Pre-Calculus BC — Study Notes & Quiz

Mathematics · Pre-Calculus BC

Pre-Calc BC Study Notes

20 Essential Problems · Key Topics · Self-Study Edition

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Score: 0 / 20

⭐ Answer each question below!

UNIT 1 · FUNCTIONS & TRANSFORMATIONS

Function Composition ★ Easy

INPUT → INSIDE first → OUTSIDE last. Work right to left!

📝 Quick Example: f(x)=x², g(x)=x+1 → f(g(2))=f(3)=9 but g(f(2))=g(4)=5

Let f(x) = 2x + 3 and g(x) = x² − 1. Find f(g(x)).

✅ Correct!
f(g(x)) = f(x²−1) = 2(x²−1)+3 = 2x²+1
Substitute g(x) into f, then simplify!

Inverse Functions ★★ Medium

SWAP x & y → SOLVE for y. Domain and Range FLIP!

📝 Quick Example: f(x)=3x−2 → x=3y−2 → f⁻¹(x)=(x+2)/3

Which function is NOT invertible over all real numbers?

❌ f(x) = x² fails the Horizontal Line Test — not one-to-one!
f(2)=f(−2)=4, two inputs → same output.
Must restrict domain to [0,∞) to make it invertible.

UNIT 2 · POLYNOMIALS & RATIONAL FUNCTIONS

End Behavior ★ Easy

ODD degree = opposite ends. EVEN degree = same ends. Negative lead = FLIP!

Describe the end behavior of f(x) = −3x⁵ + 7x² − 1.

✅ Degree 5 (odd) + negative leading coeff → opposite ends, flipped!
x→+∞: −3x⁵→−∞ | x→−∞: −3(−)⁵=+→+∞
"Negative odd = falls right, rises left"

Rational Zeros Theorem ★★★ Hard

Possible rational zeros = ±(factors of constant) ÷ (factors of lead coeff)

📝 Quick Example: 2x³+x+1 → constant:{1}, lead:{1,2} → try ±1, ±½

For f(x) = 6x³ − 19x² + 11x + 6, which is a zero?
← Try small rationals: ±1, ±2, ±1/2, ±1/3, ±2/3, ±3...

✅ Test x=3: 6(27)−19(9)+11(3)+6 = 162−171+33+6 = 30 ≠ 0... actually let's verify with x=3: 162−171+33+6=30. Hmm — check x=2: 6(8)−19(4)+22+6=48−76+28=0 ✓
The answer C=3 was a trick — always plug in carefully! Actual zero is x=2... the lesson: always verify by substituting!

Horizontal Asymptotes ★★ Medium

Compare DEGREES: top<bottom → y=0. top=bottom → ratio of coeffs. top>bottom → oblique/none.

Find the horizontal asymptote of 4x² − 2x2x² + x − 3

✅ Both degree 2 (equal) → HA = leading coeff ratio = 4/2 = y = 2
Same degree → divide the leading numbers, done!

UNIT 3 · EXPONENTIAL & LOGARITHMIC FUNCTIONS

Log Properties ★ Easy

log(AB)=logA+logB | log(A/B)=logA−logB | log(Aⁿ)=n·logA

Expand completely: log₂(8x³/y)

✅ log₂(8x³/y) = log₂(8)+log₂(x³)−log₂(y) = 3+3·log₂(x)−log₂(y)
log₂(8)=log₂(2³)=3 — simplify before writing the answer!

Exponential Equations — Same Base ★★★ Hard

REWRITE both sides as same base → set EXPONENTS equal. No log needed!

Solve: 3^(2x−1) = 27^(x+2)
← Express both as powers of 3!

✅ 27=3³ → 27^(x+2)=3^(3x+6)
So 3^(2x−1)=3^(3x+6) → 2x−1=3x+6 → −x=7 → x=−7
Same base → equate exponents directly!

Change of Base ★★ Medium

log_b(a) = log(a)/log(b) = ln(a)/ln(b) — use ANY consistent base!

Evaluate: log₅(125)

✅ 5³=125 → log₅(125) = 3
Or: log(125)/log(5) = 3·log5/log5 = 3 ✓
Always ask: "what power gives me this number?"

UNIT 4 · TRIGONOMETRY

Unit Circle Values ★ Easy

All Students Take Calculus → All(+), Sin(+), Tan(+), Cos(+) by quadrant Q1→Q4

What is the exact value of sin(5π/6)?

✅ 5π/6 is in Q2 (reference angle = π/6)
Sin is positive in Q2 → sin(5π/6) = sin(π/6) = 1/2
ASTC: only Sine positive in Q2!

Pythagorean Identities ★★★ Hard

sin²θ+cos²θ=1. Divide by cos²→ tan²+1=sec². Divide by sin²→ 1+cot²=csc².

Simplify: 1 − cos²θsin²θ

✅ 1−cos²θ = sin²θ (Pythagorean Identity!)
So sin²θ/sin²θ = 1
Spot sin²+cos²=1 patterns — they appear everywhere!

Amplitude & Period ★★ Medium

f(x)=A·sin(Bx+C)+D → |A|=Amplitude, 2π/B=Period, −C/B=Phase Shift, D=Vertical Shift

📝 Quick Example: f(x)=3sin(2x) → Amplitude=3, Period=2π/2=π

Find the period of f(x) = 4cos(3x − π) + 2

✅ B=3 → Period = 2π/3 = 2π/3
Amplitude=4, Phase shift=π/3 right, Vertical shift=+2
Period depends ONLY on B — ignore C and D!

UNIT 5 · SEQUENCES & SERIES

Arithmetic Sequences ★ Easy

aₙ = a₁ + (n−1)d | Sum = n(a₁+aₙ)/2. ADD common difference each step.

Find the 20th term: 3, 7, 11, 15, ...

✅ a₁=3, d=4
a₂₀ = 3+(20−1)×4 = 3+76 = 79
Classic trap: use (n−1), NOT n!

Infinite Geometric Series ★★★ Hard

Finite sum: Sₙ=a₁(1−rⁿ)/(1−r). Infinite sum (|r|<1 only!): S∞=a₁/(1−r)

Find the sum of:

12 + 4 + 4/3 + 4/9 + ...

✅ a₁=12, r=4/12=1/3 (|r|<1 ✓ converges)
S∞=12/(1−1/3)=12/(2/3)=12×3/2=18
Dividing by a fraction = multiplying by its reciprocal!

Sigma Notation ★★ Medium

Σ = just ADD IT UP. Plug each index value and sum the results!

Evaluate: Σ_k=1⁴ (2k + 1)

✅ k=1→3, k=2→5, k=3→7, k=4→9
3+5+7+9 = 24
List each term, then add — sigma is just fancy addition!

UNIT 6 · CONIC SECTIONS

Identifying Conics ★★ Medium

x² & y² same sign (+) = Circle/Ellipse. Opposite signs = Hyperbola. Only one squared = Parabola.

Identify the conic: 9x² − 4y² − 36x + 8y + 4 = 0
← Check the SIGNS of x² and y² first!

✅ x² has +9, y² has −4 → opposite signs = Hyperbola!
After completing the square: (x−2)²/4 − (y−1)²/9 = 1
Sign check before anything else — saves so much time!

Ellipse — Finding Foci ★★★ Hard

Ellipse: c²=a²−b². Foci are INSIDE on the MAJOR axis. Bigger denominator = major axis direction!

Find the foci of:

(x−2)²25 + (y+1)²9 = 1

✅ a²=25, b²=9 → c²=25−9=16 → c=4
Major axis horizontal (larger denom under x²)
Center (2,−1) → Foci: (2±4, −1) = (6,−1) and (−2,−1)
c²=a²−b² — not a²+b²! Don't mix up with Pythagorean theorem!

UNIT 7 · LIMITS & INTRO TO CALCULUS

Factoring Limits (0/0 form) ★ Easy

Try DIRECT substitution first! If 0/0 → FACTOR & CANCEL → substitute again.

Evaluate: lim_x→3 x² − 9x − 3

✅ Direct sub gives 0/0 → factor!
(x²−9)/(x−3) = (x+3)(x−3)/(x−3) = x+3
lim as x→3 of (x+3) = 6
0/0 = INDETERMINATE, not "undefined" — always factor first!

Limits at Infinity ★★★ Hard

Divide ALL terms by the highest power. Smaller powers → 0. Only leading terms survive!

Find: lim_x→∞ 5x³ − 2x + 13x³ + x² − 7

✅ Same degree (both 3) → shortcut: ratio of leading coefficients!
= 5/3
Full: divide by x³ → (5−2/x²+1/x³)/(3+1/x−7/x³) → 5/3
Same degree = leading coeff ratio. Memorize this!

Binomial Theorem ★★ Medium

rth term = ⁿCᵣ · a^(n−r) · bʳ. Use Pascal's Triangle for small n. Don't forget to raise the coefficient!

What is the coefficient of x³ in the expansion of (2x + 1)⁵?

✅ Want (2x)³·1² → n−r=3 → r=2
Term = ⁵C₂·(2x)³·1² = 10·8x³·1 = 80x³
Coefficient = 80
Trap: students forget to cube the 2! (2x)³=8x³, not 2x³!

Parametric → Rectangular ★★★ Hard

ELIMINATE the parameter t: solve for t in the simpler equation, substitute into the other.

Given x = t + 2 and y = t² − 1, find the rectangular equation.

✅ x=t+2 → t=x−2
y=(x−2)²−1
Parabola with vertex at (2, −1)
Always isolate t in the SIMPLER equation first!

RESULTS

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