Problem 9 - Thinking Mathematically

Question

Plot the given point in a rectangular coordinate system. \((-4,0)\)

Step-by-Step Solution

Answer

The point (-4,0) will be located 4 units to the left of the y-axis and precisely on the x-axis.

Step 1: Understand the Cartesian Coordinate System

A point in two dimensions is denoted as (x, y) in Cartesian coordinate system. Here, x is the distance of the point from y-axis (positive towards right and negative towards left) and y is the distance from the x-axis (positive upwards and negative downwards).

Step 2: Identify the Coordinates

The coordinates of the given point are (-4,0). So, the point is 4 units to the left of the y-axis (because the x-coordinate is negative) and exactly on the x-axis (because the y-coordinate is 0).

Step 3: Plot the Point

Start at the origin (where the x-axis and y-axis intersect), move 4 units to the left along the x-axis (because the x-coordinate is -4), and there you go – you’ve found the point (-4,0).

Key Concepts

Plotting Points

Plotting points on a graph is like finding coordinates on a map, allowing you to pinpoint exact locations. Each point is represented by an ordered pair \((x, y)\), where 'x' is the horizontal position and 'y' is the vertical position. The first number, 'x', tells you how far to go left or right from the origin, which is the center point of the graph at \((0,0)\). If 'x' is positive, you move right; if negative, move left.
The second number, 'y', tells you how far up or down to go from the x-axis. A positive 'y' means you move up, while a negative 'y' indicates you move down. To plot the point \((-4,0)\), start at the origin, move 4 units to the left since -4 is negative, and since the y-value is 0, there's no need to move up or down.
By breaking it down, even complex graphs start making sense.

Coordinate System

The coordinate system, often referred to as the Cartesian coordinate system, is a foundational tool used in mathematics to represent points. Named after the mathematician René Descartes, this system consists of two perpendicular lines: the horizontal x-axis and the vertical y-axis.
These axes intersect at a point called the origin \((0,0)\), which serves as the reference point for locating all other points on the plane.

• The x-axis is the horizontal line. Points move from left to right, and these movements are measured as positive or negative values.
• The y-axis is the vertical line, where movements are made upwards or downwards with positive or negative values respectively.

Every pair \((x, y)\) indicates a precise location based on how far a point is from both axes. Understanding this makes graphing and reading points on a plane much easier.

Graphing on a Plane

Graphing on a plane involves transferring theoretical points to a tangible grid, making abstract mathematical concepts visible and understandable. The plane or grid is composed of little squares intersected by two main lines: the x-axis and the y-axis forming the Cartesian coordinate system.
To successfully graph on this plane, follow simple steps. First, take any point represented as \((x, y)\). Second, identify the x-value, which tells you how many units to move to the right if positive, or to the left if negative, starting from the origin.
Next, the y-value comes into play. This number dictates moving up if positive, or down if negative, from the new position along the y-axis. After both movements, mark the point.
The point \((-4, 0)\) lies directly on the x-axis, moved four units left from the origin, showing how graphing is about precise placement and a step-by-step approach.