Problem 5 - Thinking Mathematically

Question

Plot the given point in a rectangular coordinate system. \((-3,-5)\)

Step-by-Step Solution

Answer

The point (-3,-5) is plotted by moving 3 units to the left from the origin (as the x-coordinate is negative) and five units downwards from there (as the y-coordinate is negative) in the rectangular coordinate system.

Step 1: Understand the Point's Coordinate

The given point is (-3,-5). In a coordinate point, the first number relates to the x-coordinate and the second one to the y-coordinate. In this case, the x-coordinate is -3 and the y-coordinate is -5.

Step 2: Initialize the Graph

Draw two perpendicular lines on a plane that meet at a point called the origin. One line runs horizontally (left to right) called the x-axis, and a line running vertically (down to up) called the y-axis. These lines create four quadrants on the plane.

Step 3: Plot the Point

Start at the origin (the point where the x-axis and y-axis intersect). Move along the x-axis to the left as it's a negative number until you reach -3. Then move downwards along the y-axis as it is also a negative number until you reach -5. Mark this point which is the location of (-3, -5).

Key Concepts

Plotting Points

When plotting points in a coordinate system, we turn pairs of numbers into visual representations on a graph. The first number in a pair is the x-coordinate, and the second is the y-coordinate. This tells you how far to move horizontally and vertically from the origin, which is the point (0,0) on the graph. For example, let's consider the point (-3, -5).

• The x-coordinate is -3: you move 3 units to the left because it's negative.
• The y-coordinate is -5: you go 5 units down since it is also negative.

Finding the perfect spot to place your point involves following those movements precisely. Make sure to start at the origin, which is the center of your graph. Then, by moving horizontally and then vertically, you mark the exact spot that represents your coordinate pair, like playing a connect-the-dots game.

Rectangular Coordinate System

A rectangular coordinate system is a framework that allows us to identify any point in two-dimensional space using a pair of numbers. It consists of two perpendicular number lines meeting at a point called the origin.

• The horizontal number line is called the x-axis.
• The vertical number line is known as the y-axis.

These axes divide the graph into four sections called quadrants. The quadrants are numbered counterclockwise starting from the upper right: - Quadrant I: where both x and y are positive - Quadrant II: where x is negative and y is positive - Quadrant III: where both x and y are negative - Quadrant IV: where x is positive and y is negative For the point (-3, -5), when both numbers are negative, the point lies in Quadrant III. This system helps you understand where something is located and is used everywhere in mathematics from simple geometry problems to complex data analysis.

Graphing

Graphing is the art of turning mathematical concepts into visual pictures. With graphing, complex relationships are shown in simple diagrams or plots, making them easier to understand. In our example, the graph is a plot of the point (-3, -5) on a simple coordinate plane. Graphing requires a few easy steps:

• Draw both axes to form a cross, ensuring they meet at the origin.
• Label tick marks so that each movement is clear and numbers are correctly positioned.
• Identify the location of your point by using the coordinates given.
• Plot multiple points to determine relationships, like the path a line takes, by drawing a line through related points.

By graphing, one moves not just dots, but interprets numbers into insights. The visuals make it tangible, transforming a list of numbers into a meaningful story that describes size, shape, and direction, essential especially in fields like statistics and physics.