Problem 58 - Algebra 2

Question

Graph each set of ordered pairs on a coordinate plane. \(\\{(-3,5),(-2,-4),(3,-2)\\}\)

Step-by-Step Solution

Answer

Plot points at (-3,5), (-2,-4), and (3,-2) on the graph.

Step 1: Understand the Coordinates

Each ordered pair is given in the form \((x, y)\). This means that the first number is the \(x\)-coordinate, and the second number is the \(y\)-coordinate. We will use these coordinates to plot points on a graph.

Step 2: Set Up the Coordinate Plane

Draw a vertical axis (y-axis) and a horizontal axis (x-axis), intersecting at the origin \((0, 0)\). Make sure to label each axis with appropriate positive and negative values.

Step 3: Plot the First Point (-3, 5)

To plot \((-3, 5)\), start at the origin. Move 3 units to the left on the x-axis (because it's negative) and then move 5 units up on the y-axis. Mark this point on the graph.

Step 4: Plot the Second Point (-2, -4)

To plot \((-2, -4)\), start at the origin. Move 2 units to the left on the x-axis and then move 4 units down on the y-axis (because it's negative). Mark this point on the graph.

Step 5: Plot the Third Point (3, -2)

To plot \((3, -2)\), start at the origin. Move 3 units to the right on the x-axis and then move 2 units down on the y-axis. Mark this point on the graph.

Key Concepts

Coordinate Plane

A coordinate plane is a two-dimensional space vital for mathematics and graphing activities. It is made up of two perpendicular lines called axes. The horizontal line is known as the x-axis and the vertical line is the y-axis. These axes intersect at a point called the origin, represented as (0, 0).

Each axis on the coordinate plane is marked with positive and negative numbers. The right and upward sides are positive, whereas the left and downward sides are negative. This structure allows for precise placement of points using ordered pairs.

• X-Axis: Horizontal line, marks left and right.
• Y-Axis: Vertical line, marks up and down.
• Origin: (0, 0) where the x-axis and y-axis cross.

With a coordinate plane properly set up, plotting points and tracing lines becomes straightforward and systematic.

X-Coordinate

The x-coordinate is the first number in an ordered pair like (3,-2). It tells us how far the point is from the origin along the x-axis. Imagine it as a direction: moving left or right depending on if the number is negative or positive.

When graphing a point:

• If the x-coordinate is positive, move right from the origin.
• If it's negative, move left from the origin.

The position on the x-axis guides our horizontal movement before we consider the vertical movement with the y-coordinate.

Y-Coordinate

In every ordered pair, the y-coordinate is the second number. It indicates the vertical position of a point on the y-axis. This value expresses how far up or down the point sits from the origin.

To plot:

• A positive y-coordinate means move up the number of units.
• A negative y-coordinate means move down.

By understanding the y-coordinate, we accurately complete the positioning of our point after moving horizontally based on the x-coordinate.

Together, the x and y coordinates tell the precise location of any point on the plane.

Ordered Pairs

An ordered pair, such as (-3, 5), contains a specific sequence of two numbers. This sequence is crucial as it determines the placement of points on the coordinate plane.

Ordered pairs always appear in the format (x, y). The sequence matters:

• The first number is the x-coordinate, guiding horizontal movement.
• The second number is the y-coordinate, indicating vertical movement.

This standard format ensures clarity and consistency, making it easy to plot points. By following the order, we accurately navigate the graph, ensuring each plotted point is in the correct position on the coordinate plane.