Problem 19 - Thinking Mathematically

Question

Plot the given point in a rectangular coordinate system. \((1.25,-3.25)\)

Step-by-Step Solution

Answer

The point \((1.25, -3.25)\) has been successfully plotted in the rectangular coordinate system.

Step 1: Identify the point to plot

Identify the coordinates of the given point. The given point is \((1.25, -3.25)\). This means the x-coordinate is 1.25 and the y-coordinate is -3.25.

Step 2: Draw the coordinate system

Next step involves drawing a two-dimensional coordinate system, known as the Cartesian coordinate system. This system consists of two perpendicular lines or axes - the horizontal line is called the x-axis and the vertical line is called the y-axis. The intersection of these two axes is known as the origin, where both x and y values are 0.

Step 3: Plot the point

Now, the point can be plotted. Start at the origin. Since the x-coordinate is 1.25, a right move should be made along the x-axis to 1.25. As the y-coordinate is -3.25, move down along the y-axis to -3.25. Mark this point, it represents the given coordinates \((1.25, -3.25)\).

Key Concepts

Rectangular Coordinate System

Understanding the rectangular coordinate system is fundamental in graphing points. It is a two-dimensional plane made up of two number lines that intersect at a right angle. The horizontal line is known as the x-axis, and the vertical line is the y-axis. These intersect at a point called the origin, designated as (0,0).

Each point on this plane is represented by a pair of numbers known as coordinates. The first number in the pair is the x-coordinate, and it shows the point's horizontal position relative to the origin. The second number is the y-coordinate, indicating vertical position. Points above the origin have positive y-coordinates, while those below have negative y-coordinates. Similarly, points to the right of the origin have positive x-coordinates, and those to the left have negative x-coordinates.

Cartesian Coordinate System

The Cartesian coordinate system, named after René Descartes, provides a uniform grid for plotting points. By dividing the plane into four quadrants using the x-axis and y-axis, locating any point becomes simpler.

• The first quadrant (top-right) contains points with positive x and y values.
• The second quadrant (top-left) has points with negative x values but positive y values.
• The third quadrant (bottom-left) includes points with negative x and y values.
• The fourth quadrant (bottom-right) holds points with positive x values and negative y values.

This organization aids in understanding the relationship between algebraic equations and their graphical representations.

X-axis and Y-axis

In graphing, the axes are the backbone of the coordinate system. The x-axis runs horizontally and is used to determine the left or right position of a point. The y-axis runs vertically and decides the up or down placement of a point.

Understanding Positive and Negative Axes

The x-axis has positive values to the right of the origin and negative values to the left. Conversely, the y-axis has positive values above the origin and negative values below. By knowing which direction to move along these axes, we can accurately plot any given point, such as moving right to plot a positive x-coordinate or down for a negative y-coordinate.

Graphing Coordinates

Graphing coordinates refers to the process of plotting points on the Cartesian plane according to their x and y values. To graph a point, begin by starting at the origin.

Depending on the x-coordinate, move horizontally to the right for positive and to the left for negative. Then, adjust vertically, moving up for a positive y-coordinate or down for negative. The final position where these two movements meet is where the point is plotted.

• If an x-coordinate is zero, the point lies on the y-axis.
• If a y-coordinate is zero, the point lies on the x-axis.

Practice by graphing multiple points can help solidify the understanding of coordinates and their positions on the plane.