Pre-Calculus · All Units · Exam Style

Master Quiz

20 carefully crafted multiple-choice problems spanning every core concept. College-entrance difficulty.

Functions Polynomials Exponentials Trigonometry Conics Sequences Limits

Core Concepts

U1 Functions

  • Domain & Range (interval notation)
  • Function composition: (f∘g)(x)
  • Inverse functions: f⁻¹(x)
  • Even/Odd: f(−x) = ±f(x)
  • Transformations: shifts, stretches, reflections

U2 Polynomials & Rationals

  • Factor Theorem: f(a)=0 ↔ (x−a) is factor
  • Rational Root Theorem: ±p/q
  • End behavior by leading term
  • Vertical/Horizontal asymptotes
  • Multiplicity of zeros

U3 Exp & Log

  • logb(xy) = logbx + logby
  • logb(x/y) = logbx − logby
  • logb(xⁿ) = n·logbx
  • Change of base: logbx = ln x / ln b
  • Compound: A = P(1 + r/n)^(nt)

U4 Trigonometry

  • sin²θ + cos²θ = 1 (Pythagorean)
  • Unit circle: 30°, 45°, 60° exact values
  • sin(A±B), cos(A±B) addition formulas
  • Double angle: sin2θ = 2sinθcosθ
  • Period, amplitude, phase shift

U5 Conics

  • Circle: (x−h)²+(y−k)²=r²
  • Ellipse: x²/a²+y²/b²=1
  • Parabola: y=ax²+bx+c; vertex (−b/2a, f(−b/2a))
  • Hyperbola: x²/a²−y²/b²=1

U6 Sequences & Series

  • Arithmetic: aₙ = a₁ + (n−1)d; S = n/2·(a₁+aₙ)
  • Geometric: aₙ = a₁·rⁿ⁻¹; S = a₁(1−rⁿ)/(1−r)
  • Infinite geo: S∞ = a₁/(1−r), |r|<1
  • Binomial Theorem: (a+b)ⁿ = Σ C(n,k)aⁿ⁻ᵏbᵏ

U7 Vectors & Polar

  • |v| = √(x²+y²)
  • Dot product: a·b = |a||b|cosθ
  • Polar: x=rcosθ, y=rsinθ
  • r² = x²+y², tanθ = y/x

U8 Limits & Intro Calc

  • lim(x→a) f(x) exists if L⁺ = L⁻
  • lim(x→∞) 1/xⁿ = 0
  • Squeeze Theorem
  • Continuity: f(a) defined, limit exists, equal

📌 Key Formulas to Memorize

Quad: x = (−b ± √(b²−4ac)) / 2a
sin30°: 1/2  |  cos30°: √3/2
sin45°: √2/2  |  cos60°: 1/2
e≈2.718  |  ln(e)=1
Law of Cos: c²=a²+b²−2ab·cosC
Pascal row n=4: 1 4 6 4 1

Exam Problems

1
Functions — Domain & Range
Medium
What is the domain of \( f(x) = \dfrac{\sqrt{x+3}}{x-2} \)?
2
Functions — Composition
Medium
If \( f(x)=2x+1 \) and \( g(x)=x^2-3 \), what is \( (f \circ g)(2) \)?
3
Polynomials — Zeros & Factoring
Hard
Which of the following is a factor of \( p(x)=x^3-2x^2-5x+6 \)?
4
Exponential & Logarithm
Medium
Solve for \( x \): \( \log_2(x+4) + \log_2(x-4) = 5 \)
5
Trigonometry — Unit Circle
Easy
What is the exact value of \( \cos\!\left(\dfrac{5\pi}{6}\right) \)?
6
Trigonometry — Identities
Medium
Simplify: \( \dfrac{\sin^2\theta}{1-\cos\theta} \)
7
Rational Functions — Asymptotes
Medium
Find the horizontal asymptote of \( f(x) = \dfrac{3x^2+1}{x^2-4} \).
8
Sequences — Geometric Series
Medium
What is the sum of the infinite geometric series \( 8 + 4 + 2 + 1 + \cdots \)?
9
Conics — Parabola
Medium
The vertex of \( f(x) = 2x^2 - 8x + 5 \) is located at:
10
Trigonometry — Double Angle
Hard
If \( \sin\theta = \dfrac{3}{5} \) and \( \theta \) is in Quadrant I, find \( \sin(2\theta) \).
11
Exponential — Growth/Decay
Medium
A population doubles every 12 years. Starting from 500, which expression gives the population after \(t\) years?
12
Arithmetic Sequences
Easy
The 5th term of an arithmetic sequence is 17 and the common difference is 4. What is the first term?
13
Conics — Ellipse
Hard
The ellipse \( \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1 \) has its foci at:
14
Limits — Indeterminate Forms
Hard
Evaluate: \( \displaystyle\lim_{x \to 3} \dfrac{x^2 - 9}{x - 3} \)
15
Binomial Theorem
Hard
What is the coefficient of \( x^2 \) in the expansion of \( (x+3)^5 \)?
16
Inverse Functions
Medium
If \( f(x) = \dfrac{2x+1}{x-3} \), find \( f^{-1}(x) \).
17
Polar Coordinates
Medium
Convert the polar point \( \left(4,\, \dfrac{\pi}{3}\right) \) to rectangular coordinates.
18
Logarithm — Properties
Easy
Simplify: \( \log_3 27 - \log_3 3 \)
19
Vectors — Dot Product
Hard
Find the angle between vectors \( \mathbf{u} = \langle 1, 0 \rangle \) and \( \mathbf{v} = \langle 1, 1 \rangle \).
20
Sinusoidal Functions
Hard
The function \( f(x) = 3\sin\!\left(2x - \dfrac{\pi}{4}\right) + 1 \) has period equal to:
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Answer Key & Solutions