🦕 DinoAlgebra 2
All Units · 20 Essential Problems · Master Every Topic
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Unit 1
Quadratic Functions & Equations
Vertex, Factoring, Quadratic Formula, Discriminant

📚 Key Concepts

A quadratic function has the form f(x) = ax² + bx + c where a ≠ 0.
Vertex Form: f(x) = a(x − h)² + k  |  Vertex = (h, k)
Quadratic Formula: x = (−b ± √(b²−4ac)) / (2a)
Discriminant: Δ = b² − 4ac
⭐ MEMORIZE

Δ > 0 → 2 real roots  |  Δ = 0 → 1 real root (double)  |  Δ < 0 → 2 complex roots
Axis of symmetry: x = −b / (2a)

📝 Example
Solve: x² − 5x + 6 = 0
Factor: (x − 2)(x − 3) = 0
∴ x = 2 or x = 3
🦕
Hey! Always check the discriminant first. If Δ < 0, the roots are complex — don't panic, just use i = √(−1)! Now let's try some real exam questions!
1
Unit 1 · Quadratic Functions
What is the vertex of the parabola f(x) = 2(x − 3)² + 5?
2
Unit 1 · Quadratic Formula
Using the quadratic formula, what are the solutions to x² + 4x + 8 = 0?
Unit 2
Polynomials
Polynomial Operations, Factor Theorem, Remainder Theorem

📚 Key Concepts

The Remainder Theorem: when p(x) is divided by (x − a), the remainder equals p(a).
The Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0.
Rational Root Theorem: possible rational roots = ±(factors of constant term) / (factors of leading coefficient)
End behavior: determined by the degree and leading coefficient sign.
⭐ MEMORIZE

Degree n polynomial has AT MOST n real roots and exactly n complex roots (counted with multiplicity).
If (x − r) appears an even number of times, the graph TOUCHES the x-axis at r but doesn't cross.

📝 Example
Find the remainder when p(x) = x³ − 2x + 5 is divided by (x − 2).
p(2) = (2)³ − 2(2) + 5 = 8 − 4 + 5 = 9
∴ Remainder = 9
🦕
Polynomials can be sneaky! Remember: degree tells you the max number of turns in the graph. A degree-3 polynomial can have AT MOST 2 turning points!
3
Unit 2 · Remainder Theorem
When p(x) = x³ + 3x² − x − 3 is divided by (x − 1), what is the remainder?
4
Unit 2 · Polynomial Roots
The polynomial p(x) = x⁴ − 5x² + 4 has how many distinct real roots?
Unit 3
Rational Functions
Asymptotes, Domain, Holes, Graphing

📚 Key Concepts

Vertical Asymptote: values of x where denominator = 0 (after canceling common factors)
Horizontal Asymptote: compare degrees of numerator (n) and denominator (m)
  • n < m → y = 0  |  n = m → y = (leading coeff ratio)  |  n > m → no H.A. (slant instead)
⭐ MEMORIZE

A hole occurs when a factor cancels from both numerator and denominator.
To find a hole: set the canceled factor = 0, then find the y-value by plugging x into the simplified function.

📝 Example
Find the vertical asymptote(s) of f(x) = (x + 2) / (x² − 4).
Factor denominator: x² − 4 = (x − 2)(x + 2)
After canceling (x + 2): simplified = 1/(x − 2), hole at x = −2
Vertical asymptote: x = 2 only
5
Unit 3 · Rational Functions
What is the horizontal asymptote of f(x) = (3x² + 2x) / (x² − 5)?
Unit 4
Radical Functions & Complex Numbers
Radical Equations, Complex Arithmetic, Powers of i

📚 Key Concepts

i = √(−1)  |  i² = −1  |  i³ = −i  |  i⁴ = 1 (then repeats)
Complex: a + bi  |  Conjugate: a − bi
|a + bi| = √(a² + b²)
⭐ MEMORIZE

To divide complex numbers, multiply numerator AND denominator by the conjugate of the denominator.
To find iⁿ: divide n by 4 and use the remainder (0→1, 1→i, 2→−1, 3→−i).

📝 Example
Simplify i²⁷.
27 ÷ 4 = 6 remainder 3
i³ = −i  ∴ i²⁷ = −i
6
Unit 4 · Complex Numbers
What is the simplified form of (3 + 2i)(3 − 2i)?
Unit 5
Exponential & Logarithmic Functions
Log Laws, Natural Log, Solving Exponential Equations

📚 Key Concepts

logₐ(mn) = logₐm + logₐn
logₐ(m/n) = logₐm − logₐn
logₐ(mⁿ) = n·logₐm
Change of base: logₐb = ln(b)/ln(a) = log(b)/log(a)
⭐ MEMORIZE

logₐ(aˣ) = x and aˡᵒᵍₐˣ = x (inverse property)
ln(e) = 1  |  ln(1) = 0  |  log(10) = 1  |  log(1) = 0

📝 Example
Solve for x: 2ˣ⁺¹ = 16
Write 16 = 2⁴
So 2ˣ⁺¹ = 2⁴ → x + 1 = 4 → x = 3
🦕
Logarithms are just asking "what POWER do I raise the base to?" So log₂(8) = 3 because 2³ = 8. Simple!
7
Unit 5 · Logarithms
Simplify: log₃(81) − log₃(3)
8
Unit 5 · Exponential Equations
Solve for x: 3^(2x) = 27^(x−1)
Unit 6
Sequences & Series
Arithmetic, Geometric, Sigma Notation, Infinite Series

📚 Key Concepts

Arithmetic: aₙ = a₁ + (n−1)d  |  Sₙ = n(a₁+aₙ)/2 = n[2a₁+(n−1)d]/2
Geometric: aₙ = a₁·rⁿ⁻¹  |  Sₙ = a₁(1−rⁿ)/(1−r) for r ≠ 1
Infinite geometric sum: S∞ = a₁/(1−r) when |r| < 1
⭐ MEMORIZE

An infinite geometric series converges ONLY when |r| < 1.
Common ratio r = aₙ₊₁ / aₙ (any consecutive terms).

📝 Example
Find the sum of the infinite geometric series: 8 + 4 + 2 + 1 + ···
a₁ = 8, r = 4/8 = 1/2 (|r| < 1 ✓)
S∞ = 8 / (1 − 1/2) = 8 / (1/2) = 16
9
Unit 6 · Arithmetic Sequences
The 1st term of an arithmetic sequence is 7 and the common difference is −3. What is the 10th term?
10
Unit 6 · Geometric Series
What is the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54, …?
Unit 7
Trigonometry
Unit Circle, Trig Identities, Graphs, Inverse Trig

📚 Key Concepts

sin²θ + cos²θ = 1  |  tan θ = sin θ / cos θ
1 + tan²θ = sec²θ  |  1 + cot²θ = csc²θ
Amplitude = |a|, Period = 2π/|b| for y = a·sin(bx + c) + d
⭐ MEMORIZE — Unit Circle Highlights

sin(30°) = 1/2  |  cos(60°) = 1/2  |  sin(45°) = cos(45°) = √2/2
sin(60°) = cos(30°) = √3/2  |  sin(90°) = 1  |  cos(0°) = 1

📝 Example
Find the period of y = 3·sin(2x − π).
b = 2 → Period = 2π / 2 = π
(Amplitude = 3, phase shift = π/2 to the right)
🦕
SOHCAHTOA is your best friend! Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. And always remember: all students take calculus — ALL, STUDENTS, TAKE, CALCULUS — the quadrant signs!
11
Unit 7 · Trigonometry
What is the period of y = −2·cos(3x)?
12
Unit 7 · Trig Identities
Simplify: (1 − sin²θ) / cos θ
Unit 8
Conic Sections
Circles, Ellipses, Hyperbolas, Parabolas

📚 Key Concepts

Circle: (x−h)² + (y−k)² = r²
Ellipse: (x−h)²/a² + (y−k)²/b² = 1 (a > b > 0, horizontal major axis)
Hyperbola: (x−h)²/a² − (y−k)²/b² = 1 (opens left/right)
Parabola: (x−h)² = 4p(y−k) where p = distance from vertex to focus
⭐ MEMORIZE

Ellipse: c² = a² − b² (a > b) | Hyperbola: c² = a² + b²
Eccentricity: circle e=0, ellipse 0<e<1, parabola e=1, hyperbola e>1

13
Unit 8 · Conic Sections
What are the center and radius of the circle x² + y² − 6x + 4y − 3 = 0?
Unit 9
Matrices
Matrix Operations, Determinants, Inverse, Systems

📚 Key Concepts

For 2×2 matrix A = [[a,b],[c,d]]:
det(A) = ad − bc
A⁻¹ = (1/det(A)) · [[d,−b],[−c,a]] (only if det(A) ≠ 0)
⭐ MEMORIZE

Matrix multiplication AB is defined only when columns of A = rows of B.
AB ≠ BA in general (matrix multiplication is NOT commutative).
A system Ax = b has unique solution iff det(A) ≠ 0.

📝 Example
Find det(A) if A = [[5, 2], [3, 4]].
det(A) = (5)(4) − (2)(3) = 20 − 6 = 14
14
Unit 9 · Matrices
Matrix A = [[4, 1], [2, 3]]. What is det(A)?
Unit 10
Statistics & Probability
Normal Distribution, Combinations, Permutations, Binomial Theorem

📚 Key Concepts

Permutations: P(n,r) = n! / (n−r)!
Combinations: C(n,r) = n! / (r!(n−r)!)
Binomial Theorem: (a+b)ⁿ = Σ C(n,k)·aⁿ⁻ᵏ·bᵏ for k=0 to n
⭐ MEMORIZE

Permutation = ORDER matters (P for Position).
Combination = ORDER does NOT matter (C for Choose).
The (k+1)th term in (a+b)ⁿ expansion = C(n,k)·aⁿ⁻ᵏ·bᵏ

📝 Example
Find the 3rd term in the expansion of (x + 2)⁵.
3rd term: k = 2 → C(5,2)·x³·2² = 10·x³·4 = 40x³
15
Unit 10 · Binomial Theorem
What is the coefficient of x² in the expansion of (x + 3)⁴?
Mixed Review
Mixed Practice — All Units
Systems of Equations, Inverse Functions, Inequalities, Function Transformations
🦕
The final stretch! These last 5 questions mix everything together — just like on a real exam. Stay calm, apply what you know, and show those problems who's boss! 💪
16
Systems of Equations
Solve the system: 2x + y = 7 and x − y = 2. What is the value of x?
17
Inverse Functions
If f(x) = (2x + 1) / 3, what is f⁻¹(x)?
18
Function Transformations
The function g(x) = f(x − 3) + 2 represents what transformation of f(x)?
19
Solving Inequalities
Solve: |2x − 4| ≤ 6. Which interval represents the solution?
20
Combining Concepts
If log₂(x) + log₂(x − 2) = 3, what is the value of x?
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