Problem 2 - Contemporary Precalculus

Question

Find the coordinates of the point \(P .\). \(P\) lies 4 units to the left of the \(y\) -axis and 5 units below the \(x\) -axis.

Step-by-Step Solution

Answer

Answer: The coordinates of point P are (-4, -5).

Step 1: Determine the x-coordinate of point P

Since P lies 4 units to the left of the y-axis, its x-coordinate is -4.

Step 2: Determine the y-coordinate of point P

Since P lies 5 units below the x-axis, its y-coordinate is -5.

Step 3: Write the coordinates of point P

The coordinates of point P are (-4, -5).

Key Concepts

x-coordinate

In coordinate geometry, the x-coordinate is the first number in an ordered pair \(x, y\). It represents a point's horizontal position on the Cartesian plane. Imagine the x-axis as a number line stretching left and right.

- Positive values show how far to the right of the y-axis a point is located. - Negative values show how far to the left.When a point lies to the left of the y-axis, its x-coordinate is a negative number. So, if point P is 4 units to the left of the y-axis, we understand this as shifting 4 units in the negative direction along the x-axis. Hence, the x-coordinate for point P is \(-4\).
Understanding the role of the x-coordinate is essential when pinpointing exact locations on a plane. Recognizing whether it is positive or negative helps us understand the direction in relation to the y-axis.

y-coordinate

The y-coordinate is an equally critical aspect of coordinate geometry, representing a point's vertical position in a two-dimensional space, known as the Cartesian plane. Think of the y-axis as the vertical number line.

- Positive values indicate a position above the x-axis. - Negative values imply the point is below.When a point is situated below the x-axis, the y-coordinate will be negative. In this exercise, with point P being 5 units below the x-axis, we determine its y-coordinate as \(-5\). This negative value indicates P's downward location from the x-axis.
Comprehending the y-coordinate helps define a point's specific place, especially alongside the x-coordinate, ensuring we can accurately navigate any point on the Cartesian plane.

Cartesian plane

The Cartesian plane is a foundational tool in coordinate geometry. It's a grid formed by two perpendicular number lines, the x-axis, and the y-axis. These axes divide the plane into four quadrants.

The x-axis runs horizontally and the y-axis vertically, intersecting at the origin \(0, 0\). Each point on the plane is identified with an ordered pair \(x, y\), where:

• The x-coordinate shows horizontal distance from the y-axis.
• The y-coordinate represents vertical distance from the x-axis.

The plane offers a space where precise descriptions of locations are possible. So, to find point P's location, we see it moves to \(-4, -5\), meaning 4 left along the x-axis and 5 down from the y-axis.
The essence of the Cartesian plane lies in transforming abstract numerical relationships into clear visual maps, simplifying complex mathematical problems to understandable geometric concepts.