Introduction to the Coordinate Plane, Domain, and Range

Foundations of the Coordinate Plane

The Cartesian coordinate system is established by two perpendicular lines intersecting at a central point. These lines are referred to as axes.

• The Horizontal Axis: This is called the $x$-axis.

• Incremental Division: The axes are divided into equal increments, which can be counted by units of one ($1$), two ($2$), or other increments as needed.

• Plotting Points: These increments allow for the plotting of points on a flat surface. A point is defined by its position relative to these two axes.

Ordered Pairs and Coordinates

The location of any point on a coordinate plane is represented by an ordered pair. An ordered pair is written within parentheses and identifies a specific location through two numerical values.

• The $x$-coordinate: This is the number on the left side of the ordered pair. It indicates the distance and direction to move along the horizontal $x$-axis. For example, in the ordered pair $(2, 3)$, the value $2$ indicates moving two units along the $x$-axis.

• The $y$-coordinate: This is the number on the right side of the ordered pair, separated from the $x$-coordinate by a comma. It indicates the vertical position. In the pair $(2, 3)$, the value $3$ is the $y$-coordinate.

• Representation: The standard format for an ordered pair is $(x, y)$.

Examples of Specific Points:

• A point located at $(2, 3)$ signifies moving $2$ units along the $x$-axis and $3$ units up the $y$-axis.

• A point located at $(-3, 4)$ signifies moving $-3$ units on the $x$-axis and approximately $4$ units on the $y$-axis.

• A point located at $(-3, 0)$ signifies moving $-3$ units on the $x$-axis with a vertical position of $0$.

Domain and Range of Discrete Point Sets

When a graph or data set consists of individual, distinct points, the domain and range are described as sets of specific values.

Domain

• Definition: The domain is the set of all input values, which correspond to the $x$-values the graph takes on.

• Notation: The domain is expressed using braces $\{ \}$ to denote a set.

• Set Rules:

  • The order of numbers within the braces does not matter. For example, $\{2, -3, 0\}$ is equivalent to $\{-3, 2, 0\}$.

  • If an $x$-value occurs more than once in the data set (for example, if two different points share the same $x$-coordinate), that value only needs to be listed once in the domain set.

Range

• Definition: The range is the set of all output values, which correspond to the $y$-values that the graph takes on.

• Notation: Similar to the domain, the range is written within braces $\{ \}$.

• Set Rules: Like the domain, duplicate $y$-values are listed only once, and the numerical order of the elements inside the braces is not strictly required.

Alternative Data Representations

Information regarding points can be presented in multiple formats, all of which allow for the identification of domain and range.

• Sets of Ordered Pairs: Given a set such as $\{(-3, 2), (5, 4), (9, -8), (0, y)\}$, the domain is identified by listing all first numbers in the pairs: $\{-3, 5, 9, 0\}$. The range is identified by listing all second numbers: $\{2, 4, -8, \dots \}$.

• Tables: In a standard table, the $x$-column is typically on the left and the $y$-column is on the right. To find the domain and range, one simply lists the unique values from the respective columns. For instance, if the $y$-column contains the values $-7, -7, -2, -2$, the range is recorded simply as $\{-7, -2\}$.

• Mappings: A mapping consists of two sets: the set of $x$-values on the left (the input) and the set of $y$-values on the right (the output). Arrows indicate which input connects to which output.

  • Example Mapping: If the input set contains $1, 3, 11$ and the output set contains $5, 6, 2$, the domain is $\{1, 3, 11\}$ and the range is $\{5, 6, 2\}$.

Domain and Range for Continuous Curves

A continuous curve represents a graph made of infinitely many points without any breaks. Because the points are infinite, braces and discrete lists cannot be used to represent the domain and range. Instead, interval notation is required.

Interval Notation Rules

• Directionality: Entries must always be written in order from the smallest (lowest) value to the biggest (largest) value. Writing an interval as $[\text{Largest}, \text{Smallest}]$ is mathematically incorrect.

• Brackets $[ \text{ or } ]$: These are used to indicate that the endpoint value is included in the set. This corresponds to a closed circle or dot on a graph.

• **Parentheses $( \text{ or } )$: ** These are used to indicate that the endpoint value is not included. This corresponds to an open circle on a graph. Parentheses are also always used for infinity ($\infty$) or negative infinity ($-\infty$) because these represent directions rather than specific reachable numbers.

Example Case Study 1: Finite Continuous Curve

• Domain Analysis: If the leftmost point of a curve is at $x = -4$ (marked by a closed dot) and the rightmost point is at $x = 7$ (marked by a closed dot), the domain is written as: $[-4, 7]$.

• Range Analysis: If the lowest point on the vertical axis reached by the curve is $y = -4$ and the highest point is $y = 8$, the range is written as: $[-4, 8]$.

Example Case Study 2: Infinite Continuous Curve

• Context: A curve begins at a specific point and extends upwards and to the left indefinitely.

• Domain Analysis: Moving "forever to the left" indicates that the $x$-values approach negative infinity ($-\infty$). If the furthest right the graph goes is $x = 3$ (and it includes that value), the domain is: $(-\infty, 3]$.

• Range Analysis: The lowest point the graph reaches on the $y$-axis is at height $0$. Because the graph has an arrow pointing upwards, the $y$-values extend to infinity ($\infty$). Therefore, the range is: $[0, \infty)$.

Course Administration

• Practice and Mastery: Determining domain and range, especially for continuous curves, requires repetition, memory, and practice to achieve certainty.

• Quiz 1: The first quiz for the course consists of successfully registering with the MyMathLab platform.

Questions & Discussion

Question: For the graph extending upward and to the left, would the domain be between $0$ and $3$?

Response: No. The $x$-axis is a horizontal line. As a graph moves further to the left, it passes $x = -3, -5, -10$, and so on. If it goes forever to the left, it has no "smallest" value, so we represent this as negative infinity ($-\infty$). The value $0$ is not the lowest point if the graph continues past it to the left.

Question: What is the domain of the graph extending forever to the left and stopping at $x = 3$?

Response: The domain is $(-\infty, 3]$. We use the negative infinity symbol to express "forever and ever to the left." We must put the smaller value ($-\infty$) on the left and the larger value ($3$) on the right. Parentheses are always used next to the infinity symbol.

Question: Is the range for that same graph $[0, \infty)$?

Response: Yes. The smallest $y$-value is $0$, and since the graph points upward forever, the largest value is infinity ($\infty$).