Definitive Study Guide for EOC Algebra: Sequences, Probability, and Algebraic Functions

Reference Sheet Overview and General Formulas

• Quadratic and Cubic Functions:     - The reference sheet includes formulas for the quadratic formula, quadratic functions, and cubic functions.     - To find the $x$-coordinate of the vertex for a quadratic function, use the formula:       $x = \frac{-b}{2a}$     - After finding the $x$-coordinate, plug this value back into the original function to find the corresponding $y$-coordinate.

• Logarithm and Exponential Properties:     - The natural logarithm ($\ln$) and base ($e$) cancel each other out: $\ln(e^x) = x$ and $e^{\ln(x)} = x$.     - Note: $\ln(e) = 1$.     - The reference sheet provides instructions on how to convert exponentials and contains all standard logarithm formulas.

• Finance Formulas:     - Simple Interest: Located at the top of the finance section.     - Compound Interest: $A = P(1 + \frac{r}{n})^{nt}$.     - Compounded Continuously: $A = Pe^{rt}$.

Probability and Permutations

• Intuitive Probability Example:     - Consider a set of $16$ marbles.     - If half are red and half are green ($8$ red, $8$ green), the probability of pulling a green marble is $50\%$.

• Permutations:     - Definition: A permutation is used to determine the number of ways to rearrange a specific number of items from a larger set where order matters.     - Formula Location: This formula is found on the far-left side of the POC reference sheet.     - The Formula:       $P(n, r) = \frac{n!}{(n - r)!}$       Where $n$ is the total number of items and $r$ is the number of items being selected/rearranged.

• Factorial Notation ($!$):     - A factorial is a mathematical shortcut meaning you multiply a number by every whole number below it down to $1$.     - Examples:         - $3! = 3 \times 2 \times 1 = 6$         - $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$         - $8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$         - $800! = 800 \times 799 \times 798 \dots \times 1$

• Case Study: Rearranging the word "POINTS":     - Problem: How many ways can you rearrange two letters from the word "POINTS"?     - Identification: Total letters ($n$) = $6$. Letters to rearrange at a time ($r$) = $2$.     - Setup:       $\frac{6!}{(6 - 2)!} = \frac{6!}{4!}$     - Calculation:       $\frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1}$     - The values $4, 3, 2, 1$ in the numerator and denominator cancel out.     - Result: $6 \times 5 = 30$.

Rational Expressions and Common Denominators

• Finding a Common Denominator:     - Example problem involves the denominators $x + 5$ and $2x + 10$.     - First, factor the expression: $2x + 10 = 2(x + 5)$.     - The common denominator is $2(x + 5)$.

• Addition Example:     - To add fractions, multiply the numerator and denominator of the term missing a factor so they match the common denominator.     - If adding $\frac{3x + 6}{2(x + 5)}$ and $\frac{3x - 1}{x + 5}$, you must multiply the second term by $\frac{2}{2}$.     - Equation: $3x + 6 + 2(3x - 1) = 3x + 6 + 6x - 2 = 9x + 4$.     - Final Result would be $\frac{9x - 2}{2(x + 5)}$ (based on transcript calculation).

Exponential Growth and Decay Models

• Problem Context: In the year $2020$, a village has a population of $2,000$ people with a projected annual increase of $0.8\%$.

• Growth Formula:   $A = P(1 + r)^t$   Where:   - $P$ = Initial value ($2,000$).   - $r$ = Growth rate as a decimal.   - $t$ = Time (often denoted as $n$ or $x$).

• Decimal Conversion: Converting a percentage to a decimal involves moving the decimal point two places to the left.     - $0.8\% = 0.008$

• Final Model:   $A = 2,000(1 + 0.008)^t$   $A = 2,000(1.008)^t$

Characteristics of Polynomials

• Criteria for a Polynomial:     - Exponents must be positive whole numbers.     - Non-polynomials:         - Functions with negative exponents (Rational functions).         - Functions with fractional exponents, such as the square root function: $y = \sqrt{x} = x^{\frac{1}{2}}$.         - Absolute value functions ($y = |x|$) because they contain a "sharp edge."     - Differentiability Note: A sharp edge in a graph (like absolute value) makes the function non-differentiable at that point, which distinguishes it from polynomials.

• Examples of Polynomials:     - Quadratic functions: $y = x^2$     - Cubic functions: $y = x^3$

Solving Rational Equations

• Method 1: Common Denominators (Hand Calculation):     - If all denominators are made the same across the equation, you can ignore the denominators and solve using only the numerators.     - Example: $\frac{4}{x + 2} + \frac{3}{x - 2} = \frac{10}{x^2 - 4}$.     - Step 1: Recognize $(x + 2)(x - 2) = x^2 - 4$.     - Step 2: Multiply terms to get the common denominator.     - Step 3: Solve the numerator equation: $4(x - 2) + 3(x + 2) = 10$       $4x - 8 + 3x + 6 = 10$       $7x - 2 = 10$       $7x = 12$       $x = \frac{12}{7} \approx 1.7142$

• Method 2: Technology (Desmos/Calculator):     - Enter the equation into Desmos to find the intersection or solution.     - If answer choices are in radical form, convert the radicals to decimals to see which matches the Desmos output.

Undefined Expressions and Domain Restrictions

• Definition: An expression is undefined when the denominator equals zero ($0$).

• Example: For which values is $\frac{\text{Numerator}}{x^2 - 9}$ undefined?     - Set the denominator to zero: $x^2 - 9 = 0$.     - Factoring: $(x - 3)(x + 3) = 0$.     - Solutions: $x = 3$ and $x = -3$.

• Alternative Algebraic Approach:     - $x^2 = 9$     - $\sqrt{x^2} = \sqrt{9}$     - $x = \pm 3$

Inverse Functions

• The Inverse Process:     - To find the inverse of a function, switch the $x$ and $y$ variables and solve for the new $y$.     - Example Function: $y = \sqrt{x + 6}$     - Inverse Calculation:         - Switch: $x = \sqrt{y + 6}$         - Square both sides: $x^2 = y + 6$         - Subtract $6$: $y = x^2 - 6$     - Verification: To verify, perform the process on the result to see if you return to the original function: $y = x^2 - 6 \rightarrow x = y^2 - 6 \rightarrow x + 6 = y^2 \rightarrow \sqrt{x + 6} = y$.

• Distinction: "Inverse functions" are different from "inverse variation" ($y = \frac{k}{x}$).

Simplifying Radicals: The Prism Break Method

• Method Steps:     - Break down the number inside the radical into its prime factors.     - Identify pairs (for square roots) to "break out" of the "prison" (the radical symbol).

• Example: Simplify $\sqrt{24x^4y^8}$:     - Factor $24$: $6 \times 4 \rightarrow (3 \times 2) \times (2 \times 2)$. Prime factors: $2, 2, 2, 3$.     - Factor Variables: $x^4 = (x, x), (x, x)$ and $y^8 = (y, y), (y, y), (y, y), (y, y)$.     - Pairs that "break out": One pair of $2$s, two pairs of $x$s ($x^2$), and four pairs of $y$s ($y^4$).     - Left inside the "prison": The remaining factors $3 \times 2 = 6$.     - Final Simplified Answer: $2x^2y^4\sqrt{6}$.

Absolute Value Inequalities

• The Two-Calculation Rule: To remove absolute value symbols, create two separate inequalities.     - Inequality 1: Write the expression exactly as seen without the bars.     - Inequality 2: Flip the inequality symbol and change the sign of the constant.

• GreatOR vs. Less thAND:     - Greater Than Symbols (> or $\ge$): Use "OR". The solution goes to the left of the negative value and to the right of the positive value.     - Less Than Symbols (< or $\le$): Use "AND". The solution is the area "in between" the two values (-a < x < a).

• Example: Solve |3x - 5| > 10:     - This is a "GreatOR" problem.     - Path 1: 3x - 5 > 10 \rightarrow 3x > 15 \rightarrow x > 5     - Path 2: 3x - 5 < -10 \rightarrow 3x < -5 \rightarrow x < -\frac{5}{3}     - Final Answer: x > 5 \text{ or } x < -\frac{5}{3}.

Questions & Discussion

• Question: How did we get the number $4$ in the denominator of the permutation problem?

• Answer: The formula is $n - r$. Since $n = 6$ (total letters in "POINTS") and $r = 2$ (letters chosen at a time), $6 - 2 = 4$. This resulting $4$ becomes $4!$ in the denominator.

• Question: Can you solve rational equations by multiplying the denominator across?

• Answer: Yes, if you multiply every term by the common denominator, the denominators will cancel out, leaving you with a linear equation to solve.

• Clarification on Inverse Variation: A student pointed out the confusion between "inverse functions" and "inverse variation." The instructor clarified that inverse functions involve switching $x$ and $y$, while inverse variation results in the formula $y = \frac{k}{x}$.