Problem 1 - Precalculus Essentials

Question

Plot the given point in a rectangular coordinate system. (1,4)

Step-by-Step Solution

Answer

The plotted point on the rectangular coordinate system correspond to the position (1,4)

Step 1: Identify the coordinates in the problem.

In this case, the given point is (1,4). This means that 1 is the x-coordinate and 4 is the y-coordinate.

Step 2: Plot the x-coordinate on the number line.

On your rectangular coordinate system, go to the position that corresponds to 1 on the x-axis (horizontal number line).

Step 3: Plot the y-coordinate on the number line.

From the position that corresponds to 1 on the x-axis, go to the position that corresponds to 4 on the y-axis (vertical number line). Mark the point where these positions intersect.

Key Concepts

Coordinates

Coordinates are an essential part of the rectangular coordinate system. They provide information about the position of a point in space. Each point in this system is defined by an ordered pair, typically written as \((x, y)\).

• The first number in the pair is called the x-coordinate. It tells us how far along the point is on the horizontal axis.
• The second number is the y-coordinate. This number indicates the position along the vertical axis.

To plot a point, one must understand both the x-coordinate and the y-coordinate. In our example, the coordinates \((1, 4)\) mean the point is located 1 unit along the x-axis and 4 units up the y-axis.
Understanding how to read and plot coordinates is vital for graphing points accurately in any application that uses the rectangular coordinate system.

X-Axis

The x-axis is a fundamental component of the rectangular coordinate system. It is the horizontal line that extends from left to right and is often labeled as "x". The x-axis serves as a foundation for determining the x-coordinate of any point.

• An x-coordinate tells how far a point is from the origin along the x-axis.
• Positive x-values are positioned to the right of the origin, whereas negative x-values are placed to the left.

For example, in the point \((1, 4)\), the x-coordinate is 1, which indicates that the point is 1 unit to the right of the origin. The origin is the point \((0, 0)\), where the x-axis and y-axis intersect. Knowing how to interpret and work with the x-axis is important when plotting points and graphing various mathematical functions.

Y-Axis

The y-axis is the vertical counterpart to the x-axis in the rectangular coordinate system. It runs up and down, helping to measure how high or low a point is from the origin. Points to assess their location along the y-axis use their y-coordinates.

• Positive y-values are located above the origin.
• Negative y-values can be found below the origin.

When plotting our point \((1, 4)\), the y-coordinate 4 tells us that the point is 4 units above the x-axis. Much like the x-axis, understanding the y-axis is critical for graphing equations and analyzing patterns in data. It provides a basis for comprehending vertical changes in any graphical representation.

Plotting Points

Plotting points on the rectangular coordinate system is a straightforward process once you understand the structure. Here’s a concise guide to plotting:

• Start by identifying the coordinates \((x, y)\) of your point.
• From the origin \((0, 0)\), move horizontally to the x-coordinate on the x-axis.
• Once aligned with the correct x-value, move vertically to the y-coordinate on the y-axis.
• Mark the point where these two positions meet.

For our example point \((1, 4)\), begin at the origin, move 1 unit right, and then 4 units up. Mark the intersection to plot the point. Practice plotting regularly, as it aids in visualizing mathematical concepts and solving graph-related problems successfully.