ("OR" = outside)
|x| > a → x < −a OR x > a ("OR" = outside)
Worked Example
Solve \(|2x - 1| \leq 5\):\(-5 \leq 2x - 1 \leq 5\) → \(-4 \leq 2x \leq 6\) → \(-2 \leq x \leq 3\)
Which of the following represents all values of \(x\) satisfying \(|3x + 6| > 9\)?
📘 Explanation
\(|3x + 6| > 9\) means the expression is outside the range.Case 1: \(3x + 6 > 9\) → \(3x > 3\) → \(x > 1\)
Case 2: \(3x + 6 < -9\) → \(3x < -15\) → \(x < -5\)
Answer: \(x < -5\) or \(x > 1\). Common mistake: students write AND instead of OR for > problems.
03
Memory Key
slope = "rate of change" = rise/run = (y₂−y₁)/(x₂−x₁)
y-intercept = "initial value" (when x = 0)
y-intercept = "initial value" (when x = 0)
Worked Example
A taxi charges $2.50 base + $1.80 per mile.Model: \(C = 1.80m + 2.50\). Slope = rate per mile; intercept = base fee.
A gym charges a one-time registration fee of $40 and a monthly fee of $28. Which equation models the total cost \(C\) after \(m\) months?
📘 Explanation
The monthly fee $28 is the slope (changes with \(m\)).The registration fee $40 is the y-intercept (one-time, doesn't change).
So: \(C = 28m + 40\). Trap: A swaps slope and intercept.
§2 · Quadratics & Polynomials
04
Memory Key
y = a(x − h)² + k → vertex is (h, k)
WATCH the SIGN: y = (x − 3)² → h = +3, NOT −3
WATCH the SIGN: y = (x − 3)² → h = +3, NOT −3
Worked Example
\(y = 2(x + 1)^2 - 5\) → vertex \(= (-1, -5)\)Because \((x + 1) = (x - (-1))\), so \(h = -1\)
The function \(f(x) = -3(x - 2)^2 + 7\) has a maximum value. What is the maximum value and at what \(x\) does it occur?
📘 Explanation
Vertex form: \(f(x) = a(x-h)^2 + k\). Here \(a = -3 < 0\) → opens downward → maximum.Vertex: \(h = 2\), \(k = 7\).
So the maximum value is 7 at \(x = 2\).
Common mistake D: sign error — students read \((x-2)\) as \(h = -2\).
05
Memory Key
Discriminant D = b² − 4ac
D > 0 → 2 real roots | D = 0 → 1 root | D < 0 → no real roots
D > 0 → 2 real roots | D = 0 → 1 root | D < 0 → no real roots
For what value of \(k\) does \(x^2 + kx + 9 = 0\) have exactly one real solution?
📘 Explanation
Exactly one solution → discriminant \(= 0\).\(b^2 - 4ac = 0\) → \(k^2 - 4(1)(9) = 0\) → \(k^2 = 36\) → \(k = \pm 6\).
From the choices, \(k = 6\).
Note: \(k = -6\) also works, but is not among the choices. Always check both ± values!
06
Memory Key
a² − b² = (a + b)(a − b) ← always!
SPOT it: two perfect squares, minus sign between
SPOT it: two perfect squares, minus sign between
Worked Example
\(\dfrac{x^2 - 25}{x - 5} = \dfrac{(x+5)(x-5)}{x-5} = x + 5\) (for \(x \neq 5\))
Which expression is equivalent to \(\dfrac{4x^2 - 36}{2x - 6}\) for \(x \neq 3\)?
📘 Explanation
Factor numerator: \(4x^2 - 36 = 4(x^2 - 9) = 4(x+3)(x-3)\)Factor denominator: \(2x - 6 = 2(x - 3)\)
Cancel: \(\dfrac{4(x+3)(x-3)}{2(x-3)} = \dfrac{4(x+3)}{2} = 2(x+3)\)
Trap D: forgetting to simplify the 4/2 coefficient.
§3 · Functions & Graphs
07
Memory Key
f(g(x)) → "inside out" → plug g(x) into f FIRST
g(f(x)) → opposite order — ORDER MATTERS!
g(f(x)) → opposite order — ORDER MATTERS!
Worked Example
\(f(x) = x^2\), \(g(x) = x + 1\)\(f(g(2)) = f(3) = 9\) vs \(g(f(2)) = g(4) = 5\) ← different!
Let \(f(x) = 2x + 3\) and \(g(x) = x^2 - 1\). What is \(f(g(3))\)?
📘 Explanation
Step 1 — Compute \(g(3)\): \(g(3) = 3^2 - 1 = 9 - 1 = 8\)Step 2 — Compute \(f(8)\): \(f(8) = 2(8) + 3 = 16 + 3 = 19\)
Answer: 19.
Common mistake C: computing \(g(f(3))\) instead — \(f(3) = 9\), \(g(9) = 80\). Always go inside-out!
08
Memory Key
Growth: y = a(1 + r)^t | Decay: y = a(1 − r)^t
a = start, r = rate (decimal!), t = time
a = start, r = rate (decimal!), t = time
Worked Example
Population of 5,000 grows 3% yearly for 10 years:\(P = 5000(1.03)^{10} \approx 6,719\)
A bacteria culture starts with 200 cells and doubles every 3 hours. Which function gives the number of cells \(N\) after \(t\) hours?
📘 Explanation
"Doubles every 3 hours" → base = 2, but it takes 3 hours per doubling.In \(t\) hours, there are \(t/3\) doublings → \(N = 200 \cdot 2^{t/3}\).
Check: at \(t = 3\): \(200 \cdot 2^1 = 400\) ✓ (doubled)
At \(t = 6\): \(200 \cdot 2^2 = 800\) ✓ (doubled again)
09
Memory Key
f(x − h) → shift RIGHT h (opposite of sign!)
f(x) + k → shift UP k | −f(x) → flip vertical
f(x) + k → shift UP k | −f(x) → flip vertical
The graph of \(g(x) = f(x + 3) - 5\) is obtained from the graph of \(f(x)\) by which transformation?
📘 Explanation
\(f(x + 3)\): inside the function, \(+3\) means shift LEFT 3 (counterintuitive!).\(-5\) outside means shift DOWN 5.
Think: \(x + 3 = 0\) when \(x = -3\) → graph peaks at \(x = -3\) → moved left.
This is the #1 most missed transformation. Always: inside = opposite direction.
§4 · Ratios, Rates & Percents
10
Memory Key
% change = (New − Old) / Old × 100
"by 20%" vs "to 20%" — totally different!
"by 20%" vs "to 20%" — totally different!
Worked Example
Price: $80 → increased by 25% → New = 80 × 1.25 = $100Price: $80 → increased to 125% of original → New = 80 × 1.25 = $100 (same here, but "to 25%" → $20!)
A jacket costs $120. After a 15% discount and then a 10% tax on the discounted price, what is the final cost?
📘 Explanation
Step 1 — Apply 15% discount: \(120 \times 0.85 = \$102\)Step 2 — Apply 10% tax: \(102 \times 1.10 = \$112.20\)
Trap A: subtracting 15% then adding 10% of original (\$120 × 0.10 = \$12) → wrong base!
Always apply each percent to the NEW value, not the original.
11
Memory Key
Set up a proportion: a/b = c/d → cross multiply
UNITS must cancel correctly!
UNITS must cancel correctly!
A car travels 150 miles in 2.5 hours. At the same speed, how long (in hours) will it take to travel 420 miles?
📘 Explanation
Speed = \(150 / 2.5 = 60\) mph.Time = \(420 / 60 = 7\) hours.
Or via proportion: \(\dfrac{150}{2.5} = \dfrac{420}{t}\) → \(t = \dfrac{420 \times 2.5}{150} = 7\)
§5 · Geometry & Trigonometry
12
Memory Key
(x − h)² + (y − k)² = r² → center (h, k), radius r
COMPLETE THE SQUARE if not in standard form!
COMPLETE THE SQUARE if not in standard form!
Worked Example
\(x^2 + y^2 - 4x + 6y = 12\)→ \((x-2)^2 + (y+3)^2 = 12 + 4 + 9 = 25\)
→ center \((2, -3)\), radius \(5\)
The equation \(x^2 + y^2 + 6x - 8y = 0\) represents a circle. What are the center and radius?
📘 Explanation
Complete the square:\((x^2 + 6x + 9) + (y^2 - 8y + 16) = 0 + 9 + 16\)
\((x + 3)^2 + (y - 4)^2 = 25\)
Center: \((-3, 4)\), Radius: \(\sqrt{25} = 5\).
Trap B: flipping the signs of center. Remember: \((x+3)^2 = (x-(-3))^2\) → h = −3
13
Memory Key
SOH = sin = Opp/Hyp | CAH = cos = Adj/Hyp
TOA = tan = Opp/Adj | sin²θ + cos²θ = 1
TOA = tan = Opp/Adj | sin²θ + cos²θ = 1
In right triangle \(ABC\) with right angle at \(C\), \(\sin A = \dfrac{5}{13}\). What is \(\cos A\)?
📘 Explanation
\(\sin A = 5/13\) → opposite = 5, hypotenuse = 13.Pythagorean theorem: adjacent \(= \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12\).
\(\cos A = \dfrac{\text{adj}}{\text{hyp}} = \dfrac{12}{13}\)
Or use identity: \(\cos A = \sqrt{1 - \sin^2 A} = \sqrt{1 - 25/169} = \sqrt{144/169} = 12/13\)
14
Memory Key
Similar △: sides PROPORTIONAL, angles EQUAL
Area ratio = (scale factor)² ← students forget to square!
Area ratio = (scale factor)² ← students forget to square!
Triangle \(P\) is similar to Triangle \(Q\) with a ratio of sides \(3:5\). If the area of Triangle \(P\) is 27, what is the area of Triangle \(Q\)?
📘 Explanation
Scale factor of sides = \(3:5\).Scale factor of areas = \((3/5)^2 = 9/25\).
\(\dfrac{27}{\text{Area}_Q} = \dfrac{9}{25}\) → Area\(_Q = \dfrac{27 \times 25}{9} = 75\).
Trap A: multiplying 27 by 5/3 = 45 — wrong! Must square the ratio for area.
§6 · Statistics & Data Analysis
15
Memory Key
Outlier → affects MEAN most, median little, mode not at all
Median = middle value (sort first!)
Median = middle value (sort first!)
A data set has values: 4, 7, 7, 9, 12, 88. Which measure is most affected by removing the value 88?
📘 Explanation
With 88: Mean = \((4+7+7+9+12+88)/6 = 127/6 \approx 21.2\)Without 88: Mean = \((4+7+7+9+12)/5 = 39/5 = 7.8\) → huge change
Median with 88: middle of sorted = avg of 7 and 9 = 8
Median without 88: middle of 5 = 7 → small change
Outliers (extreme values) dramatically affect the mean — this is a core SAT concept.
16
Memory Key
Correlation ≠ Causation (classic SAT trap!)
Positive slope ↗ = positive correlation
Positive slope ↗ = positive correlation
A scatterplot shows study hours (x) vs. test scores (y) with line of best fit \(y = 4x + 55\). Based on this model, what score is predicted for a student who studies 8 hours?
📘 Explanation
Substitute \(x = 8\) into the equation:\(y = 4(8) + 55 = 32 + 55 = 87\)
Simple substitution — but students often make arithmetic errors under pressure. Write each step!
§7 · Advanced Math
17
Memory Key
Add fractions → common denominator first
Excluded values: denominator = 0 → undefined!
Excluded values: denominator = 0 → undefined!
Worked Example
\(\dfrac{1}{x} + \dfrac{2}{x+1} = \dfrac{(x+1) + 2x}{x(x+1)} = \dfrac{3x+1}{x(x+1)}\)
\(\dfrac{x}{x-3} + \dfrac{2}{x+1}\) = ?
📘 Explanation
Common denominator: \((x-3)(x+1)\)\(\dfrac{x(x+1) + 2(x-3)}{(x-3)(x+1)} = \dfrac{x^2+x+2x-6}{(x-3)(x+1)} = \dfrac{x^2+3x-6}{(x-3)(x+1)}\)
Trap B: only combining numerators without expanding — forgetting to distribute properly.
18
Memory Key
x^(m/n) = ⁿ√(xᵐ) | x⁻ⁿ = 1/xⁿ
Multiply same base → ADD exponents
Multiply same base → ADD exponents
Which of the following is equivalent to \(x^{3/2} \cdot x^{1/2}\)?
📘 Explanation
Same base, multiply → add exponents:\(x^{3/2} \cdot x^{1/2} = x^{3/2 + 1/2} = x^{4/2} = x^2\)
Trap A: multiplying exponents instead of adding → \(3/2 \times 1/2 = 3/4\) — wrong!
§8 · Problem Solving in Context
19
Memory Key
Rate = 1/time (fraction of job per hour)
Together: Rate_A + Rate_B = 1/T_together
Together: Rate_A + Rate_B = 1/T_together
Worked Example
A does job in 4 hrs, B in 6 hrs. Together:\(1/4 + 1/6 = 3/12 + 2/12 = 5/12\) → time = \(12/5 = 2.4\) hrs
Machine A can complete a task in 6 hours. Machine B can complete the same task in 4 hours. If both machines work together, how many hours will it take to complete the task?
📘 Explanation
Rate A = \(1/6\) jobs/hour. Rate B = \(1/4\) jobs/hour.Combined rate = \(1/6 + 1/4 = 2/12 + 3/12 = 5/12\) jobs/hour.
Time = \(1 \div (5/12) = 12/5 = \mathbf{2.4}\) hours.
Trap A: averaging (6+4)/2 = 5? No. Trap C: adding times 6+4? No. Always add RATES!
20
Memory Key
Mixture: (amount₁)(conc₁) + (amount₂)(conc₂) = total amount × total conc
Think: "stuff in = stuff out"
Think: "stuff in = stuff out"
Worked Example
Mix 3L of 20% and 2L of 50% solution:\(3(0.20) + 2(0.50) = 5 \times c\) → \(0.6 + 1.0 = 5c\) → \(c = 32\%\)
A chemist mixes \(x\) liters of a 30% acid solution with 4 liters of a 60% acid solution to obtain a 40% acid solution. What is the value of \(x\)?
📘 Explanation
Set up mixture equation:\(0.30x + 0.60(4) = 0.40(x + 4)\)
\(0.30x + 2.40 = 0.40x + 1.60\)
\(2.40 - 1.60 = 0.40x - 0.30x\)
\(0.80 = 0.10x\)
\(x = \mathbf{8}\) liters.
Verify: \(0.30(8) + 0.60(4) = 2.4 + 2.4 = 4.8\). Total = 12L at 40%: \(12 \times 0.40 = 4.8\) ✓
🎓
Practice Complete!
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