Problem 1 - Essential Calculus Early Transcendentals

Question

Suppose you start at the origin, move along the \(x\) -axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?

Step-by-Step Solution

Answer

The coordinates are (4, -3).

Step 1: Starting at the Origin

At the beginning, you start at the origin of a coordinate plane. The coordinates of the origin are (0,0).

Step 2: Moving Along the X-axis

You move along the x-axis a distance of 4 units in the positive direction. This means you add 4 to the x-coordinate: (0+4, 0) = (4, 0).

Step 3: Moving Downward

From your current position (4,0), move downward a distance of 3 units. Moving downward affects the y-coordinate. Subtract 3 from the y-coordinate: (4, 0-3) = (4, -3).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can define the position of points by using a pair of numerical values, often called coordinates. This concept is central to understanding much of geometry. It consists of an x-axis, which runs horizontally, and a y-axis, which runs vertically. Both axes intersect at the origin, which is the point (0,0).
The coordinate plane is often referred to as the Cartesian plane, named after the mathematician René Descartes. Each point on this plane can be identified using two numbers - the first representing its distance along the x-axis, and the second representing its distance along the y-axis. This system makes it easier to graph equations and understand spatial relationships between different points.

Cartesian Coordinates

Cartesian Coordinates are used to determine the exact location of a point on the coordinate plane. These coordinates are expressed as pairs (x, y), where 'x' is the horizontal value and 'y' is the vertical value.
This system allows for precise placement of points on a plane, which is crucial for solving many problems in geometry and other branches of mathematics. For example, in the problem you started at the origin (0,0), moved to (4,0) along the x-axis, and finally ended at (4,-3) after moving downward. This shows how each movement directly impacts the Cartesian coordinates of a point.
Understanding these coordinates helps in visualizing and solving complex geometrical problems by simplifying how we represent and interact with points in space.

Movement in Geometry

Movement in geometry refers to how points, lines, and figures can shift on the coordinate plane. These movements include translations, rotations, and reflections. In the context of this exercise, movement is described in terms of horizontal and vertical shifts.
When moving along the x-axis or y-axis, you adjust the respective coordinate. For example, moving 4 units on the x-axis adds to the x-value, while moving downward adjusts the y-value negatively, as seen when changing from (4,0) to (4,-3).

• Horizontal movements affect the x-coordinate. Moving right leads to an increase, while moving left causes a decrease.
• Vertical movements affect the y-coordinate. Moving up increases the value, while moving down decreases it.

These simple rules govern how figures are manipulated within the coordinate plane, providing a basis for more complex transformations.