Problem 3 - Contemporary Precalculus

Question

Find the coordinates of the point \(P .\). \(P\) lies 3 units above the \(x\) -axis and on the same vertical line as (-6,7).

Step-by-Step Solution

Answer

Answer: The coordinates of point P are (-6, 3).

Step 1: Determine the x-coordinate of point P

Since P is on the same vertical line as (-6,7), the x-coordinate of P will be the same as the x-coordinate of the given point, which is -6. So, the x-coordinate of point P is -6.

Step 2: Determine the y-coordinate of point P

P lies 3 units above the x-axis. Since the x-axis has a y-coordinate of 0, the y-coordinate of P will be 3 units above 0, which is 3.

Step 3: Write the coordinates of point P

Now that we have both the x and y coordinates of point P, we can write its coordinates as (-6, 3). Therefore, point P has coordinates of (-6, 3).

Key Concepts

Understanding the x-axis

The x-axis is an essential component of coordinate geometry. It forms the horizontal axis in the Cartesian coordinate system. Imagine it like an endless line that runs left to right through the origin, dividing the plane into two parts. The x-axis helps locate points horizontally. Each point on this axis has its y-coordinate equal to zero.

In our exercise, point P is described in relation to its position above the x-axis. This means when we say a point is "3 units above the x-axis," its y-coordinate is 3, since it's 3 steps upwards from 0. The x-coordinate remains as it was given, making it independent of vertical changes.

Remember, the x-axis is vital in determining how horizontally far a point might be. It influences the x-coordinate which, in our example for point P, was derived from the existing condition of being vertically aligned with another point (-6,7). So, point P shares the same x-coordinate as point (-6,7), which is -6.

Exploring the y-axis

The y-axis is the vertical axis in a two-dimensional coordinate plane. It runs up and down, standing perpendicular to the x-axis. Every point lying on the y-axis has an x-coordinate of zero. The y-axis assists in finding out how high or low something is on the plane.

Our exercise focuses on this concept by describing point P's position as 3 units above the x-axis. The y-axis plays a role here because the 3 units above are measured along this vertical line. Being a prominent feature in geometry, the y-axis helps you assess how far above or below a point is from the horizontal x-axis.

For point P, the vertical distance from the x-axis was given as 3 units, making its y-coordinate 3. So, understanding the y-axis helps in understanding the vertical component of any point on the coordinate plane.

Defining coordinates

Coordinates are pairs of numbers that define the exact position of a point on the coordinate plane. They come in the form \(x, y\), where the first number is the x-coordinate, indicating horizontal position, and the second is the y-coordinate, showing vertical position.

Coordinates provide clear direction to locate a point without ambiguity. In this exercise, we sought to establish the coordinates for point P. We knew point P had the same x-coordinate as point (-6,7), which is -6, due to sharing the same vertical line. Its y-coordinate was 3 because it was 3 units above the x-axis.

This results in the coordinates of point P being (-6, 3). Coordinates are essential tools in geometry since they allow students to pinpoint exact locations on the coordinate grid by manipulating x and y values accurately.