Problem 1 - Precalculus: Mathematics for Calculus

Question

The point that is 3 units to the right of the y-axis and 5 units below the x-axis has coordinates (_____ ,_________)

Step-by-Step Solution

Answer

The point has coordinates (3, -5).

Step 1: Understand the Axes

The y-axis is vertical, and moving to the right means increasing the x-coordinates. The x-axis is horizontal, and moving below the x-axis means decreasing y-coordinates (indicating a negative y-value).

Step 2: Move Right Along the X-axis

Starting from the y-axis, move 3 units to the right. Since moving right increases the x-coordinate, the new x-coordinate is 3.

Step 3: Move Down Along the Y-axis

Starting from the x-axis, move 5 units down. Moving down decreases the y-coordinate, so the new y-coordinate is -5.

Step 4: Combine Coordinates

Combine both results: the x-coordinate is 3, and the y-coordinate is -5. Thus, the coordinates of the point are (3,-5).

Key Concepts

x-coordinate

In coordinate geometry, the x-coordinate indicates how far a point is along the horizontal axis, known as the x-axis. It shows the point's horizontal position on the Cartesian plane.
When you move to the right from the y-axis, the x-coordinate increases. Conversely, moving to the left decreases it. This concept is crucial for solving problems like the one given, where determining the correct x-coordinate involves understanding movement along the x-axis.

For example:

• Moving 3 units to the right from the origin gives an x-coordinate of +3 because you're increasing in the positive direction along the x-axis.
• If a point starts at (0,0) and moves 4 units left, the x-coordinate would be -4 since it’s now 4 units in the negative direction.

Always start counting from the y-axis when determining how far to move in the x-direction.

y-coordinate

The y-coordinate in coordinate geometry shows how far a point is on the vertical line of the y-axis. This vertical component describes the position of the point above or below the x-axis.
When a point moves up from the x-axis, the y-coordinate becomes more positive. Conversely, if the point moves down, the y-coordinate becomes more negative.

Let's clarify with examples:

• If you move 5 units below the origin, your y-coordinate is -5. This movement shows a decrease since you're moving in the negative direction of the y-axis.
• In contrast, moving 2 units above the x-axis will give a y-coordinate of +2, indicating you are moving upward in the positive direction.

Always consider the x-axis as your starting point when calculating vertical movement on the Cartesian plane.

cartesian plane

The Cartesian plane is a two-dimensional grid defined by a horizontal x-axis and a vertical y-axis. It creates a framework where every point can be identified by an ordered pair of coordinates, \(x,y\).
When working within the Cartesian plane, each point's location is determined by referencing these axes. The x-coordinate tells you how far left or right to go, while the y-coordinate indicates how far up or down you should move. This dual coordinate system allows precise location of points.

Here are some key aspects:

• The intersection of the x-axis and y-axis is called the origin. It has coordinates (0,0) and is the central reference point.
• The plane is divided into four quadrants. Each quadrant represents a different combination of positive and negative values for x and y coordinates.
• Understanding this system makes graphing not only possible but also easy, enabling you to plot points or interpret the position of points accurately.

By mastering the concepts of the Cartesian plane, students can solve problems more confidently and understand more advanced geometry topics.