Problem 76 - Introductory Algebra for College Students

Question

Exercises \(75-77\) will help you prepare for the material covered in the next section. From \((0,1),\) move 2 units down and 3 units to the right. What point do you obtain?

Step-by-Step Solution

Answer

The point obtained after moving 2 units down and 3 units to the right from the initial position (0,1) is (3, -1).

Step 1: Identify Initial Position

The initial point is given as (0,1). The first value, 0, is the x-coordinate and the second value, 1, is the y-coordinate. This is in a Cartesian coordinate system where the x-coordinate (denoted by a horizontal line) signifies left and right movement while the y-coordinate (denoted by a vertical line) signifies up and down movement.

Step 2: Apply Vertical Movement

It's given to move 2 units down. In the Cartesian coordinate system, down is represented by a subtraction from the y-coordinate. Initial y-coordinate is 1. Thus, after moving 2 units down, the new y-coordinate becomes \(1 - 2 = -1\).

Step 3: Apply Horizontal Movement

Next, it's given to move 3 units to the right. In the Cartesian coordinate system, right is represented by an addition to the x-coordinate. Initial x-coordinate is 0. Thus, after moving 3 units right, the new x-coordinate becomes \(0 + 3 = 3\).

Step 4: Identify Final Position

Now, combining the new x and y coordinates, the final point after the movements can be identified. The final point based on the movements would thus be (3, -1).

Key Concepts

Graphical Point Representation

Graphical point representation is a critical concept within the Cartesian coordinate system. It allows us to visually interpret numerical data and understand the location of a point in space. For instance, consider the point (0,1) from the exercise. Here, the first number, known as the x-coordinate, aligns with the horizontal axis, while the second number, the y-coordinate, aligns with the vertical axis.

When we plot this point on graph paper or a whiteboard, we start at the origin, where the two axes meet at the coordinate (0,0), and from there, we make our moves to locate the point. In the case of (0,1), one would remain in the same place horizontally and move one unit upward vertically. Representing points graphically helps in creating a vivid mental image of their location which is indispensable for learners to grasp geometric concepts and execute operations involving those points.

Coordinate Plane

The coordinate plane is the two-dimensional surface where we draw graphs and plot points, lines, shapes, and functions. It's comprised of two number lines that intersect at a right angle: the horizontal x-axis and the vertical y-axis. The plane is divided into four quadrants, allowing for the representation of a wide range of values, both positive and negative.

In the context of the exercise, from the initial location (0,1), we must consider the provided instructions 'move 2 units down and 3 units to the right' to find the new location on this plane. The coordinate plane is vital in ensuring that we can translate these instructions into a new point by considering the direction and magnitude of the movements.

Horizontal and Vertical Movements

In the Cartesian coordinate system, the horizontal and vertical movements or shifts are fundamental actions. Horizontally, we move along the x-axis; to the right for positive changes, to the left for negative. Vertically, movements along the y-axis involve going up for positive changes and down for negative ones.

In the exercise, it details a movement of '3 units to the right', increasing the x-coordinate from 0 to 3, and '2 units down', which decreases the y-coordinate from 1 to -1. It's crucial to understand that these movements are independent; horizontal shifts don't affect the y-coordinate, and vertical shifts don't change the x-coordinate. Keeping this independence in mind is necessary to accurately navigate the coordinate plane and achieve precise point representation.