🦕 Algebra 2 with Dino

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Welcome to Dino's Algebra 2 Adventure!

Hi! I'm Dino 🦕 and I'll guide you through 20 exam-style Algebra 2 problems covering every major topic. Review the concepts, then take the timed quiz!

📈Linear & Absolute Value

🔢Quadratic Functions

🧩Polynomials

🌿Radical Functions

➗Rational Functions

📊Exponential Functions

🪵Logarithms

🔄Sequences & Series

📐Trigonometry

🔵Conic Sections

Unit 1

Linear & Absolute Value Functions

Solving equations, inequalities, and transformations

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Dino says:

Absolute value means distance from zero — it's always non-negative! When solving |ax + b| = c (where c > 0), split into TWO cases: ax + b = c OR ax + b = −c. Easy-dino-sy! 🎉

📌 Must Know

Absolute Value Equations & Inequalities

|ax + b| = c → ax + b = c OR ax + b = −c
|ax + b| < c → −c < ax + b < c
|ax + b| > c → ax + b > c OR ax + b < −c

⭐ Key Facts to Memorize

|x| ≥ 0 always — distance is never negative
If c < 0, |expression| = c has NO solution
Vertex form of absolute value: f(x) = a|x − h| + k, vertex at (h, k)

🔍 Worked Example

Solve: |2x − 3| = 7

Case 1: 2x − 3 = 7 → 2x = 10 → x = 5
Case 2: 2x − 3 = −7 → 2x = −4 → x = −2
Answer: x = 5 or x = −2

Unit 2

Quadratic Functions

Vertex form, standard form, discriminant, quadratic formula

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Dino says:

The discriminant b² − 4ac is your crystal ball 🔮! Positive → 2 real roots, Zero → 1 real root (double), Negative → 2 complex roots. Always check this before solving!

📌 Must Know

Quadratic Formula & Discriminant

x = (−b ± √(b² − 4ac)) / 2a (Quadratic Formula)
Vertex: x = −b/(2a)
Vertex form: f(x) = a(x − h)² + k

⭐ Key Facts to Memorize

Discriminant D = b² − 4ac: D > 0 → 2 real roots; D = 0 → 1 real root; D < 0 → 2 complex roots
Sum of roots = −b/a; Product of roots = c/a
Completing the square: ax² + bx + c → a(x + b/2a)² + c − b²/(4a)

🔍 Worked Example

Find the number of real solutions of 2x² − 4x + 5 = 0.

D = (−4)² − 4(2)(5) = 16 − 40 = −24
D < 0 → No real solutions (2 complex roots)

Unit 3

Polynomials

Factor theorem, remainder theorem, end behavior, zeros

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Dino says:

The Remainder Theorem: when you divide f(x) by (x − k), the remainder equals f(k). And if f(k) = 0, then (x − k) is a factor — that's the Factor Theorem! 🦴

📌 Must Know

Factor & Remainder Theorems + End Behavior

Remainder Theorem: f(x) ÷ (x−k) → remainder = f(k)
Factor Theorem: (x−k) is a factor ↔ f(k) = 0
Rational Root Theorem: possible roots = ±(factors of constant)/(factors of leading coeff)

⭐ End Behavior Rules

Even degree, positive leading coeff → both ends go UP (↑ ↑)
Even degree, negative leading coeff → both ends go DOWN (↓ ↓)
Odd degree, positive leading coeff → left DOWN, right UP (↓ ↑)
Odd degree, negative leading coeff → left UP, right DOWN (↑ ↓)

🔍 Worked Example

Find the remainder when f(x) = x³ − 2x² + 4x − 1 is divided by (x − 2).

Remainder = f(2) = (2)³ − 2(2)² + 4(2) − 1
= 8 − 8 + 8 − 1 = 7
Remainder = 7

Unit 4

Radical & Rational Exponent Functions

Simplifying radicals, solving radical equations, domain

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Dino says:

After solving a radical equation, always check for extraneous solutions! Squaring both sides can introduce fake answers. Dino learned this the hard way 😅

📌 Must Know

Radical Equations & Rational Exponents

x^(m/n) = (ⁿ√x)^m = ⁿ√(x^m)
To solve √(f(x)) = g(x): square both sides, then CHECK answers
Domain of √f(x): f(x) ≥ 0

🔍 Worked Example

Solve: √(3x + 4) = x − 2

Square both sides: 3x + 4 = (x−2)² = x² − 4x + 4
0 = x² − 7x → x(x − 7) = 0 → x = 0 or x = 7
Check x = 0: √(4) = 2, but 0 − 2 = −2. Extraneous!
Check x = 7: √(25) = 5, and 7 − 2 = 5. ✓
x = 7

Unit 5

Rational Functions

Asymptotes, holes, domain, simplifying

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Dino says:

A hole occurs when the same factor cancels from numerator AND denominator. A vertical asymptote occurs when the denominator = 0 but the factor does NOT cancel. Different things! 🔍

📌 Must Know

Asymptotes & Holes

Vertical Asymptote: set denominator = 0 (after canceling)
Horizontal Asymptote: compare degrees of numerator (n) and denominator (d):
n < d → y = 0 | n = d → y = lead coeff ratio | n > d → none (oblique)

Unit 6

Exponential & Logarithmic Functions

Growth/decay, log properties, solving exponential equations

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Dino says:

Log rules are your best friends! log(AB) = log A + log B, log(A/B) = log A − log B, log(Aⁿ) = n·log A. Master these and exponential equations become dino-simple! 🦕

📌 Must Know

Log Properties & Change of Base

log_b(xy) = log_b(x) + log_b(y)
log_b(x/y) = log_b(x) − log_b(y)
log_b(x^n) = n · log_b(x)
Change of Base: log_b(x) = ln(x)/ln(b) = log(x)/log(b)
b^x = y ↔ log_b(y) = x

🔍 Worked Example

Solve: 3^(2x−1) = 27

27 = 3³, so: 3^(2x−1) = 3³
2x − 1 = 3 → 2x = 4 → x = 2
x = 2

Unit 7

Sequences & Series

Arithmetic, geometric, sum formulas

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Dino says:

Arithmetic sequences ADD the same number each time (common difference d). Geometric sequences MULTIPLY by the same number each time (common ratio r). Know which type you have first! 🦕

📌 Must Know

Arithmetic & Geometric Formulas

Arithmetic: a_n = a_1 + (n−1)d S_n = n(a_1 + a_n)/2
Geometric: a_n = a_1 · r^(n−1) S_n = a_1(1−r^n)/(1−r)
Infinite Geometric (|r|<1): S = a_1/(1−r)

Unit 8

Trigonometry

Unit circle, trig functions, identities, inverse trig

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Dino says:

Memorize the unit circle — it's the backbone of trig! At angle θ on the unit circle: the point is (cos θ, sin θ). Remember SOH-CAH-TOA for right triangles! 📐

📌 Must Know

Key Trig Identities & Values

sin²θ + cos²θ = 1 tan θ = sin θ/cos θ
1 + tan²θ = sec²θ 1 + cot²θ = csc²θ
sin(30°) = 1/2 sin(45°) = √2/2 sin(60°) = √3/2
cos(30°) = √3/2 cos(45°) = √2/2 cos(60°) = 1/2

Unit 9

Conic Sections

Circles, parabolas, ellipses, hyperbolas

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Dino says:

The key to conics is completing the square to get them into standard form. Then identify center, radius, vertices, foci from the equation. You've got this! 🔵

📌 Must Know

Standard Forms of Conic Sections

Circle: (x−h)² + (y−k)² = r²
Parabola (vertical): x² = 4py (focus at (0,p))
Ellipse: (x−h)²/a² + (y−k)²/b² = 1 (a > b, horizontal major axis)
Hyperbola: (x−h)²/a² − (y−k)²/b² = 1

Unit 10

Complex Numbers & Systems

Operations with complex numbers, systems of equations

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Dino says:

Remember: i = √(−1), i² = −1, i³ = −i, i⁴ = 1. The pattern repeats every 4 powers! For complex conjugates: (a+bi)(a−bi) = a² + b². 🧮

📌 Must Know

Complex Number Operations

i = √(−1), i² = −1, i³ = −i, i⁴ = 1
(a+bi) + (c+di) = (a+c) + (b+d)i
(a+bi)(c+di) = (ac−bd) + (ad+bc)i
Modulus: |a+bi| = √(a² + b²)

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Ready for the Quiz? 🦕

20 exam-style multiple choice questions. You have 40 minutes. Good luck!

📋 Complete Solutions & Explanations