Problem 12 - Precalculus with Limits

Question

In Exercises 11-14, find the coordinates of the point. The point is located eight units below the \( x \)-axis and four units to the right of the \( y \)-axis.

Step-by-Step Solution

Answer

The coordinates of the point are (4, -8).

Step 1: Determine the x-coordinate

The point is four units to the right of the y-axis, which puts it into the positive area of the x-axis. Thus, the x-coordinate of the point is 4.

Step 2: Determine the y-coordinate

The point is eight units below the x-axis which locates it into the negative sector of the y-axis. Therefore, the y-coordinate of the point is -8.

Step 3: Report the coordinates of the point

Coordinates on a Cartesian plane are written as an ordered pair in the form (x, y). So putting it all together, the coordinates are (4, -8).

Key Concepts

Understanding the x-axis

The x-axis is a crucial component of the Cartesian coordinate system. Imagine a horizontal line that goes from left to right. This line is the x-axis. It's used to help locate points in a two-dimensional space.
Here are some key facts about the x-axis:

• The x-axis is always horizontal.
• It divides the plane into two halves: the top half (positive) and the bottom half (negative).
• Points that are above or below the x-axis have positive or negative y-coordinates, respectively.
• The x-coordinate of any point tells us how far to move left or right from the origin.

In our problem, a point located eight units below the x-axis means that its y-coordinate is negative since below corresponds to the negative side of the y-axis.

Exploring the y-axis

The y-axis is another essential line in the Cartesian coordinate system. This axis is vertical and helps in locating points in the plane, just like the x-axis.
Here are some things you should know about the y-axis:

• The y-axis runs vertically from bottom to top.
• It divides the plane into two halves: the right half (positive) and the left half (negative).
• The y-axis itself is where all points have an x-coordinate of zero.
• The y-coordinate of any point tells us how far to move up or down from the origin.

In the given exercise, the point is located four units to the right of the y-axis, indicating a positive x-coordinate.

Deciphering ordered pairs

An ordered pair is a set of numbers used to locate a point on a plane in the Cartesian coordinate system. It is composed of two numbers, usually written in the form \((x, y)\).Some key points about ordered pairs include:

• The first number, 'x', tells how far along the x-axis the point is.
• The second number, 'y', tells how far along the y-axis the point is.
• Ordered pairs are crucial for pinpointing exact locations on the coordinate plane.
• The origin, where the x-axis and y-axis intersect, is represented by the ordered pair \((0, 0)\).

Thus, the ordered pair \((4, -8)\) from the exercise tells us that the point is four units to the right of the y-axis and eight units below the x-axis.