Problem 75 - Introductory Algebra for College Students

Question

Exercises \(75-77\) will help you prepare for the material covered in the next section. From \((0,-3),\) move 4 units up and 1 unit to the right. What point do you obtain?

Step-by-Step Solution

Answer

After moving 4 units up and 1 unit to the right from the point (0,-3) in a Cartesian plane, the point obtained is (1, 1).

Step 1: Identify the starting point

Start from the given point (0, -3) which is made up of the x and y coordinates.

Step 2: Move up by 4 units

In a Cartesian plane, moving up equates to an increase in the y-coordinate. This means adding 4 to the y-coordinate of the current point. Hence, from (0, -3), after moving up, the point becomes (0, -3+4) = (0, 1).

Step 3: Move 1 unit to the right

Moving to the right is associated with an increase in the x-coordinate. So, add 1 to the x-coordinate of the current point. From (0, 1), after moving to the right, the point becomes (0+1, 1) = (1, 1).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves studying geometric figures using a coordinate system. This system allows you to represent points, lines, and other shapes on a plane using pairs of numbers called coordinates. The beauty of coordinate geometry lies in its ability to quantify and solve geometric problems algebraically.

• In simple terms, you use numbers to describe the location of points.
• It translates geometric shapes into an algebraic form.
• You can perform operations such as addition and multiplication to study properties such as distance, midpoint, gradients, and intersections.

Imagine having to describe the location of a treasure on a map; coordinate geometry is much like that but with a much more universal and precise system. It's essential for fields such as physics, engineering, computer graphics, and even navigation.

Cartesian Coordinate System

The Cartesian coordinate system is a foundation for coordinate geometry and is used to display points in a plane. Developed by René Descartes, it consists of two perpendicular axes: the horizontal axis (usually called the x-axis) and the vertical axis (y-axis). These axes divide the plane into four quadrants.

• The point where they intersect is the origin, labeled as (0, 0).
• The position of any point in the plane is determined by an ordered pair of numbers, (x, y).
• The x-coordinate tells you how far to move horizontally from the origin, and the y-coordinate how far to move vertically.

It’s like a game of battleship where you call out coordinates to target hits. Understanding this system is crucial because all the plotting of points, lines, and shapes in coordinate geometry relies on the Cartesian coordinate system.

Plotting Points

Plotting points is an essential skill in coordinate geometry. It's like pinpointing specific locations on your grid map (the Cartesian plane). To successfully plot a point, you need to follow a set of straightforward steps:

Moving along the x-axis

Your journey to plotting a point always starts at the origin. From there, you move left or right along the horizontal x-axis depending on whether the x-coordinate is negative or positive.

Moving along the y-axis

After your horizontal move, you then shift your position up or down along the vertical y-axis, again based on whether the y-coordinate is negative or positive.By moving a precise number of units in each direction as defined by the coordinates, you reach the exact location to plot your point. For example, to plot the point (1, 1), you would move 1 unit to the right (positive x-direction) and then 1 unit up (positive y-direction). This systematic method of plotting ensures accuracy in representing geometric figures and is fundamental to understanding more complex concepts in coordinate geometry.