Algebra I EOC Exhaustive Study Guide

Function Notation and Data Analysis

Function notation is a specialized way of expressing mathematical relationships where $f(x)$ is essentially a fancy way of saying "y". When evaluating an expression like $f(3)$, the student is determining the value of $y$ when $x$ is $3$; this is calculated by substituting the value into the function and simplifying. This can also be identified graphically. For instance, if one searches for $f(2)$ on a graph where $y = -2$ when $x = 2$, then $f(2) = -2$. Conversely, to find $x$ when $f(x) = -1$, one must locate where the graph passes through the $y$-value of $-1$, which suggests an $x$-value of approximately $0.5$. Key components of these relationships include the domain, which is the list of all $x$ values, and the range, which is the list of all $y$ values, both of which should be enclosed in braces. Roots are defined as the list of $x$-intercepts, or the points where $y = 0$. In any given relationship, identifying the independent versus dependent variable is crucial. This can be decided by placing quantities into a sentence where the independent variable stands alone and the dependent variable "depends on" it. Time is almost always used as the independent $x$ variable, plotted on the horizontal axis, while the dependent variable $y$ is on the vertical axis.

Statistical Representations and Trends

Data can be organized and visualized through various charts and plots. A stem and leaf plot arranges data like ${70, 52, 58, 45, 59, 52, 75, 47, 46}$ by finding the least and greatest values, writing the stems (the tens place) in a column, and arranging the leaves (the ones place) from smallest to largest. An explanation, such as $5|2 = 52$, must be provided. Correlations describe how data sets relate: a positive correlation occurs when both sets increase together, whereas a negative correlation indicates that one set decreases as the other increases. No correlation exists if data sets are unrelated. A trend line on a scatter plot helps clarify these correlations, with the line of best fit acting as the most accurate trend line. To find a trend line, one identifies the $y$-intercept ($b$) and the slope ($m$) to write an equation in slope-intercept form, $y = mx + b$.

Measures of Center and Data Spread

Central tendencies summarize a data set through the mode, median, and mean. Given a list like ${5, 6, 6, 7, 10, 11, 12, 14, 15, 20}$, the mode is the most common number, which is $6$. A set may have multiple modes or no mode if all numbers are unique. The median is the middle number ($10.5$ for this set), found by listing numbers in order and averaging the two center digits if necessary. The mean is the average ($10.6$), calculated by dividing the sum of all items by the total number of items. The range is the measure of spread, representing the difference between the largest and smallest numbers ($15$). Changes to data affect these measures differently. In percentage increases, all measures of center and the range increase. However, in a flat or constant increase, the measures of center increase while the range stays the same because the gap between the adjusted highest and lowest values remains constant.

Box and Whisker Plots and Outliers

Box and whisker plots visualize data distribution by dividing it into four groups. The construction involves drawing a line graph with equal intervals and placing dots for the minimum and maximum values. Further dots are placed for the median and the medians of the first and second halves of the data. A box is then drawn around the two middle quartiles. Extreme data items, or outliers, are designated with an asterisk ($*$). For a sample set like ${20, 36, 58, 45, 59, 55, 75, 35, 35}$, this method provides a clear picture of the concentration and spread of values.

Numerical Classifications and Mathematical Properties

Real numbers ($R$) are categorized into rational and irrational groups. The subset of natural or counting numbers ($N$) includes ${1, 2, 3, 4, …}$, while whole numbers ($W$) add zero to that set (${0, 1, 2, 3, 4, …}$). Integers ($I$) include positive and negative whole numbers. Rational numbers ($R$) are any that can be expressed as a fraction $\frac{a}{b}$, whereas irrational numbers ($Q$) cannot, such as radicals or non-repeating decimals. When ordering real numbers, it is best to convert all formats to decimal form for easy comparison; for example, 2.01 < 2.1. Fundamental properties guide algebraic manipulation: the Commutative Property ($a + b = b + a$; $ab = ba$) and Associative Property ($(a + b) + c = a + (b + c)$; $(ab)c = a(bc)$) handle arrangement and grouping. The Distributive Property is $a(b + c) = ab + ac$, and Identity Properties state $a + 0 = a$ and $a \times 1 = a$. Inverse Properties involve adding the opposite ($a + (-a) = 0$) or multiplying by the reciprocal or multiplicative inverse ($a \times \frac{1}{a} = 1$). Properties of zero dictate that $a \times 0 = 0$, $\frac{0}{a} = 0$, and that division by zero is undefined.

Integer Operations and Order of Operations

When performing arithmetic on integers, same signs in addition result in keeping that sign, while different signs require subtracting the smaller absolute value from the larger and keeping the sign of the larger number. Subtraction should be converted to addition using $a - b = a + (-b)$ or $a - (-b) = a + b$. For multiplication and division of two numbers, same signs result in a positive answer and different signs result in a negative answer. For more than two numbers, an odd count of negative signs leads to a negative product, while an even count results in a positive one. Calculations must follow the order of operations: first Parentheses (inside out), then Exponents, followed by Multiplication or Division (left to right), and finally Addition or Subtraction (left to right). Detailed examples include evaluating $6^2 - 4 \cdot 3 = 36 - 12 = 24$ or solving complex bracketed terms like $4^2 \cdot 2 + [7 - (3 - 5)] = 16 \cdot 2 + [7 - (-2)] = 32 + 9 = 41$. For polynomials, operations are restricted to combining like terms and carefully managing signs during subtraction.

Equations, Proportions, and Literal Isolation

Solving equations requires performing inverse operations in reverse order of PEMDAS. One-step equations like $3x = 27$ require division ($x = 9$), while literal equations like solving for $t$ in $\frac{abt}{3w} + 2 = c$ require systematic isolation: subtracting $2$, multiplying by $3w$, and dividing by $ab$ to reach $t = \frac{3wc - 6w}{ab}$. Multi-step equations involve moving variables to one side and constants to the other. A proportion is an equation showing two equal ratios where cross products are equal; for instance, $\frac{4}{9} = \frac{12}{36}$ because $36 = 36$. If $\frac{x}{25} = \frac{27}{9}$, then $9x = 675$, resulting in $x = 75$. Percentage problems require converting percents to decimals by moving the decimal two places left. Percent change is found by taking the difference between the new and original amounts and dividing by the original. Percent equations are written as read: "is" means $=$, "of" means $imes$, and "what" is the variable $x$.

Slope and Linear Equations

Slope ($m$) is the ratio of rise over run, or $m = \frac{y_2 - y_1}{x_2 - x_1}$. Slopes can be positive ($/$), negative ($\backslash$), zero ($0$ for horizontal lines such as $y = 8$), or undefined (for vertical lines such as $x = 8$). Parallel lines share the same slope, while perpendicular lines have negative reciprocal slopes. Equations are predominantly written in slope-intercept form, $y = mx + b$, where $b$ is the $y$-intercept. For instance, given $m = -2$ and point $(4, 2)$, one can find $b = 10$ to write $y = -2x + 10$. Standard form is expressed as $Ax + By = C$, where $x$ is positive, no fractions exist, and the slope is defined as $\frac{-A}{B}$. Graphing symbols include open circles for $<$ and $>$ and closed circles for $\le$ and $\ge$. When solving inequalities, the sign must be reversed if multiplying or dividing by a negative number.

Systems of Equations and Absolute Value

Systems can be solved using substitution, ideal for when one variable is easily isolated, or elimination, where equations are multiplied by factors to cancel variables. For example, in a system with $15x - 5y = 30$ and $y = 2x + 3$, substituting $y$ results in $x = 9$ and $y = 21$, giving the coordinate solution $(9, 21)$. Systems are classified as inconsistent if lines are parallel (no solution), or consistent. Consistent systems are either independent (one intersection) or dependent (same line, infinite solutions). Absolute value equations and inequalities are solved twice, once for the positive version and once for the negative, with the inequality sign reversed for the negative case. For example, to solve $|3x - 2| = 10$, one sets up $3x - 2 = 10$ and $3x - 2 = -10$.

Parent Functions and Geometric Transformations

The study of functions includes parent functions: 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 ($y = \frac{1}{x} + k$), and Square Root ($y = a\sqrt{x}$). Transformation constants include $h$ for horizontal shifts, $k$ for vertical shifts, $a$ for stretches, and $m$ for slope. A negative sign before $a$ reflects the graph upside down. In absolute value, a positive $k$ moves the graph up, a negative $k$ moves it down, a positive inside $h$ moves it left, and a negative inside $h$ moves it right. Thus, $y = |x - 2| + 3$ represents a shift right $2$ and up $3$.

Exponential Growth, Sequences, and Laws of Exponents

Exponential functions follow $y = ab^x$, where $a$ is the starting amount and $b$ is the multiplier. For percentage changes, add or subtract the rate from $1$; a $5\%$ increase is a multiplier of $1.05$, and a $1.2\%$ decrease is $0.988$. Sequences are categorized as arithmetic (adding a common difference $d$) or geometric (multiplying by a common ratio $r$). Arithmetic formulas include recursive ($a_n = a_{n-1} + d$) and explicit ($a_n = a_1 + d(n - 1)$). Geometric formulas include recursive ($a_n = a_{n-1} \cdot r$) and explicit ($a_n = a_1 \cdot r^{n-1}$). Laws of exponents define operations: Product of Powers ($x^m \cdot x^n = x^{m+n}$), Power of a Power ($(x^m)^n = x^{mn}$), Power of a Product ($(xy)^n = x^n y^n$), and Zero Power ($x^0 = 1$). Negative exponents move the base to the denominator ($x^{-n} = \frac{1}{x^n}$). Scientific notation formats numbers as $a \times 10^n$. Multiplicative operations add exponents ($(3 \times 10^3)(4 \times 10^5) = 1.2 imes 10^9$), while division subtracts them.

Specialized Word Problems and Questions

Word problems utilize specific formulas such as the "DERT" formula ($\text{Distance} = \text{Rate} \times \text{Time}$). For ratio problems like $4:5:9$, add the parts to create a denominator ($18$) to find each measure. Coin problems involve quantity and value equations (e.g., $d + q = 250$ and $0.10d + 0.25q = 39.25$). Age problems compare past, present, or future relative values. Traveling with or against wind/current uses $(r + w)t = d$ or $(r - w)t = d$. For sequences, if $u(0) = 3$ and $u(n + 1) = u(n) + 7$, then $d = 7$ and $a_1 = 10$. To find $u(5)$, use $10 + (7 \cdot 4) = 38$. If seeking $n$ such that $u(n) = 367$, the calculation is $367 = 3 + 7n$, identifying $n = 52$. For a geometric sequence where $u(0) = 2$ and $u(n + 1) = 3u(n)$, then $r = 3$ and $a_1 = 6$, resulting in the explicit formula $a = 6 \cdot r^{n-1}$. To succeed on exams, students should skip difficult problems to return to them later, write down every step rather than doing math mentally, and take deep breaths if panicking. If time permits, redoing problems without referencing previous work ensures accuracy. This guide was produced by Mary Shoemaker, © 2008.