Problem 14 - Algebra 1

Question

Plot and label the ordered pairs in a coordinate plane. $$A(5,2), B(4,3), C(-2,-4)$$

Step-by-Step Solution

Answer

Plot the points on the coordinate plane by starting at the origin, then moving according to the x-coordinate (horizontal) and y-coordinate (vertical) of each of the ordered pairs: A(5,2), B(4,3), and C(-2,-4).

Step 1: Identify the Ordered Pairs

Identify the ordered pairs. In a pair, the first number represents the position on the x-axis (horizontal), and the second number represents the position on the y-axis (vertical). Slowly moving from the origin (point at the center, (0,0)), the ordered pairs are: A is (5, 2): move 5 steps to the right and 2 steps up from the origin. B is (4, 3): move 4 steps to the right and 3 steps up from the origin. C is (-2, -4): move 2 steps to the left and 4 steps down from the origin.

Step 2: Plotting Points on the Coordinate Plane

Now plot the points on the x-y plane according to the given ordered pairs. Start at the origin, then move according to the x-coordinate (left or right), then move according to the y-coordinate (up or down), and draw a dot representing each point.

Step 3: Label the Points

Finally, label each point with its corresponding letter (A, B, or C) so that you know which point is which.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It is formed by the intersection of two perpendicular number lines, the horizontal x-axis, and the vertical y-axis, which meet at a point called the origin.

In order to plot a point in this plane, you need to understand its layout. The x-axis runs left to right, and the y-axis runs top to bottom. Each axis is labeled with both positive and negative numbers, with the origin marked as (0,0). The plane is divided into four quadrants where the axes meet. The top-right quadrant is the first, followed by the top-left as the second, bottom-left as the third, and bottom-right as the fourth.

When plotting points on a coordinate plane, it is essential to move in the correct direction from the origin. Positive x-values are found to the right of the origin, while negative x-values are to the left. For y-values, positive numbers are above the origin, whereas negative numbers are below it.

Ordered Pairs

Ordered pairs are essential in understanding how to navigate a coordinate plane. Each ordered pair represents the exact location of a point on the plane and consists of an 'x' value followed by a 'y' value, typically written as (x, y).

The x-value, or abscissa, tells you how far to move horizontally from the origin and in which direction - to the right for positive values and to the left for negative ones. The y-value, or ordinate, informs you about the vertical movement from the origin, up for positive values and down for negative ones.

Let's take the ordered pair (5,2) as an example. Here, the number 5 is the x-coordinate, which means you move 5 units to the right from the origin. The number 2 is the y-coordinate, indicating that from that new position, you move 2 units up. As a tip, always start by moving horizontally and then vertically from the origin to locate your point accurately on the coordinate plane.

X-Axis and Y-Axis

Understanding the x-axis and y-axis is crucial for correctly using a coordinate plane. The x-axis is horizontal and is associated with the width or the left and right direction from the central point, the origin. The y-axis is vertical and deals with the height, or the up and down direction from the origin.

The x and y-axes are used to determine the position of a point in two dimensions. If you have a positive x-value, you will move to the right of the origin. If the x-value is negative, you move to the left. Similarly, for the y-axis, positive values mean moving up from the origin, while negative values mean moving down.

Quadrants and Axes

The intersecting axes create four quadrants which are used to group the ordered pairs based on the signs of their coordinates. Here's how it is divided:

• Quadrant I: x > 0, y > 0
• Quadrant II: x < 0, y > 0
• Quadrant III: x < 0, y < 0
• Quadrant IV: x > 0, y < 0

These quadrants are important as they help quickly identify which section of the plane a point lies. Knowing how to read the x-axis and y-axis will give you the ability to accurately plot any ordered pair provided.