Comprehensive Algebra 1 and Algebra 2 Study Guide

Real Number System & Number Types

• Natural or Counting Numbers:

  • Definition: Numbers used for counting items starting from $1$.
  • Elements: 1, 2, 3, 4, 5\ndots

• Whole Numbers:

  • Definition: Natural numbers including zero.
  • Elements: 0, 1, 2\ndots

• Integers:

  • Definition: Whole numbers along with their negative counterparts.
  • Elements: $-5, -3, -1, 0, 1, 2, 3, 4$

• Rational Numbers:

  • Definition: Numbers that can be expressed as a quotient or fraction of two integers, including repeating decimals and simplified fractions.
  • Examples: $0.141414\dots$, $0.23$, $\frac{1}{2}$, $\frac{14}{5}$

• Irrational Numbers:

  • Definition: Decimals that cannot be simplified to a fraction, non-terminating and non-repeating numbers.
  • Examples: $\pi$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{11}$, $\frac{4}{7}$, $e$

Arithmetic Rules for Signed Numbers & Absolute Value

• Adding and Subtracting Signed Numbers:

  • Same Sign: Add the numbers together and keep the common sign.
    • $5 + 3 = 8$
    • $-5 - 3 = -8$
    • $-5 + (-3) = -8$
    • $5 - (-3) = 5 + (+3) = 8$
    • $5 + (+3) = 8$
  • Different Sign: Subtract the smaller magnitude from the larger magnitude. The sign of the answer is determined by the sign of the larger number.
    • $-5 + 3 = -2$ (The larger number is negative, so the answer is $-2$)
    • $5 - 3 = 2$ (The larger number is positive, so the answer is $2$)
    • $-5 - (-3) = -5 + (+3) = -2$ (Different sign subtraction, larger number is negative, answer is $-2$)
    • $-5 + (+3) = -2$ (Different sign subtraction, larger number is negative, answer is $-2$)
    • $5 + (-3) = 2$ (Different sign subtraction, sign on the bigger number is positive, answer is $2$)

• Multiplying & Dividing Signed Numbers:

  • Same Sign: Outcome is always POSITIVE ($+$).
    • $6 \times 7 = +42$
    • $-6 \times -7 = +42$
    • $42 \div 7 = +6$
    • $-42 \div -7 = +6$
  • Different Sign: Outcome is always NEGATIVE ($-$).
    • $6 \times -7 = -42$
    • $-7 \times 6 = -42$
    • $42 \div -7 = -6$
    • $42 \div -6 = -7$

• Absolute Value:

  • Definition: Represents the DISTANCE from ZERO on a number line.
  • Property: Absolute value is ALWAYS POSITIVE ($+++$).
  • Examples:
    • $|8| = +8$
    • $|125| = +125$
    • $|-8| = +8$
    • $|-125| = +125$
  • Negative Outcome Exception: The only possible negative outcome occurs if a negative sign is situated out in FRONT of the absolute value expression.
    • $-|-8| = -8$
    • $-|-125| = -125$

Order of Operations & Algebraic Expression Evaluation

• Order of Operations (PEMDAS):

  • P: Please (Parentheses)
  • E: Excuse (Exponents)
  • M: My (Multiplication - performed from left to right alongside division)
  • D: Dear (Division - performed from left to right alongside multiplication)
  • A: Aunt (Addition - performed from left to right alongside subtraction)
  • S: Sally (Subtraction - performed from left to right alongside addition)

• Evaluating Algebraic Expressions:

  • Procedure: Substitute (plug in) the numerical values given for each variable into the algebraic expression, then calculate the outcome.
  • Example 1: Evaluate $x^2 - 5y$ for $x = 2$ and $y = -1$
    • $x^2 - 5y = (2)^2 - 5(-1) = 4 - (-5) = 9$
  • Example 2: Evaluate $3x^2 + 2y$ for $x = 5$ and $y = -4$
    • $3x^2 + 2y = 3(5)^2 + 2(-4) = 75 - 8 = 67$

• Distributive Property:

  • Rule: Distribute (or multiply) the term outside the parentheses times EACH TERM INSIDE THE PARENTHESES.
  • Example 1: $5(x + 1) = 5 \cdot x + 5 \cdot 1 = 5x + 5$
  • Example 2: $y(2y - 3) = y \cdot 2y + y \cdot (-3) = 2y^2 - 3y$
  • Example 3: $12(a^2 + 5b) = 12 \cdot a^2 + 12 \cdot 5b = 12a^2 + 60b$

Variable Operations & Simplification

• Adding and Subtracting Variables:

  • Rule: Only add and subtract LIKE TERMS (terms sharing exact variable bases and exponents).
  • Examples of Like Terms:
    • $4x, -10x, 100x, -3x$
    • $4ab, -10ab, 100ab, -3ab$
    • $4y, -10y, 100y, -3y$
    • $4x^2, -10x^2, 100x^2, -3x^2$
  • Operations:
    • $4x - 10x = -6x$
    • $4x - 10y = \text{NOT POSSIBLE}$ (Unlike terms cannot be added or subtracted)
    • $4ab + 100ab = 104ab$
    • $4y - 10y + 100y = 94y$
    • $4x^2 + 100x = \text{NOT POSSIBLE}$ (Different exponents mean terms are unlike)
    • $4x^2 - 3x^2 = 1x^2$

• Multiplying and Dividing Variables:

  • Rule: UNLIKE terms CAN be multiplied and divided. Multiply and divide whole numbers separately from the variables.
  • Exponent Addition Rule: ADD exponents when multiplying variables.
  • Exponent Subtraction Rule: SUBTRACT exponents when dividing variables.
  • Examples:
    • $2x \cdot 3x^2 = 2 \cdot 3 \cdot x^{1+2} = 6x^3$
    • $5y^4 \cdot 6x^2 y^3 = 5 \cdot 6 \cdot x^2 \cdot y^{4+3} = 30x^2 y^7$
    • $\frac{12a^3}{4a} = \frac{12}{4} \cdot a^{3-1} = 3a^2$
    • $\frac{8x^5}{2x^3} = 4x^{5-3} = 4x^2$
    • $\frac{18x^2 y^3}{3y} = 6x^2 y^{3-1} = 6x^2 y^2$

Exponent Rules & Properties

• Fundamental Laws of Exponents:

  • Zero Exponent Rule: $x^0 = 1$
  • Product Rule: ADD exponents when multiplying like bases: $x^a \cdot x^b = x^{a+b}$
  • Quotient Rule: SUBTRACT exponents when dividing like bases: $\frac{x^a}{x^b} = x^{a-b}$
  • Product Power Rule: $(xy)^a = x^a y^a$
  • Power of a Power Rule: MULTIPLY exponents when raised to another exponent: $(x^a)^b = x^{a \cdot b}$

• Negative Exponent Rules:

  • Rule: If a term has a negative exponent and is in the numerator, move it to the denominator to become positive. If the term with the negative exponent is in the denominator, move it to the numerator to become positive.
  • Formulas: $x^{-a} = \frac{1}{x^a}$ and $\frac{1}{x^{-a}} = x^a$

• Comprehensive Examples:

  • Negative base multiplication: $(-2) \cdot (-2) \cdot (-2) = -8$
  • Raising a power to a power: $(4x^3)^2 = 4^2 \cdot x^{3 \cdot 2} = 16x^6$
  • Raising a higher power: $(8y^5)^3 = 8^3 \cdot y^{5 \cdot 3} = 512y^{15}$
  • Negative base raised to power: $(-2a^3)^2 = (-2)^2 \cdot a^{3 \cdot 2} = 4a^6$
  • Negative exponent conversion: $(-2a^3)^{-2} = \frac{1}{(-2a^3)^2} = \frac{1}{(-2)^2 \cdot a^{3 \cdot 2}} = \frac{1}{4a^6}$

Operations with Fractions

• Simplifying Fractions:

  • Procedure: Find a number that can be divided evenly into both numerator and denominator. Continue this process until no further division is possible; at that point, the fraction is simplified.
  • Examples:
    • $\frac{10 \div 5}{25 \div 5} = \frac{2}{5}$
    • $\frac{6 \div 2}{8 \div 2} = \frac{3}{4}$
    • $\frac{8 \div 8}{24 \div 8} = \frac{1}{3}$
    • $\frac{42 \div 6}{72 \div 6} = \frac{7}{12}$

• Multiplying Fractions:

  • Procedure: Multiply top times top (numerators), and bottom times bottom (denominators).
  • Examples:
    • $\frac{2}{3} \times \frac{5}{7} = \frac{10}{21}$
    • $\frac{5}{6} \times \frac{4}{7} = \frac{20}{42} = \frac{20 \div 2}{42 \div 2} = \frac{10}{21}$

• Dividing Fractions:

  • Procedure: Flip the second fraction (take its reciprocal) and multiply across.
  • Examples:
    • $\frac{3}{5} \div \frac{2}{7} = \frac{3}{5} \times \frac{7}{2} = \frac{21}{10}$
    • $\frac{8}{9} \div \frac{7}{4} = \frac{8}{9} \times \frac{4}{7} = \frac{32}{63}$
    • $\frac{5}{6} \div \frac{4}{7} = \frac{5}{6} \times \frac{7}{4} = \frac{35}{24}$

Radical Properties & Complex Numbers

• Properties of Radicals:

  • Product Rule for Radicals: $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$
  • Rational Exponent Forms: $\sqrt{x} = x^{\frac{1}{2}}$ and $\sqrt[n]{x} = x^{\frac{1}{n}}$
  • Example: $\sqrt[3]{8x} = \sqrt[3]{8} \cdot \sqrt[3]{x} = 2x^{\frac{1}{3}}$

• Complex Numbers:

  • Definition of Imaginary Unit: $\sqrt{-1} = i$
  • Evaluating Powers of $i$: Take out the largest factor of $4$ (since $i^4 = 1$) and evaluate the remaining power using the standard pattern.
  • Examples:
    • $i^4 = 1$
    • $i^{18} = i^{16} \cdot i^2 = 1 \cdot (-1) = -1$
    • $i^{21} = i^{20} \cdot i^1 = 1 \cdot i = i$
    • $i^{35} = i^{32} \cdot i^3 = 1 \cdot (-i) = -i$

Linear Equations, Lines, & Graphing Properties

• Forms of Linear Equations & Slope:

  • Slope-Intercept Form: $y = mx + b$
  • Point-Slope Form: $y - y_1 = m(x - x_1)$
  • Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$
  • Key Variables: $m = \text{slope}$, $b = \text{y-intercept}$, and $(x_1, y_1), (x_2, y_2) = \text{given points}$.

• Special Slope Conditions:

  • Vertical Lines: Any line with the equation $x = 3$, $x = -9$, or $x = \text{ANY NUMBER}$ has an UNDEFINED SLOPE.
  • Horizontal Lines: Any line with a ZERO SLOPE, such as $y = 5$ or $y = -3$, is a horizontal line.

• Graphing Inequality Boundaries:

  • Strict Inequalities ($<$, $>$): Represented by a DOTTED LINE.
  • Inclusive Inequalities ($\le$, $\ge$): Represented by a SOLID LINE.

• Conic Section Shapes:

  • Hyperbola
  • Parabola

Factoring Techniques

• Factoring Greatest Common Factor (GCF):

  • Question: Is there a number or term that can be factored out of all terms?
  • Formula: $ab + bc = b(a + c)$ with $\text{GCF} = b$
  • Examples:
    • $4x + 10xy = 2x(2 + 5y)$ with $\text{GCF} = 2x$
    • $12abc - 3ab + 6ac = 3a(4bc - b + 2c)$ with $\text{GCF} = 3a$
    • $8x^2 + 4x - 2x^3 = 4x(2x + 1 - 2x^2)$ with $\text{GCF} = 4x$

• Factoring Difference of Squares:

  • Applicability: Works when there are two squared terms being subtracted.
  • Formula: $a^2 - b^2 = (a + b)(a - b)$
  • Examples:
    • $x^2 - y^2 = (x + y)(x - y)$
    • $4x^2 - 25y^2 = (2x + 5y)(2x - 5y)$
    • $100j^4 - 16k^2 = (10j^2 + 4k)(10j^2 - 4k)$
    • $81x^2 - y^2 = (9x + y)(9x - y)$

• Factor by Grouping:

  • Procedure:
    1. Group the first two terms and the last two terms.
    2. Factor out the GCF of both pairs, which should result in the same binomial.
    3. Re-write with the GCF terms as one binomial and the repeated binomial as the other.
  • Example: Factor $x^3 + 2x^2 - 3x - 6$
    • Group first two and last two: $(x^3 + 2x^2) + (-3x - 6)$
    • Factor GCFs: $x^2(x + 2) - 3(x + 2)$
    • Combine outer GCFs and repeated binomial: $(x^2 - 3)(x + 2)$

• Factoring Cubes (SOAP Mnemonic):

  • SOAP Rule: Same sign, Opposite sign, Always Positive.
  • Difference of Cubes Formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
  • Sum of Cubes Formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
  • Examples:
    • Factor $8x^3 - 125y^3$:
      • Cube root of each term gives $a = 2x$ and $b = 5y$
      • Result: $(2x - 5y)(4x^2 + 10xy + 5y^2)$
    • Factor $27j^3 + 64$:
      • Cube root of each term gives $a = 3j$ and $b = 4$
      • Result: $(3j + 4)(9j^2 - 12j + 16)$

• Factoring Trinomials ($a = 1$):

  • Procedure: Find factors of the $c$ term that sum up to the $b$ term.
  • Examples:
    • $x^2 - x - 6$: Factors of $-6$ that sum to $-1$ are $-3, 2$. Factored pair: $(x + 2)(x - 3)$.
    • $x^2 - 2x - 8$: Factors of $-8$ that sum to $-2$ are $-4, 2$. Factored pair: $(x + 2)(x - 4)$.
    • $y^2 + 12y + 20$: Factors of $20$ that sum to $12$ are $10, 2$. Factored pair: $(y + 10)(y + 2)$.
    • $a^2 - 6a + 9$: Factors of $9$ that sum to $-6$ are $-3, -3$. Factored pair: $(a - 3)(a - 3)$ or $(a - 3)^2$

• Factoring Trinomials when $a \ne 1$:

  • Procedure:
    1. Multiply the $a$ term by the $c$ term ($a \cdot c$).
    2. Find factors of $a \cdot c$ that sum to the $b$ term.
    3. Split the middle term $b$ into these two terms.
    4. Factor by GROUPING.
  • Example: Factor $2x^2 + 9x - 18$
    • Calculate $a \cdot c = 2 \cdot (-18) = -36$
    • Find factors of $-36$ that sum to $9$: $12, -3$
    • Split middle term $9x$ into $12x - 3x$: $2x^2 + 12x - 3x - 18$
    • Group and factor GCFs: $2x(x + 6) - 3(x + 6)$
    • Combine outer terms: $(2x - 3)(x + 6)$

Completing the Square

• General Steps:

  1. Factor the equation so the $a$ term equals $1$
  2. Add spaces for the needed value to complete the square on both sides of the equation.
  3. Calculate the needed value using $\left(\frac{b}{2}\right)^2$
  4. Factor into a squared binomial, then solve for $x$ by taking the square root.

• Example 1 (Quadratic Equation):

  • Solve $6x^2 + 12x - 54 = 0$
  • Factor so $a = 1$: $6(x^2 + 2x + \underline{\hspace{0.5cm}}) = 54 + \underline{\hspace{0.5cm}}$
  • Add needed value $\left(\frac{2}{2}\right)^2 = 1$ (scaled by 6 on the right side):
  • $6(x^2 + 2x + 1) = 54 + 6$
  • $6(x + 1)^2 = 60$
  • $(x + 1)^2 = 10$

• Example 2 (Hyperbola Equation Rearrangement):

  • Given $4x^2 - 25y^2 - 8x - 100y - 196 = 0$
  • Group variables: $4x^2 - 8x - 25y^2 - 100y = 196$
  • Set up square completion: $4(x^2 - 2x + \underline{\hspace{0.5cm}}) - 25(y^2 - 4y + \underline{\hspace{0.5cm}}) = 196 + \underline{\hspace{0.5cm}} + \underline{\hspace{0.5cm}}$
  • Fill in values: $4(x^2 - 2x + 1) - 25(y^2 - 4y + 4) = 196 + 4 + (-100)$
  • Factor into standard binomial form: $4(x + 3)^2 - 25(y - 2)^2 = 100$

Logarithmic Functions & Rules

• Definition & Logarithmic Conversion:

  • $A^b = C \iff \log_A C = b$

• Logarithmic Rules:

  • Product Rule: $\log_x(y \cdot z) = \log_x y + \log_x z$
  • Quotient Rule: $\log_x\left(\frac{y}{z}\right) = \log_x y - \log_x z$
  • Power Rule: $\log_x(y^r) = r \cdot \log_x y$
  • Logarithm of Base: $\log_x x = 1$
  • Logarithm of One: $\log_x 1 = 0$

• Natural Logarithm Property:

  • The exact same rules apply to natural logarithms, $\ln$.

Functions & Their Properties

• Domain & Range:

  • Domain: All $x$-values.
  • Range: All $y$-values.

• Function Criteria:

  • Definition: A relation where $x$-values CANNOT repeat.

• One-to-One Functions:

  • Passes both the vertical and horizontal line tests.
  • Each input has a unique, different output.
  • No $x$'s or $y$'s repeat.

• Inverse Functions:

  • Notation: Original function $f(x)$, Inverse function $f^{-1}(x)$.
  • Definition: The inverse is the reflection of a function over the line $y = x$
  • Method to find: Switch $x$ and $y$ in the equation, then solve for $y$ to get the new equation.
  • Example:
    • Given original equation: $y = 2x + 3$
    • Switch $x$ and $y$: $x = 2y + 3$
    • Isolate $y$: $x - 3 = 2y \implies y = \frac{x - 3}{2}$

• Even and Odd Functions:

  • Even Function:
    • Condition: $f(-x) = f(x)$
    • Symmetry: Exhibits Y-AXIS SYMMETRY.
  • Odd Function:
    • Condition: $f(-x) = -f(x)$
    • Symmetry: Exhibits ORIGIN SYMMETRY.

Permutations & Combinations

• Permutation:

  • Key Rule: ORDER MATTERS.
  • Definition: Choosing $r$ items from $n$ choices where repetition is not allowed and sequence matters.
  • Notation / Formula: ${_n P_r} = \frac{n!}{(n-r)!}$
  • Real-World Context: Locker combination ("What's your locker permutation?").

• Combination:

  • Key Rule: ORDER IS NOT IMPORTANT.
  • Definition: "$n$ choose $r$". Selecting $r$ items from $n$ choices regardless of order.
  • Notation / Formula: ${_n C_r} = \frac{n!}{r!(n-r)!}$
  • Examples:
    • Lottery winning numbers.
    • Choosing 3 scoops from 5 ice cream flavors: $n = 5, r = 3$
    • Choosing 4 people from a group of 10: $n = 10, r = 4$