Problem 1 - Thinking Mathematically

Question

In Exercises 1-20, plot the given point in a rectangular coordinate system. \((1,4)\)

Step-by-Step Solution

Answer

The point \((1,4)\) is plotted on the Cartesian coordinate system by starting at the origin, moving 1 unit right along the x-axis, and then 4 units upwards along the y-axis. The final position is our point \((1,4)\).

Step 1: Understand the Coordinates

The provided point is an ordered pair in the form \((x,y)\) given as \((1,4)\), where '1' is the x-coordinate and '4' is the y-coordinate. The x-coordinate represents the horizontal distance from the origin while the y-coordinate represents the vertical distance from the origin.

Step 2: Plotting the X-Coordinate

Start at the origin (0,0), move 1 unit to the right along the x-axis. This is because the x-coordinate is 1, and it's positive so we move to the right.

Step 3: Plotting the Y-Coordinate

From the position after step 2, move 4 units upwards along the y-axis. This is because the y-coordinate is 4, and it's positive so we move up.

Step 4: Mark The Point

Mark the point and label it as \((1,4)\). The marked point represents the location of \((1,4)\) on the Cartesian coordinate system.

Key Concepts

Plotting Points

Plotting points in a rectangular coordinate system is essential to graphing in mathematics. A point is defined using an ordered pair of numbers, such as (1,4). These numbers provide the exact location of the point in the system.

To plot points:

• Start at the origin, which is the intersection of the x-axis and the y-axis, often denoted as (0,0).
• Use the x-coordinate to determine how far to move horizontally. A positive x-value signifies moving right, while a negative value indicates moving left.
• Next, use the y-coordinate to move vertically. A positive y-value means moving upwards and a negative value means moving downwards.
• Finally, place a mark at the intersection of these movements and label the point accordingly, for instance, (1,4).

Plotting points accurately helps visualize equations and understand spatial relationships within the coordinate system.

Ordered Pairs

In a coordinate system, every point is represented by an ordered pair, usually in the form (x, y). The essence of ordered pairs is the specific sequence they follow, where the x-value always comes before the y-value. This sequence is crucial since it specifies the point's precise location on the graph.

An ordered pair denotes a relationship:

• The x-coordinate tells you the position along the horizontal axis (how far from the center you are moving left or right).
• The y-coordinate indicates the position along the vertical axis (how far up or down you are going from the horizontal line).

With our example, the ordered pair (1,4) identifies exactly one unique spot on the graph, granting a clear way of distinguishing different points. This ordering ensures there is no confusion about the point's actual place in the coordinate plane, which would happen if they were switched around.

Cartesian Coordinate System

The Cartesian coordinate system is a fundamental element in plotting points and graphing equations. It consists of two perpendicular lines, forming a plane divided into four quadrants.

Essential features of the Cartesian coordinate system:

• The horizontal axis is the x-axis, where positive values stretch to the right and negative values to the left.
• The vertical axis is the y-axis, with positive values extending upwards and negatives heading downwards.
• Where these axes cross each other at (0,0) is called the origin.

This system allows us to represent visual and numerical relationships clearly. By employing ordered pairs and plotting points, we can create graphs of lines, curves, and various shapes, making the abstract concepts of algebra and geometry tangible and understandable. This system is named after the French mathematician René Descartes, who developed this form of graphing.