Problem 6 - Prealgebra

Question

Graph each of the following ordered pairs. $$(-3,4)$$

Step-by-Step Solution

Answer

Graph the point (-3, 4) by moving 3 units left and 4 units up from the origin.

Step 1: Understanding the Coordinate Plane

A coordinate plane consists of a horizontal axis (the x-axis) and a vertical axis (the y-axis). Each point on the plane is represented by an ordered pair (x, y), where 'x' is the horizontal position and 'y' is the vertical position.

Step 2: Identifying the Components of the Ordered Pair

The ordered pair is (-3, 4). This means you have an x-coordinate of -3 and a y-coordinate of 4.

Step 3: Placing the Point on the X-axis

Start at the origin (0, 0) on the x-axis. Since the x-coordinate is -3, move three units to the left because negative values indicate leftward movement.

Step 4: Placing the Point on the Y-axis

From the position at x = -3, use the y-coordinate of 4 to move four units upward on the y-axis. This indicates the location on the y-axis corresponding to y = 4.

Step 5: Marking the Ordered Pair on the Plane

After moving -3 units on the x-axis and 4 units on the y-axis, you arrive at the position (-3, 4). Mark this location with a dot or a small cross, representing the ordered pair on the graph.

Key Concepts

Ordered Pairs

When working on the coordinate plane, ordered pairs play a crucial role. An ordered pair is made up of two coordinates, written in the form \((x, y)\). This structure provides a specific location in the coordinate system.

- The first number in the pair represents the x-coordinate.- The second number signifies the y-coordinate.

Each ordered pair pinpoints a unique spot on the plane, similar to providing specific directions to a destination. If you imagine giving directions, the x and y values are your coordinates, just like telling someone to "go 3 steps east and 4 steps north." The precision of ordered pairs ensures clarity in locating any point within the two-dimensional space of a graph.

X-Coordinate

The x-coordinate in an ordered pair like \((-3, 4)\) gives the horizontal position of a point on the coordinate plane. Often, it is the first number in the ordered pair notation.

- If this number is positive, move to the right on the x-axis.- If it is zero, the point is located right on the vertical y-axis.- If it is negative, as in our example, move to the left.

Understanding the x-coordinate helps create a mental image of where to start or place your point horizontally, providing the first step in plotting any position on a graph.

Y-Coordinate

Once you've positioned yourself horizontally using the x-coordinate, the y-coordinate guides your vertical placement on the coordinate plane. In an ordered pair like \((-3, 4)\), the y-coordinate is the second number, helping you figure out how high or low to place the point.

- Positive values mean you move upward.- A zero value places you directly on the horizontal x-axis.- Negative values indicate a movement downward.

Think of this as adjusting your altitude on a graph where the sky represents positive numbers and the ground represents negative. With the y-coordinate, combined with the x-coordinate, you precisely identify the target location on the grid.