🦕 Algebra 2 with Dino | Interactive Math Workbook

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Unit 1

Functions & Transformations

DINO SAYS 🦕

Functions are like a dino's menu — each input gets exactly ONE output! Let's roar through transformations!

🔑 Key Concept: Function Transformations

For a parent function \(f(x)\), the transformed form \(y = a \cdot f(b(x-h)) + k\) shifts the graph:
h = horizontal shift, k = vertical shift, a = vertical stretch/flip, b = horizontal stretch/compress

⚡ Memorize These!

Vertical Shift Up k

\(y = f(x) + k\)

Reflection over x-axis

\(y = -f(x)\)

Horizontal Shift Right h

\(y = f(x - h)\)

📝 Example

Describe the transformation: \(y = -2(x+3)^2 - 1\) from \(y = x^2\)

→ \(a = -2\): vertical stretch by 2, reflected over x-axis

→ \(h = -3\): shift left 3 units

→ \(k = -1\): shift down 1 unit

✓ Vertex at (-3, -1), opens downward

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Unit 2

Quadratic Functions

DINO SAYS 🦕

Quadratics make a parabola — like a dino's tail arc! The Quadratic Formula solves ANY quadratic!

🔑 Quadratic Forms

Standard: \(y = ax^2 + bx + c\)
Vertex: \(y = a(x-h)^2 + k\), vertex at \((h, k)\)
Factored: \(y = a(x-r_1)(x-r_2)\), roots at \(r_1, r_2\)

⚡ Must-Know Formulas

Quadratic Formula

\(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Discriminant

\(\Delta = b^2 - 4ac\) → \(\Delta>0\): 2 real roots · \(\Delta=0\): 1 real root · \(\Delta<0\): no real roots

Vertex x-coordinate

\(x = -\dfrac{b}{2a}\)

📝 Example

Solve: \(2x^2 - 5x - 3 = 0\)

\(x = \dfrac{5 \pm \sqrt{25 + 24}}{4} = \dfrac{5 \pm 7}{4}\)

\(x = 3\) or \(x = -\dfrac{1}{2}\)

✓ x = 3 or x = −1/2

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Unit 3

Polynomials & Division

🔑 Remainder & Factor Theorems

Remainder Theorem: When \(p(x)\) is divided by \((x-a)\), remainder \(= p(a)\)
Factor Theorem: \((x-a)\) is a factor of \(p(x)\) if and only if \(p(a) = 0\)
Rational Root Test: Possible rational roots \(= \pm\dfrac{p}{q}\) where \(p\mid\) constant, \(q\mid\) leading coefficient

⚡ Memorize

Sum/Difference of Cubes

\(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)

Fundamental Theorem of Algebra

A degree-n polynomial has exactly n roots (counting multiplicity, complex)

📝 Example

Is \((x-2)\) a factor of \(p(x)=x^3-6x^2+11x-6\)?

\(p(2) = 8 - 24 + 22 - 6 = 0\) ✓

✓ Yes, (x-2) is a factor

⚗️

Unit 4

Radical & Rational Functions

🔑 Key Rules

Rational Exponent: \(a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}\)
Asymptotes: Vertical: set denominator = 0; Horizontal: compare degrees
Extraneous Solutions: Always CHECK solutions when solving radical/rational equations!

⚡ Horizontal Asymptote Rules

deg num < deg den → HA: \(y = 0\)

deg num = deg den → HA: \(y = \dfrac{\text{leading coeff}}{\text{leading coeff}}\)

deg num > deg den → No HA (oblique asymptote)

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Unit 5

Exponential & Logarithms

DINO SAYS 🦕

Logs and exponentials are INVERSES — just like a dino eating and... un-eating? Master the log laws and you're unstoppable!

🔑 Log Laws (The Big 4!)

Logs and exponentials are inverse functions: \(\log_b(b^x) = x\) and \(b^{\log_b x} = x\)

⚡ Must Memorize: Log Laws

Product Rule

\(\log_b(mn) = \log_b m + \log_b n\)

Quotient Rule

\(\log_b\!\left(\dfrac{m}{n}\right) = \log_b m - \log_b n\)

Power Rule

\(\log_b(m^p) = p \cdot \log_b m\)

Change of Base

\(\log_b x = \dfrac{\ln x}{\ln b} = \dfrac{\log x}{\log b}\)

📝 Example

Solve: \(3^{2x-1} = 27\)

\(3^{2x-1} = 3^3 \Rightarrow 2x-1 = 3 \Rightarrow x = 2\)

✓ x = 2

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Unit 6

Conic Sections

🔑 The 4 Conics

Circle: \((x-h)^2 + (y-k)^2 = r^2\)
Parabola: \((x-h)^2 = 4p(y-k)\) or \((y-k)^2 = 4p(x-h)\)
Ellipse: \(\dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1\)
Hyperbola: \(\dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1\)

⚡ Quick ID Tips

Same sign, same coeff → Circle

Same sign, diff coeff → Ellipse

Opposite signs → Hyperbola

One variable squared → Parabola

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Unit 7

Sequences & Series

⚡ All Formulas

Arithmetic nth term

\(a_n = a_1 + (n-1)d\)

Arithmetic Sum

\(S_n = \dfrac{n}{2}(a_1 + a_n)\)

Geometric nth term

\(a_n = a_1 \cdot r^{n-1}\)

Geometric Sum (finite)

\(S_n = a_1 \cdot \dfrac{1-r^n}{1-r}\)

Infinite Geometric (|r|<1)

\(S_\infty = \dfrac{a_1}{1-r}\)

Algebra 2
with Dino! 🦕

🎯 20 Practice Problems

📋 Answer Key & Explanations

Algebra 2with Dino! 🦕

🎯 20 Practice Problems

📋 Answer Key & Explanations

Algebra 2
with Dino! 🦕