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Pre-Calculus
Ultimate Self-Study Worksheet 📐
20 Core Problems · Multiple Choice · Instant Feedback
Topics: Functions · Polynomials · Trig · Logs · Conics

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📈Section 1 — Functions & Transformations
"SOLVE" = Shift · Out/In · Left/Right · Vertical · Everything
f(x±h) → LEFT/RIGHT shift | f(x)±k → UP/DOWN shift | af(x) → VERTICAL stretch | f(bx) → HORIZONTAL squeeze
Given f(x) = x², describe g(x) = 2(x−3)² + 1
→ Right 3, Up 1, Vertical stretch ×2
Vertex moves from (0,0) → (3, 1)
1
Easy
Which transformation maps f(x) = √x to g(x) = √(x+4) − 2? #transformation
Inside the function: (x+4) means shift LEFT 4 (opposite sign!).
Outside: −2 means shift DOWN 2.
KEY: Inside = opposite direction. Outside = same direction.
2
Tricky!
If f(x) = x² and g(x) = f(−x), which statement is TRUE? #reflection
f(−x) reflects over the y-axis. But f(−x) = (−x)² = x² = f(x).
So g(x) = f(x) — the parabola is symmetric, they look identical!
KEY: Even functions satisfy f(−x) = f(x) → y-axis symmetry.
3
Medium
The domain of h(x) = √(5 − 2x) is: #domain
Square roots need radicand ≥ 0.
5 − 2x ≥ 0 → 5 ≥ 2x → x ≤ 5/2 = 2.5
KEY: √(expression) → set expression ≥ 0, solve for x.
🔢Section 2 — Polynomials & Rational Functions
"ZERO-FAR" = Zeros · End behavior · Factored form · Asymptotes · Remainder theorem
Rational Root Test: possible roots = ±(factors of constant) / (factors of leading coeff)
End behavior of f(x) = −3x⁴ + 2x − 1:
Even degree + Negative leading → BOTH ends go DOWN ↓↓
Remember: look at the LEADING TERM only!
4
Easy
What is the remainder when p(x) = x³ − 4x + 6 is divided by (x − 2)? #remainder-theorem
Remainder Theorem: remainder = p(2)
p(2) = (2)³ − 4(2) + 6 = 8 − 8 + 6 = 6
KEY: Divide by (x−a) → plug in x = a to get remainder instantly!
5
Tricky!
The rational function r(x) = (x² − 9) / (x² − x − 6) has a hole (removable discontinuity) at x = #rational-functions
Factor: (x²−9) = (x+3)(x−3) and (x²−x−6) = (x−3)(x+2)
Common factor (x−3) cancels → hole at x = 3
Remaining denominator (x+2) → vertical asymptote at x = −2
KEY: HOLE = canceled factor. ASYMPTOTE = remaining denominator = 0.
6
Medium
How many positive real zeros can f(x) = x⁴ − 3x³ + x² + x − 2 have, according to Descartes' Rule? #Descartes-rule
Sign changes in f(x): +, −, +, +, −
Changes at positions: +→−, −→+, +→− = 3 sign changes
→ Possible positive zeros: 3 or 1 (subtract 2 each time)
KEY: Count sign changes in f(x). That's max positive zeros. Reduce by 2 for other possibilities.
🔵Section 3 — Trigonometry
"ASTC" = All · Sin · Tan · Cos (which are positive per quadrant I→IV)
"All Students Take Calculus" 🎓 | Unit circle: memorize 0°, 30°, 45°, 60°, 90°
sin²θ + cos²θ = 1  |  1 + tan²θ = sec²θ  |  1 + cot²θ = csc²θ
Period of y = 3sin(2x − π) + 1:
A = 3 (amplitude) · B = 2 · Period = 2π/B = 2π/2 = π
Phase shift = π/2 to the RIGHT · Midline: y = 1
7
Easy
In what quadrant is the angle θ if sin θ < 0 and cos θ > 0? #ASTC #quadrants
cos > 0 → right side → Quadrant I or IV
sin < 0 → below x-axis → Quadrant III or IV
Intersection: Quadrant IV ✓
KEY: ASTC — QI: all +, QII: sin+, QIII: tan+, QIV: cos+
8
Medium
What is the period of f(x) = tan(3x)? #period #tangent
Period of tan(bx) = π/b (NOT 2π/b like sine/cosine!)
Period = π/3
KEY: sin/cos period = 2π/b · tan/cot period = π/b (half as long!)
9
Tricky!
Simplify: (sin²x − 1) / cos x #identities
sin²x − 1 = −(1 − sin²x) = −cos²x (Pythagorean identity)
So: −cos²x / cos x = −cos x ✓
KEY: When you see (sin²x ± 1), think Pythagorean identity: sin²+cos²=1
10
Medium
arcsin(−1/2) = ? (in degrees, −90° ≤ θ ≤ 90°) #inverse-trig
arcsin outputs angles in [−90°, 90°] only.
sin(30°) = 1/2, so sin(−30°) = −1/2
→ arcsin(−1/2) = −30° ✓
KEY: arcsin range is [−π/2, π/2] · arccos is [0, π] · arctan is (−π/2, π/2)
📊Section 4 — Exponential & Logarithmic Functions
"PQRS" = Product rule · Quotient rule · Reciprocal (change of base) · Sum becomes product
log(AB) = logA + logB · log(A/B) = logA − logB · log(Aⁿ) = n·logA
logb(x) = ln(x)/ln(b)  |  bx and logb(x) are inverses  |  e ≈ 2.718
11
Easy
Solve: log₂(x) = 5 #logarithms
log₂(x) = 5 means 2⁵ = x
2⁵ = 32 → x = 32 ✓
KEY: logb(x) = y ↔ by = x (convert log to exponential form)
12
Tricky!
Expand: log₃(9x⁴/√y) #log-properties
log₃(9) = log₃(3²) = 2
log₃(x⁴) = 4log₃x
log₃(√y) = log₃(y^½) = ½log₃y → subtract (division = subtraction)
Answer: 2 + 4log₃x − ½log₃y ✓
KEY: Mult→add, Divide→subtract, Exponent→multiply out front
13
Medium
If an investment doubles using A = Pert, with r = 0.06, approx how many years to double? #exponential-growth
Set 2P = Pe0.06t → 2 = e0.06t → ln(2) = 0.06t
t = ln(2)/0.06 ≈ 0.693/0.06 ≈ 11.55 years ✓
SHORTCUT "Rule of 72": 72 ÷ interest rate% ≈ doubling time → 72/6 = 12 ✓
⭕Section 5 — Conic Sections
"CHEP" = Circle · Hyperbola · Ellipse · Parabola
Circle: (x−h)²+(y−k)²=r² | Ellipse: x²/a²+y²/b²=1 | Hyperbola: x²/a²−y²/b²=1 | Parabola: x²=4py
14
Easy
What is the center and radius of (x+3)² + (y−5)² = 49? #circle
Standard form: (x−h)²+(y−k)² = r²
(x−(−3))² + (y−5)² = 49 → h = −3, k = 5, r = √49 = 7
KEY: Signs flip! (x+3) means h = −3 · (y−5) means k = 5 · r = √(right side)
15
Tricky!
The equation 4x² − 9y² = 36 represents which conic? #conics #identification
Divide by 36: x²/9 − y²/4 = 1
This is x²/a² − y²/b² = 1 → Hyperbola, x-axis transverse (opens left/right) ✓
KEY: Both squared? Same sign→ellipse/circle, Different sign→hyperbola. One squared→parabola.
16
Medium
The foci of ellipse x²/25 + y²/9 = 1 are at: #ellipse #foci
a² = 25, b² = 9 (a > b, so major axis is x-axis)
c² = a² − b² = 25 − 9 = 16 → c = 4
Foci on x-axis: (±4, 0) ✓
KEY: c² = a² − b² for ellipse. Foci go along major axis (bigger denominator wins).
🔗Section 6 — Sequences, Systems & Miscellaneous
"SAG" = Sum formula · Arithmetic (add d) · Geometric (multiply r)
Arithmetic Sₙ = n/2·(a₁+aₙ) · Geometric Sₙ = a₁(1−rⁿ)/(1−r) · Infinite Geo S = a₁/(1−r) only if |r|<1
17
Medium
The sum of the infinite geometric series 4 + 2 + 1 + ½ + ... is: #infinite-series #geometric
r = 2/4 = 1/2 (|r| < 1 so sum exists ✓)
S∞ = a₁/(1−r) = 4/(1−½) = 4/(½) = 8 ✓
KEY: Infinite geo sum works ONLY if |r| < 1. Formula: S = first term ÷ (1 − ratio)
18
Tricky!
Using the binomial theorem, the coefficient of x³y² in the expansion of (x + y)⁵ is: #binomial-theorem
Coefficient = C(5,2) = C(n, power of y) = 5!/(2!·3!) = 10 ✓
(x+y)⁵ = ... + C(5,2)x³y² + ... = 10x³y²
KEY: C(n,k) = n! / (k!(n−k)!) · Use row 5 of Pascal's Triangle: 1 5 10 10 5 1
19
Medium
For f(x) = (2x+3)/(x−1), what is f⁻¹(x) (the inverse function)? #inverse-functions
Swap x and y: x = (2y+3)/(y−1)
Solve for y: x(y−1) = 2y+3 → xy−x = 2y+3 → xy−2y = x+3 → y(x−2) = x+3
→ y = (x+3)/(x−2) ✓
KEY: Step 1: swap x,y. Step 2: solve for y. Step 3: that's f⁻¹(x).
20
Tricky!
Which value of k makes 2x² + kx + 8 = 0 have exactly ONE real solution? #discriminant
One real solution → discriminant = 0: b²−4ac = 0
k² − 4(2)(8) = 0 → k² = 64 → k = ±8 ✓
KEY: Discriminant b²−4ac: >0 two real roots · =0 one root (tangent) · <0 no real roots