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✦ Interactive Study Guide ✦

Dino Algebra 2

20 Exam-Style Questions Β· All Units Covered Β· Animated Dinosaur Tutor Β· Timer Β· Score Report

30:00
πŸ¦– Unit 1 Β· Polynomial Functions
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Rex the Dino
Hey there! I'm Rex πŸ¦– β€” your Algebra 2 guide! Polynomials are expressions with multiple terms involving powers of $x$. Let's roar through the rules together!
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Unit 1
Polynomial Functions

A polynomial function has the form $f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$ where all exponents are non-negative integers.

πŸ”‘ Key Formulas
Degree: highest exponent
Leading coefficient: $a_n$ (determines end behavior)
Factor Theorem: $(x-c)$ is a factor ↔ $f(c)=0$
Remainder Theorem: $f(c)$ = remainder when $f(x) \div (x-c)$
Rational Root Theorem: possible roots = $\pm\dfrac{p}{q}$, where $p \mid a_0$, $q \mid a_n$
⭐ Memorize!
End behavior: even degree, same ends; odd degree, opposite ends
Positive leading coeff β†’ right end goes UP; Negative β†’ right end goes DOWN
A polynomial of degree $n$ has at most $n$ real roots and $n-1$ turning points
✏️ Example
Given $f(x)=x^3-6x^2+11x-6$, find all roots.
Rational Root Theorem β†’ try $x=1$: $f(1)=1-6+11-6=0$ βœ“
Divide: $f(x)=(x-1)(x^2-5x+6)=(x-1)(x-2)(x-3)$
Roots: x = 1, 2, 3
πŸ¦• Unit 2 Β· Quadratics
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Dino Dot
Quadratics are everywhere β€” from dinosaur parabolic jumps to flying meteorites! The discriminant tells us how many roots we have. Remember $b^2-4ac$!
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Unit 2
Quadratic Equations & Inequalities

A quadratic has the form $ax^2+bx+c=0$. Methods to solve: factoring, completing the square, quadratic formula, or graphing.

πŸ”‘ Key Formulas
Quadratic Formula: $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$
Vertex Form: $f(x)=a(x-h)^2+k$, vertex at $(h,k)$
Discriminant $\Delta = b^2-4ac$: >0 two real roots, =0 one real root, <0 two complex roots
Sum of roots: $-\dfrac{b}{a}$  |  Product of roots: $\dfrac{c}{a}$
⭐ Memorize!
$\Delta > 0$: two distinct real roots  |  $\Delta = 0$: one repeated root  |  $\Delta < 0$: complex conjugate roots
Axis of symmetry: $x = -\dfrac{b}{2a}$
✏️ Example
Solve $2x^2 - 5x - 3 = 0$
$\Delta = 25 + 24 = 49$
$x = \dfrac{5 \pm 7}{4}$
x = 3 or x = βˆ’1/2
πŸ‰ Unit 3 Β· Complex Numbers & Roots
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Drago
$i = \sqrt{-1}$ is my secret power! Complex numbers live in two dimensions. The complex conjugate theorem says complex roots always come in pairs like me and my twin!
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Unit 3
Complex Numbers & Radical Expressions

The imaginary unit $i = \sqrt{-1}$, so $i^2=-1$. Complex numbers have the form $a+bi$.

πŸ”‘ Key Formulas
Powers of $i$: $i^1=i,\ i^2=-1,\ i^3=-i,\ i^4=1$ (cycle repeats)
Complex Conjugate: $(a+bi)(a-bi)=a^2+b^2$
Modulus: $|a+bi|=\sqrt{a^2+b^2}$
Radical: $\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}$; rationalize by multiplying conjugate
⭐ Memorize!
To find $i^n$: compute $n \mod 4$, then use the cycle: 1β†’i, 2β†’βˆ’1, 3β†’βˆ’i, 0β†’1
If polynomial has real coefficients, complex roots come in conjugate pairs
✏️ Example
Simplify $\dfrac{3+2i}{1-i}$
Multiply by conjugate: $\dfrac{(3+2i)(1+i)}{(1-i)(1+i)} = \dfrac{3+3i+2i+2i^2}{1+1}$
$= \dfrac{3+5i-2}{2} = \dfrac{1+5i}{2}$
$\dfrac{1}{2}+\dfrac{5}{2}i$
🦎 Unit 4 · Exponential Functions
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Expo Lizard
I grow SUPER fast because I'm exponential! $e \approx 2.718$ is Euler's number β€” the base of natural growth. Bacteria, populations, radioactive decay β€” all exponential!
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Unit 4
Exponential Functions & $e$

Exponential functions: $f(x)=ab^x$ where $b>0, b\ne1$. Growth: $b>1$. Decay: $0<b<1$.

πŸ”‘ Key Formulas
Compound Interest: $A=P\!\left(1+\dfrac{r}{n}\right)^{nt}$
Continuous Growth: $A=Pe^{rt}$
Laws: $b^x\cdot b^y=b^{x+y}$; $\dfrac{b^x}{b^y}=b^{x-y}$; $(b^x)^y=b^{xy}$; $b^{-x}=\dfrac{1}{b^x}$
⭐ Memorize!
$e \approx 2.71828$ β€” base of natural exponential and natural log
Exponential growth: $y$-intercept = initial value; never touches $x$-axis (asymptote at $y=0$)
✏️ Example
$\$1000$ invested at 5% compounded continuously for 3 years.
$A = 1000\,e^{0.05\times 3}=1000\,e^{0.15}$
β‰ˆ $1161.83
🌿 Unit 5 · Logarithmic Functions
🌿
Log the Fern Dino
Log is the inverse of exponential. If $b^y=x$, then $\log_b x=y$. I use logs to undo exponents β€” think of me as the "unpacker"!
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Unit 5
Logarithmic Functions

$\log_b x = y \Leftrightarrow b^y = x$. Natural log: $\ln x = \log_e x$. Common log: $\log_{10} x$.

πŸ”‘ Key Properties
$\log_b(mn)=\log_b m + \log_b n$
$\log_b\!\left(\dfrac{m}{n}\right)=\log_b m - \log_b n$
$\log_b(m^p)=p\log_b m$
Change of Base: $\log_b a = \dfrac{\ln a}{\ln b} = \dfrac{\log a}{\log b}$
⭐ Memorize!
$\log_b 1 = 0$ and $\log_b b = 1$ for any valid base $b$
Domain of $\log_b x$: $x > 0$; vertical asymptote at $x=0$
$b^{\log_b x}=x$ and $\log_b(b^x)=x$ (inverse relationship)
✏️ Example
Solve $\log_2(x-1)+\log_2(x+1)=3$
$\log_2\!\left[(x-1)(x+1)\right]=3$ β†’ $(x^2-1)=8$
$x^2=9$ β†’ $x=3$ (reject $x=-3$, domain)
x = 3
🦴 Unit 6 · Rational Functions
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Unit 6
Rational Functions

Rational functions: $f(x)=\dfrac{p(x)}{q(x)}$ where $p,q$ are polynomials. Key: asymptotes and holes!

πŸ”‘ Finding Asymptotes
Vertical asymptote: $q(x)=0$ (after cancellation)
Horizontal asymptote: compare degrees of $p$ and $q$ Β· deg p < deg q: $y=0$ Β· deg p = deg q: $y=\dfrac{\text{leading coeff of }p}{\text{leading coeff of }q}$ Β· deg p > deg q: no horizontal asymptote (oblique)
Hole: common factor cancelled from numerator and denominator
⭐ Memorize!
BOBO BOTNA EATS DC: Bottom Over = 0, Bottom same = ratio, Exponent bigger = Dig Calculator (oblique/slant)
✏️ Example
Find asymptotes of $f(x)=\dfrac{2x^2-8}{x^2-4}$
Factor: $\dfrac{2(x^2-4)}{x^2-4}=2$ β€” actually a hole at $x=\pm2$, the function simplifies to $y=2$
No asymptotes; holes at $x=2$ and $x=-2$; $y=2$
🌊 Unit 7 · Sequences & Series
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Croco
Sequences are patterns of numbers β€” I count dino footsteps! Arithmetic adds a constant, geometric multiplies by one. Sigma notation $\sum$ is just fancy addition!
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Unit 7
Sequences & Series
πŸ”‘ Arithmetic
$a_n = a_1 + (n-1)d$
$S_n = \dfrac{n}{2}(a_1+a_n) = \dfrac{n}{2}[2a_1+(n-1)d]$
πŸ”‘ Geometric
$a_n = a_1 \cdot r^{n-1}$
$S_n = a_1\cdot\dfrac{1-r^n}{1-r}$ (finite)  |  $S_\infty = \dfrac{a_1}{1-r}$ when $|r|<1$
⭐ Memorize!
Arithmetic: constant DIFFERENCE $d$  |  Geometric: constant RATIO $r$
Infinite geometric sum only exists when $|r| < 1$
✏️ Example
Sum of infinite geometric series with $a_1=12$, $r=\tfrac{1}{3}$
$S_\infty = \dfrac{12}{1-\frac{1}{3}} = \dfrac{12}{\frac{2}{3}} = 18$
S = 18
πŸŒ™ Unit 8 Β· Trigonometry
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Star Dino
SOH-CAH-TOA! And on the unit circle, coordinates are $(\cos\theta, \sin\theta)$. The unit circle is my star map β€” memorize the key angles!
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Unit 8
Trigonometric Functions
πŸ”‘ Key Identities
Pythagorean: $\sin^2\theta+\cos^2\theta=1$; $1+\tan^2\theta=\sec^2\theta$; $1+\cot^2\theta=\csc^2\theta$
Double Angle: $\sin 2\theta=2\sin\theta\cos\theta$; $\cos 2\theta=\cos^2\theta-\sin^2\theta$
Law of Sines: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$
Law of Cosines: $c^2=a^2+b^2-2ab\cos C$
⭐ Memorize! Unit Circle
$0Β°$: $(1,0)$   $30Β°$: $(\frac{\sqrt3}{2},\frac{1}{2})$   $45Β°$: $(\frac{\sqrt2}{2},\frac{\sqrt2}{2})$   $60Β°$: $(\frac{1}{2},\frac{\sqrt3}{2})$   $90Β°$: $(0,1)$
Period of $\sin$ and $\cos$: $2\pi$; Period of $\tan$: $\pi$
ASTC (All Students Take Calculus): signs of trig functions by quadrant
✏️ Example
Find all solutions to $2\sin^2 x - \sin x - 1 = 0$ on $[0, 2\pi)$
Factor: $(2\sin x+1)(\sin x-1)=0$ β†’ $\sin x = -\frac{1}{2}$ or $\sin x = 1$
$\sin x=-\frac{1}{2}$: $x=\frac{7\pi}{6}, \frac{11\pi}{6}$   $\sin x=1$: $x=\frac{\pi}{2}$
$x = \dfrac{\pi}{2}, \dfrac{7\pi}{6}, \dfrac{11\pi}{6}$
πŸ₯š Unit 9 Β· Conic Sections
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Unit 9
Conic Sections
πŸ”‘ Standard Forms
Circle: $(x-h)^2+(y-k)^2=r^2$, center $(h,k)$, radius $r$
Parabola: $y=a(x-h)^2+k$ (vertical) or $x=a(y-k)^2+h$ (horizontal)
Ellipse: $\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$ ($a>b$: horizontal major axis)
Hyperbola: $\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$ (horizontal) or $\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1$ (vertical)
⭐ Memorize!
Discriminant $B^2-4AC$ of $Ax^2+Bxy+Cy^2+\ldots$: <0 ellipse/circle; =0 parabola; >0 hyperbola
Ellipse foci: $c^2=a^2-b^2$  |  Hyperbola foci: $c^2=a^2+b^2$
✏️ Example
Identify the conic: $4x^2+9y^2=36$
Divide by 36: $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$
Ellipse, center origin, $a=3$ (x-axis), $b=2$ (y-axis)
🎲 Unit 10 · Probability & Statistics
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Unit 10
Probability, Permutations & Combinations
πŸ”‘ Counting & Probability
Permutations: $P(n,r)=\dfrac{n!}{(n-r)!}$ (order matters)
Combinations: $C(n,r)=\binom{n}{r}=\dfrac{n!}{r!(n-r)!}$ (order doesn't matter)
Binomial Theorem: $(a+b)^n=\sum_{k=0}^n\binom{n}{k}a^{n-k}b^k$
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
⭐ Memorize!
Permutation: "Particular order" β†’ $P$; Combination: "Choose" β†’ $C$
Binomial Theorem: Pascal's Triangle gives coefficients for small $n$
✏️ Example
Expand $(x+2)^4$
Coefficients from Pascal's Row 4: 1, 4, 6, 4, 1
$x^4+4(x^3)(2)+6(x^2)(4)+4(x)(8)+16$
$x^4+8x^3+24x^2+32x+16$
πŸ¦–
Rex
RAWR! You've reviewed all 10 units! Time to prove your dino brain! The quiz has 20 exam-style questions. Good luck β€” I believe in you! πŸ’ͺ