Question

Sketch the following sets of points in the \(x-y\) plane. $$ \\{(x, y): x \in[-1,1], y=1\\} $$

Short answer

The graph is a straight line on the y=1 line, starting from x=-1 and ending at x=1. The endpoints at (-1,1) and (1,1) are included in the set.

Step-by-step solution

Understanding the x-Interval

The term \(x \in [-1,1]\) means that the x-values can range from -1 to 1, inclusive. In other words, x is greater than or equal to -1, and less than or equal to 1.

Understand the y-Condition

The expression \(y = 1\) indicates that for all x-values in the given interval, the y-value remains constant at 1. This represents a straight horizontal line on the y=1 line across the x range [-1,1].

Sketching the Graph

To represent this on the \(x-y\) plane, draw a horizontal line at \(y = 1\) from \(x = -1\) to \(x = 1\). Be sure to indicate that the line stops at these points, as the set does not include values of x less than -1 or greater than 1. This could be done by drawing a solid dot at the points \((-1,1)\) and \((1,1)\) to indicate that these points are included in the set.

Key concepts

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a fundamental concept in mathematics, especially when dealing with geometry and algebra. It is composed of two number lines that intersect at a right angle. The horizontal line is referred to as the x-axis, and the vertical line is known as the y-axis. The point where they intersect is called the origin, denoted as the point (0, 0).

Each point on this plane can be identified by an ordered pair of numbers, (x, y), which corresponds to a horizontal value (x) and a vertical value (y). For instance, (-1, 1) means you move 1 unit left (negative direction on the x-axis) and 1 unit up (positive direction on the y-axis) from the origin. The coordinate plane allows for visual representation of equations and sets of points, making it easier to understand the relationship between variables.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. In the example provided, \(x \in [-1,1]\), the square brackets indicate that the endpoints -1 and 1 are included in the set. This is known as a closed interval, where both endpoints are part of the set.

To represent when endpoints are not included, parentheses are used, denoting an open interval. For instance, if the set did not include -1 and 1, it would be written as \(x \in (-1,1)\). It's important to grasp this notation as it specifies precisely which numbers belong to a set without listing them all individually.

Horizontal Line in Graph

A horizontal line in a graph represents a situation where a variable (y, in this case) remains constant regardless of the value of the other variable (x). In the problem given, the condition \(y = 1\) dictates that the y-value is fixed at 1. No matter what x-value you choose within the interval [-1, 1], the y-coordinate will always be 1.

This creates a visual representation in the form of a straight line parallel to the x-axis. When we consider varying values of x from -1 to 1, all these points will lie on the same horizontal plane. Being comfortable with identifying such conditions helps decipher the nature of relations and functions in graph form.

Graph Representation

Graph representation transforms numerical and algebraic data into a visual format, making it more accessible and easier to comprehend. By sketching the appropriate graph, you can illustrate the relationship between variables more concretely. In our example, the set of points lies on a straight line, which is a graphical representation of all points where x is between -1 and 1, and y is constantly 1.

Graphs can come in multiple forms, including lines, curves, and discrete points. To faithfully represent our set, we draw a horizontal line from the point (-1, 1) to (1, 1) and mark these endpoints with solid dots to denote that the line segment includes these boundaries. This physical model allows us to better analyze and understand sets and functions.