Problem 14 - Precalculus with Limits

Question

In Exercises 11-14, find the coordinates of the point. The point is on the \( x \)-axis and 12 units to the left of the \( y \)-axis.

Step-by-Step Solution

Answer

The coordinates of the point are (-12, 0).

Step 1: Analyze Information

First, check where the point lies. The exercise says the point is on the x-axis and 12 units to the left of the y-axis. That means it's not above or below the x-axis as it's sitting exactly on it which makes its y-coordinate 0.

Step 2: Find the X-Coordinate

Since the point is 12 units to the left of the y-axis, this means the x-coordinate value is -12. It's negative because it's to the left side of the y-axis.

Step 3: Combining the Coordinates

Now that we have the x and y coordinates, these can be combined to give the coordinates of the point. So, the point is (-12, 0).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface that can help visualize and locate points in space easily. Imagine it as a grid laid out like a flattened piece of graph paper, stretching infinitely in all directions. This grid has two number lines that intersect at a right angle. Each point on this plane is defined by two numbers - these numbers are called "coordinates" and they come in pairs. One number tells you how far to move horizontally, while the other tells you how far to move vertically.

The two number lines that help form the coordinate plane are the X-axis and Y-axis. These axes break down the plane into four quarters, known as quadrants, but for now, our focus will be on understanding how these axes work together to pinpoint any location on the grid. The point where these two lines cross is called the origin, where the coordinates are (0,0).

X-axis

The X-axis is the horizontal number line in the coordinate plane. It runs left and right through the plane, like the horizon on a landscape. If you are trying to plot a point and need to know how far left or right to move, the X-axis is your guide. The center of the X-axis, where it meets the Y-axis, is called the origin.

• If you move to the right along the X-axis, the values increase positively.
• If you move to the left along the X-axis, the values decrease and become negative.

For instance, in our exercise, the point lies on the X-axis. It is 12 units to the left of the Y-axis, giving it an X-coordinate of -12. This means you start from the origin (0,0) and move 12 spaces to the left, which results in a negative value because it's on the left side of the Y-axis.

Y-axis

The Y-axis differs from the X-axis because it is vertical and runs up and down the coordinate plane. Think of it as the flagpole that stands at the point where the X-axis and Y-axis intersect. When plotting coordinates, the Y-axis determines how far up or down to move from the origin.

• Moving up from the origin along the Y-axis yields positive values.
• Moving down from the origin yields negative values.

In our exercise, however, the point lies directly on the X-axis. This means its position on the Y-axis is zero. It does not move up or down from the X-axis, confirming a Y-coordinate of 0. Thus, any point lying on the X-axis will have a Y-coordinate of zero, as it does not deviate vertically.

Cartesian Coordinates

Cartesian coordinates are critical for understanding positions on the coordinate plane. Each point on this plane is defined using a pair of numbers, which we refer to as the Cartesian coordinates. These numbers tell how far and in what direction to move from the origin.

• The first number in the pair is the X-coordinate, which tells how far to move left or right from the origin.
• The second number is the Y-coordinate, which tells how far to move up or down.

This pair, noted as (x, y), determines the direction and distance on the plane. In our exercise, the coordinates (-12, 0) tell us that the point is 12 units to the left along the X-axis and does not move up or down, hence the zero in the Y-position. Visualizing this as a journey from the origin can help: start at zero, count 12 steps left, and stop there. This is how the Cartesian system helps us communicate an exact position in space.