Algebra EOC Study Guide

Functions and Coordinate Graphing

• Function Notation Fundamentals     * $f(x)$ is a symbolic representation of the $y$ value.     * $f(3)$ asks the question: "What is $y$ when $x$ is $3$?"     * To solve, substitute $3$ into the function for every $x$ value and simplify.

• Finding f(x) from a Graph     * Example: Find $f(2)$ on a graph. This requires identifying the $y$ value at the point where the $x$-coordinate is $2$. In the provided example, when $x = 2$, $y = -2$, so $f(x) = -2$.     * Example: Find $x$ when $f(x) = -1$. This requires identifying the $x$ value where the $y$-coordinate is $-1$. In the example, this occurs at approximately $x = 0.5$.

• Domain and Range     * Uses braces for notation: $\{ \}$.     * Domain: The complete list (or set) of all $x$ values.     * Range: The complete list (or set) of all $y$ values.     * Roots: The list of $x$-intercepts, specifically where $y = 0$.

• Independent versus Dependent Variables     * Independent Variable ($x$): Represented on the horizontal axis. Time is almost always an independent variable when mentioned in a problem.     * Dependent Variable ($y$): Represented on the vertical axis.     * Sentence structure for identification: "(Independent Variable) is. (Dependent Variable) depends on what."

• Coordinate Graphing Layout     * Points are written as $(x, y)$.     * Quadrant I (QI): Top right.     * Quadrant II (QII): Top left.     * Quadrant III (QIII): Bottom left.     * Quadrant IV (QIV): Bottom right.     * Movement: Negative values move left or down; positive values move right or up.

Data Representation and Statistics

• Stem and Leaf Plots     * Used to organize data chronologically and by frequency.     * Data set: $70, 52, 58, 45, 59, 52, 75, 47, 46$.     * Step 1: Find the least and greatest values.     * Step 2: Write the stems in a vertical column.     * Step 3: Arrange the "leaves" from smallest to largest.     * Step 4: Provide an explanation/key. Example: $5 | 2 = 52$.

• Correlations     * Positive Correlation: Both data sets generally increase together.     * Negative Correlation: One data set decreases as the other set increases.     * No Correlation: Data sets show no visible relationship.     * A Trend Line on a scatter plot helps visualize correlation. The Line of Best Fit is the most accurate possible trend line.     * Finding a Trend Line: Identify the $y$-intercept ($b$), find the slope ($m$), and write the equation in slope-intercept form ($y = mx + b$).

• Central Tendencies and Spread     * Data set: $5, 6, 6, 7, 10, 11, 12, 14, 15, 20$     * Mode: The most common number ($6$). There can be multiple modes or no mode if all numbers are distinct.     * Median: The middle number ($10.5$). List numbers in order; if there are two middle numbers, find their average.     * Mean: The average ($10.6$). Sum all numbers and divide by the count.     * Range: The difference between the largest and smallest values ($20 - 5 = 15$).

• Percentage versus Flat Changes     * Percentage Increases:         * Measures of Center (Mean, Median, Mode) all increase.         * Measures of Spread (Range) increase.     * Flat or Constant Increase:         * Measures of Center (Mean, Median, Mode) increase.         * Measures of Spread (Range) has No Change because the low and high values shift by the same amount, maintaining the same difference.

• Box and Whisker Plots     * Divides data into four groups to show spread.     * Data set: $20, 36, 58, 45, 59, 55, 75, 35, 35$     * Step 1: Draw a line graph with equal intervals.     * Step 2: Place dots for the smallest and largest values.     * Step 3: Place a dot for the median.     * Step 4: Place dots for the median of the first and second halves (quartiles).     * Step 5: Draw a box around the middle two quartiles.     * Step 6: Use an asterisk ($*$) to mark extreme data items (outliers).

Algebraic Manipulation and Number Systems

• Isolating a Single Variable     * To "solve for" a variable means to isolate it. Follow the same rules as solving equations: perform identical operations on both sides.     * Example: Solve for $t$ in $\frac{abt}{3w} + 2 = c$         1. Subtract $2$: $\frac{abt}{3w} = c - 2$         2. Multiply by $3w$: $abt = 3w(c - 2)$         3. Distribute: $abt = 3wc - 6w$         4. Divide by $ab$: $t = \frac{3wc - 6w}{ab}$

• Order of Operations     1. Parentheses: Work inside out using $[]$, $()$, and $\{\}$.     2. Exponents.     3. Multiplication or Division: Left to right.     4. Addition or Subtraction: Left to right.     * Division Bar: Acts as a grouping symbol.     * Example: $4^2 \times 2 + [7 - (3 - 5)] = 16 \times 2 + [7 - (-2)] = 32 + [9] = 41$. (Transcript calculation check: $32 + [7 - 4]$ listed in example is inconsistent, but provides a result of $35$).

• Real Number Classification     * Natural Numbers ($N$): Counting numbers $\{1, 2, 3, 4, ...\}$.     * Whole Numbers ($W$): Count starting from zero $\{0, 1, 2, 3, ...\}$.     * Integers ($Z$): Positive and negative whole numbers and zero $\{..., -2, -1, 0, 1, 2, ...\}$.     * Rational Numbers ($R$): Any number that can be expressed as a fraction $\frac{a}{b}$. Includes natural, whole, and integers.     * Irrational Numbers ($Q$): Numbers that cannot be written as fractions (e.g., non-repeating decimals and radicals).     * Real Numbers ($R$): The set encompassing all rational and irrational numbers.

• Integer Operations     * Addition:         * Same signs: Add and keep the sign.         * Different signs: Subtract the smaller absolute value from the larger and keep the sign of the larger.     * Subtraction: Change to addition using $a - b = a + (-b)$ or $a - (-b) = a + b$.     * Multiplication/Division:         * Two numbers: Same signs = Positive; Different signs = Negative.         * Multiple numbers: Odd count of negatives = Negative; Even count of negatives = Positive.

• Algebraic Properties     * Commutative: $a + b = b + a$ and $ab = ba$.     * Associative: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$.     * Distributive: $a(b + c) = ab + ac$.     * Identity: $a + 0 = a$; $a \times 1 = a$.     * Inverse: $a + (-a) = 0$; $a \times \frac{1}{a} = 1$. The term $\frac{1}{a}$ is the reciprocal or multiplicative inverse.     * Zero Properties: $a \times 0 = 0$; $\frac{0}{a} = 0$ ($a \neq 0$). Division by zero is undefined.

• Polynomial Operations     * Combine like terms only. Pay close attention to signs during subtraction.     * Example: $(4x^2 + 3x + 2) - (2x^2 - 3x + 7) = 2x^2 + 6x - 5$.     * Distributive examples: $-2(5a - 4) = -10a + 8$; $2x(5x - 4) = 10x^2 - 8x$.

Equations and Inequalities

• Basic Equations     * One-Step: If $3x = 27$, then $x = 9$. If $\frac{x}{3} = 18$, then $x = 54$.     * Two-Step: Solve in reverse order of operations (Addition/Subtraction first, then Multiplication/Division).     * Multi-Step:         1. Distribute to eliminate parentheses.         2. Combine like terms.         3. Move variables to one side (ideally move the smaller variable or negative variables).         4. Solve remaining equations.     * Big Tip: Eliminate fractions by multiplying all terms by the common denominator.

• Proportions     * An equation where two ratios are equal. Cross products are equal: If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$.

• Literal Equations     * Treat variables as if they were numbers. Solve for a specific letter.     * Example: $A = hw$, solving for $w$ results in $w = \frac{A}{h}$.     * Example: $ax + r = 7$, solving for $x$ results in $x = \frac{7 - r}{a}$.

• Percents     * Conversions:         * Percent to Decimal ($P \rightarrow D$): Move decimal two places left.         * Decimal to Percent ($D \rightarrow P$): Move decimal two places right.         * Percent to Fraction ($P \rightarrow F$): Put number over $100$ and reduce. If decimal is involved, use $1000$ or $10,000$ as denominator accordingly.         * Fraction to Decimal ($F \rightarrow D$): Divide numerator by denominator.     * Percent Equations: Use the translated logic: "is" means "$\text{=}$", "of" means "$\times$", "what" means "$x$".     * Percent Change: $\text{Percent Change} = \frac{\text{Difference (Subtract amounts)}}{\text{Original amount}}$.

• Inequalities     * Symbols: Open circle for $<$ and $>$; closed circle for $\leq$ and $\geq$.     * Critical Rule: When multiplying or dividing by a negative number, REVERSE the inequality sign.     * Absolute Value Inequalities: Solve twice (use positive and negative values). When using the negative value, reverse the inequality.

Parent Functions and Transformations

• Function Types     * Linear: $y = mx + b$     * Absolute Value: $y = a|x - h| + k$     * Exponential: $y = a \cdot b^x$     * Quadratic: $y = a(x - h)^2 + k$     * Rational: $y = \frac{a}{x} + k$ (Note: transcript says $y + k$)     * Square Root (Radical): $a\sqrt{x}$

• Variable meanings in transformations     * $h$: Horizontal shift.     * $k$: Vertical shift.     * $a$: Stretch.     * $m$: Slope.     * A negative sign before $a$ indicates the function is reflected upside down.

• Absolute Value Details     * $+k$ (outside bars): Move up.     * $-k$ (outside bars): Move down.     * $+h$ (inside bars): Move left.     * $-h$ (inside bars): Move right.     * Example: $y = |x - 2| + 3$ (Right 2, Up 3).

• Exponential Functions Details     * $a$: Beginning amount.     * $b$: Rate or multiplier.     * $x$: Number of periods.     * Percentage adjustments: Add increase to $1$ (5% increase$= 1.05$) or subtract decrease from $1$ (1.2 decrease$= 0.988$).

Sequences and Word Problems

• Standardized Study Examples (OSPI Standards)     * Arithmetic Problem: Given $u(0) = 3$ and $u(n+1) = u(n) + 7$.         * The common difference $d = 7$.         * If $u(0) = 3$, then $a_1 - d = 3$, meaning the first term $a_1 = 10$.         * Formula: $a_n = 10 + 7(n - 1)$.         * To find $n$ where $u(n) = 367$: $367 = 3 + 7n \rightarrow 364 = 7n \rightarrow n = 52$.     * Geometric Problem: Given $u(0) = 2$ and $u(n+1) = 3u(n)$.         * The common ratio $r = 3$.         * Starting amount $a = 6$ (from $2 \times 3$).         * Formula: $a_n = 6(3)^{n-1}$.         * $u(4) = 6(3)^3 = 162$.

• Arithmetic Sequences     * Defined by adding a common difference ($d$). To find $d$, subtract the 1st term from the 2nd.     * Recursive Form: $a_n = a_{n-1} + d$.     * Explicit Form: $a_n = a_1 + d(n - 1)$.     * $d$ represents the slope of a line.

• Geometric Sequences     * Defined by multiplying by a common ratio ($r$). To find $r$, divide the 2nd term by the 1st.     * Recursive Form: $a_n = a_{n-1} \times r$.     * Explicit Form: $a_n = a_1 \cdot r^{n-1}$.

• Word Problem Formulas     * DERT: $\text{Distance} = \text{Rate} \times \text{Time}$.     * Ratio problems (4:5:9 type): Add the ratios to create a total denominator ($4+5+9=18$). Each part corresponds to a fraction (e.g., $\frac{4}{18}, \frac{5}{18}, \frac{9}{18}$) multiplied by the total.     * Coin Problems: Use two equations—one for the quantity of items ($d + q = 250$) and one for monetary value ($0.10d + 0.25q = 39.25$).     * Age Problems: Represent ages in the past/future and set up equations based on descriptions (e.g., "Father is 32 years older").     * Wind/Current: Speed with wind is $r + w$; speed against wind is $r - w$. Formula: $(r \pm w)t = d$.     * Digit Problems: Two-digit numbers are represented as $10t + u$ (tens + ones). Reversing digits is $10u + t$.

Systems of Equations

• Classifications     * Inconsistent System: Lines are parallel and never intersect. There is no solution. Algebraic outcome: $0 = 6$ (false statement).     * Consistent Independent System: Lines intersect at one point. One solution.     * Consistent Dependent System: Equations represent the same line. Infinitely many solutions. Algebraic outcome: $0 = 0$ (true statement).

• Methods of Solving     * Substitution: Use when one variable is already isolated or easy to isolate. Substitute that variable’s expression into the other equation.     * Elimination: Use when substitution is difficult. Multiply equations by factors to create opposite coefficients (like $2x$ and $-2x$), then add the equations together to eliminate a variable.

• Inequality Systems     * Shade above the line for $>$ or $\geq$.     * Shade below the line for $<$ or $\leq$.

Exponents and Scientific Notation

• Laws of Exponents     * Product of Powers: $x^m \cdot x^n = x^{m+n}$.     * Power of a Power: $(x^n)^m = x^{nm}$.     * Power of a Product: $(xy)^n = x^n y^n$.     * Negative Answer Rule: Even powers result in positive answers; odd powers of negative numbers stay negative.     * Quotient of Powers: $\frac{x^m}{x^n} = x^{m-n}$.     * Zero Power: $x^0 = 1$.     * Negative Exponents: $x^{-n} = \frac{1}{x^n}$.

• Scientific Notation     * Format: $A \times 10^n$, where $0 < A < 10$.     * Positive exponent ($n$): Move decimal left. Negative exponent ($-n$): Move decimal right.     * Operations: When multiplying, add exponents. When dividing, subtract exponents. Adjust the coefficients to ensure they remain between $0$ and $10$.

Slope and Linear Equations

• Slope Properties     * Formula: $m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$.     * Positive Slope: Upward right trend.     * Negative Slope: Downward right trend.     * Zero Slope ($m = 0$): Horizontal line ($y = c$).     * Undefined Slope: Vertical line ($x = c$).     * Parallel Lines: Slopes are identical.     * Perpendicular Lines: Slopes are opposite reciprocals (e.g., $m = 3$ and $m = -\frac{1}{3}$).

• Forms of Linear Equations     * Slope-Intercept: $y = mx + b$. $b$ is the $y$-intercept.     * Standard Form: $Ax + By = C$. Slope is $m = -\frac{A}{B}$. Rule: $x$ must be positive, and there are no fractions.

Test Taking Tips

• Skip difficult problems and return later.
• Write down all steps; do not perform work mentally.
• Use relaxation techniques (stop, close eyes, deep breath) if panicking.
• Re-do problems from scratch if finished early; do not just review previous work step-by-step.