Comprehensive Algebra I EOC Study Notes

Function Notation and Graphical Interpretation

Function notation $f(x)$ serves as a specialized way to represent the dependent variable $y$. In the context of an expression like $f(3)$, the mathematical inquiry is seeking the value of $y$ when the horizontal coordinate $x$ is exactly $3$. To solve this, one must substitute the number $3$ into the given function for every instance of the variable $x$ and simplify the resulting expression. When interpreting functions from a graph, finding a specific value such as $f(2)$ involves locating $2$ on the $x$-axis and identifying the corresponding height on the $y$-axis. For instance, if the graph indicates a point at $(2, -2)$, then $f(2) = -2$. Conversely, to find $x$ when $f(x) = -1$, one identifies the $x$-coordinate where the graph's vertical position is $-1$. In provided examples, this value is approximately $0.5$.

Data Organization: Domain, Range, and Variables

In algebra, the Domain is defined as the comprehensive list of all possible $x$ values for a function, while the Range is the list of all associated $y$ values. These lists are typically enclosed in braces $\{\}$. Roots are defined specifically as the list of $x$-intercepts, occurring at the precise points where the vertical value $y = 0$. When distinguishing between independent and dependent variables, a useful linguistic test is the sentence structure: "(Independent Variable) is… (Dependent Variable) depends on what." If time is mentioned in a word problem, it is almost universally categorized as the independent variable and plotted on the $x$-axis (horizontal). The dependent variable is plotted on the $y$-axis (vertical).

Statistical Representations: Stem and Leaf and Box and Whisker Plots

A Stem and Leaf plot is an organized chart used to arrange numerical data efficiently. For a dataset like $70, 52, 58, 45, 59, 52, 75, 47, 46$, the first step is identifying the least and greatest values. The "stems" (the tens digits) are written in a vertical column, and the "leaves" (the units digits) are arranged horizontally from smallest to largest. An explanation or key must be provided, such as $5 | 2 = 52$. Box and Whisker plots show how data is spread across four groups using quartiles. The construction involves six steps: drawing a line graph with equal intervals, placing dots for the minimum and maximum values, placing a dot for the median, adding dots for the medians of the lower and upper halves (quartiles), drawing a box around the middle two quartiles, and using an asterisk $*$ to denote extreme data items or outliers.

Correlations and Linear Trends

Correlations describe the relationship between datasets. A positive correlation occurs when both sets of data generally increase together. A negative correlation is observed when one set decreases as the other increases. If the data points suggest no discernible pattern, there is no correlation. A scatter plot utilizes a trend line to visualize these correlations clearly. The "Line of Best Fit" is the most accurate of all possible trend lines. To establish a trend line equation in the slope-intercept form $y = mx + b$, one must first identify the $y$-intercept ($b$) and then calculate the slope ($m$).

Central Tendencies and Data Changes

Central tendencies include the Mode, Median, and Mean. The Mode is the most common number in a set; a set can have multiple modes or no mode if all numbers are unique. The Median is the middle number when data is listed in order; if there are two middle numbers, their average provides the median. The Mean is the calculated average, found by adding all items and dividing by the total count. The Range is the difference between the largest and smallest values. When a data set undergoes a percentage increase, all measures of center (Mean, Median, Mode) and measures of spread (Range) increase. However, during a flat or constant increase, the measures of center increase, but the Range remains unchanged because both the maximum and minimum values are shifted by the same constant amount, preserving the difference.

Parent Functions and Algebraic Transformations

Algebraic functions are categorized by parent forms that dictate their basic shape. A Linear function follows $y = mx + b$. An Absolute Value function is represented by $y = a|x - h| + k$. Exponential functions follow $y = a^x$, and Quadratic functions follow $y = a(x - h)^2 + k$. Rational functions are expressed as $y = \frac{1}{x} + k$, and Square Root or Radical functions follow $y = a\times\text{root}(x)$. In these equations, $h$ represents a horizontal shift, $k$ represents a vertical shift, $a$ represents a stretch, and $m$ denotes slope. A negative sign before the $a$ coefficient indicates an upside-down orientation. For absolute value transformations, a positive number outside the bars moves the graph up, while a negative moves it down. Inside the bars, a positive number moves the graph left, and a negative moves it right. For example, $y = |x - 2| + 3$ shifts the parent function right $2$ units and up $3$ units, while $y = |x + 5| - 6$ shifts it left $5$ units and down $6$ units.

Exponential Growth and Word Problems

Exponential functions of the form $y = ab^x$ use $a$ as the initial amount and $b$ as the rate or multiplier, while $x$ represents the periods the multiplier is applied. Percentage changes must be adjusted from the base of $1$. For example, a $5\%$ increase results in a rate of $1.05$, whereas a $1.2\%$ decrease results in a rate of $0.988$. For word problems, the "DERT" formula ($\text{Distance} = \text{Rate} \times \text{Time}$) is foundational. "Consecutive" refers to numbers in an unbroken sequence. When modeling scenarios, the starting value is the $y$-intercept, and terms like "ratio" or "per" indicate the slope coefficient for $x$. A negative slope is used when a resource is being consumed or "used up." For ratio problems such as "$4:5:9$" involving the angles of a triangle, one method is to sum the ratios ($4 + 5 + 9 = 18$) and use this sum as a denominator to find each individual angle measure.

Sequences: Arithmetic and Geometric

Sequences can be described through recursive or explicit formulas. Arithmetic sequences involve adding a common difference ($d$). To find $d$, one subtracts the first term from the second. The Recursive form is $a_n = a_{n-1} + d$, and the Explicit form is $a_n = a_1 + d(n - 1)$. The common difference $d$ is equivalent to the slope of a line. Geometric sequences involve multiplying by a common ratio ($r$), found by dividing the second term by the first. Its Recursive form is $a_n = a_{n-1} \times r$ and the Explicit form is $a_n = a_1 \times r^{n - 1}$. A specific OSPI standard example involves $u(0) = 3$ and $u(n + 1) = u(n) + 7$. Here, the slope is $7$. If $u(0) = 3$, then $a_1$ (the first term where $n=1$) would be $10$ because $a_1 - d = u(0)$. To find $n$ such that $u(n) = 367$, the equation $367 = 10 + 7(n - 1)$ is solved, yielding $n = 52$. Note that the value was changed from $361$ on the state website to ensure an integer answer.

Numeric Systems and Order of Operations

The hierarchy of real numbers includes: Natural/Counting Numbers ($N \ni \{1, 2, 3, ...\}$), Whole Numbers ($W \ni \{0, 1, 2, ...\}$), and Integers ($I \ni \{..., -2, -1, 0, 1, 2, ...\}$). Rational Numbers ($R$) are those that can be written as a fraction $\frac{a}{b}$. Irrational Numbers ($Q$) cannot be fractions and include non-repeating decimals and radicals. Real Numbers include all rational and irrational values. The order of operations follows: 1) Parentheses (inside out), 2) Exponents, 3) Multiply or Divide (Left to Right), and 4) Add or Subtract (Left to Right). A division bar acts as a grouping symbol. For integers, addition rules vary by sign: same signs require adding and keeping the sign; different signs require subtracting and keeping the sign of the larger absolute value. For multiplication and division, like signs result in a positive, while different signs result in a negative. For multiple numbers, an odd count of negatives results in a negative product, and an even count results in a positive.

Algebraic Properties and Polynomials

Mathematical properties govern equation manipulation. The Commutative Property states $a + b = b + a$ and $ab = ba$. The Associative Property states $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$. The Distributive Property is $a(b + c) = ab + ac$. Identity properties include $a + 0 = a$ and $a \times 1 = a$. Inverse properties include $a + (-a) = 0$ and $a \times \frac{1}{a} = 1$, where $\frac{1}{a}$ is the reciprocal. Regarding properties of zero: $a \times 0 = 0$, $\frac{0}{a} = 0$ (where $a \neq 0$), and division by zero is undefined. When adding or subtracting polynomials, only like terms are combined. For subtraction, it is critical to distribute the negative sign across the entire polynomial being subtracted. For example, $(4x^2 + 3x + 2) - (2x^2 - 3x + 7)$ simplifies to $2x^2 + 6x - 5$.

Linear Equations: Slope and Forms

Slope ($m$) is defined as $\frac{\text{rise}}{\text{run}}$, calculated via the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. A line slanting up has a positive slope, down is negative, horizontal is zero ($y = \text{constant}$), and vertical is undefined ($x = \text{constant}$). Parallel lines possess identical slopes, while perpendicular lines have negative reciprocal slopes. Slope-Intercept Form is $y = mx + b$, where $b$ is the vertical intercept. Standard Form is written as $Ax + By = C$, where $x$ is positive, no fractions exist, and the slope is defined as $-\frac{A}{B}$. To graph inequalities, an open circle is used for $<$ or $>$, and a closed circle for $\leq$ or $\geq$. When solving inequalities, the sign must be reversed if multiplying or dividing by a negative number.

Systems of Equations and Absolute Values

Systems of equations can be solved via Substitution or Elimination. Substitution is preferred when one variable is already isolated or easy to isolate. For instance, if $y = 2x + 3$ and $15x - 5y = 30$, substituting the expression for $y$ leads to $x = 9$ and $y = 21$. Elimination involves multiplying equations by factors to create additive opposites ($10x$ and $-10x$), then adding the equations to remove a variable. Absolute Value equations and inequalities require solving twice: once for the positive version of the expression and once for the negative version. In the case of inequalities like $|x - 6| > 2$, the problem breaks into $x - 6 > 2$ or $x - 6 < -2$, necessitating the reversal of the inequality symbol for the negative case.