Algebra 1 Unit 3: Linear Functions Comprehensive Vocabulary Guide

Sequence: An ordered list of numbers that follows a specific mathematical pattern or rule. Each number in the group is identified as an element or a member of the set.
Term of a Sequence: An individual value within a sequence. These are indexed by their position using a subscript, where the first term is written as $a_{1}$ , the second as $a_{2}$ , and any general term as $a_{n}$ .
Arithmetic Sequence: A numerical sequence in which the difference between any two consecutive terms remains constant throughout. This suggests a linear growth or decay pattern.
Geometric Sequence: A numerical sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero constant. This reflects exponential growth or decay.
Common Difference: The constant value, denoted by the variable $d$ , that is added to each term in an arithmetic sequence to yield the following term. It is calculated as $d = a_{n} - a_{n-1}$ .
Common Ratio: The constant factor, denoted by the variable $r$ , that is multiplied by each term in a geometric sequence to yield the following term. It is calculated as $r = \frac{a_{n}}{a_{n-1}}$ .

Explicit Formula: A mathematical rule that allows for the calculation of the value of any term $a_{n}$ in a sequence directly based on its position $n$ without needing to know the preceding terms.
- Arithmetic explicit formula: $a_{n} = a_{1} + (n - 1)d$
- Geometric explicit formula: $a_{n} = a_{1} \times r^{n-1}$
Recursive Formula: A rule that defines each term of a sequence based on the value of the term or terms that come before it. A recursive formula must always include a starting value, such as $a_{1}$ .
- Arithmetic recursive formula: $a_{n} = a_{n-1} + d; a_{1} = \text{initial value}$
- Geometric recursive formula: $a_{n} = a_{n-1} \times r; a_{1} = \text{initial value}$

Relation: Any set of ordered pairs $(x, y)$ . A relation describes a connection or mapping between two sets of data.
Linear Function: A specific type of function whose graph yields a straight line. It has a constant rate of change (slope) and is typically expressed in the slope-intercept form: $f(x) = mx + b$
Function Notation: A method of expressing a mathematical function where the name of the function (often $f$ , $g$ , or $h$ ) is combined with the input variable enclosed in parentheses. For example, $f(x)$ is read as " $f$ of $x$ " and represents the output value for the input $x$ .
One-to-One Function: A specialized function in which every element in the domain corresponds to exactly one unique element in the range, and every element in the range corresponds to exactly one unique element in the domain. It passes both the Vertical Line Test and the Horizontal Line Test.

Domain: The set of all possible input values (typically independent variables or $x$ -values) for which a relation or function is defined.
Range: The set of all possible output values (typically dependent variables or $y$ -values) that result from applying the function's rule to the values in the domain.
Continuous: Data or functions represented by an unbroken curve or line on a graph. This implies that values can include every possible real number within an interval, including decimals and fractions.
Discrete: Data or functions consisting of distinct, separate points on a graph. Discrete values represent countable items where intermediate values (e.g., between $1$ and $2$ ) are not possible.

Correlation: A statistical relationship between two variables, indicating how closely they change together. Correlation can be measured in terms of both strength and direction.
Positive Association / Positive Correlation: A relationship where both the independent and dependent variables increase or decrease together. On a scatter plot, the trend line has a positive slope.
Negative Association / Negative Correlation: A relationship where an increase in one variable corresponds to a decrease in the other. On a scatter plot, the trend line has a negative slope.
No Association: A scenario where there is no detectable pattern or relationship between the variables, and the data points appear scattered randomly.
Causation: The principle that one event (the independent variable) is the direct cause of the occurrence of another event (the dependent variable). Caution must be exercised, as correlation between two variables does not automatically imply causation.
Correlation Coefficient: A numerical value, represented by the variable $r$ , that quantifies the strength and direction of the linear relationship between two variables. The value ranges from $-1$ (perfect negative correlation) through $0$ (no correlation) to $1$ (perfect positive correlation).

Linear Regression: A statistical modeling technique used to find the linear equation (the line of best fit) that characterizes the relationship between variables as accurately as possible.
Line of Best Fit: A straight line that traverses a scatter plot of data points in a way that minimizes the overall distance from all points to the line. It serves as a mathematical model for the data set.
Trend Line: An alternative name for a line of best fit, used to visualize the general movement or direction of data points over time or across variables.
Residual: The numerical difference between an actual observed data point and the value predicted by the line of best fit. It is calculated as: $\text{Residual} = y_{\text{observed}} - y_{\text{predicted}}$
Interpolation: The process of estimating or predicting a value within the range of the existing data points used to create the model.
Extrapolation: The process of predicting a value outside the range of the given data set. This is generally considered less reliable than interpolation because the trend may not continue indefinitely.