Problem 49 - Algebra 1: Concepts and Skills

Question

Plot and label the ordered pairs in a coordinate plane. $$ A(-1,-2), B(-4,5), C(0,2) $$

Step-by-Step Solution

Answer

The points A(-1,-2), B(-4,5), and C(0,2) have been plotted in the Cartesian plane.

Step 1: Draw the Coordinate Plane

Begin by drawing the Cartesian plane. This consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). They intersect at a point known as the origin (0,0). Be sure to label the axes and the origin.

Step 2: Plot Point A(-1,-2)

To plot point A(-1,-2), start at the origin, move one unit to the left along the x-axis (since x=-1) and then move two units down along the y-axis (since y=-2). Mark this point and label it as 'A'.

Step 3: Plot Point B(-4,5)

To plot point B(-4,5), start at the origin, move four units to the left along the x-axis (since x=-4) and then move five units up along the y-axis (since y=5). Mark this point and label it as 'B'.

Step 4: Plot Point C(0,2)

To plot point C(0,2), start at the origin, you don't need to move along the x-axis (since x=0), just move two units up along the y-axis (since y=2). Mark this point and label it as 'C'.

Key Concepts

Understanding Ordered Pairs

Ordered pairs are a fundamental component of working on the Cartesian plane. An ordered pair is written in the form

• \((x, y)\)

where \(x\) is the horizontal value, and \(y\) is the vertical value.
The order of the numbers is significant. The first number always refers to the x-coordinate, which tells us how far to move horizontally from the origin.
The second number indicates the y-coordinate and shows how far to move vertically.

**Why Order Matters:**

• If you switch the positions of the two numbers, you will end up in a completely different location on the coordinate plane. Thus, maintaining the correct order is essential.
• The values of an ordered pair can be positive, negative, or zero, and each has a specific meaning.

Understanding ordered pairs helps in precisely locating points within a coordinated system, thus providing a basis for plotting on the Cartesian plane.

Exploring the Cartesian Plane

The Cartesian plane, named after mathematician René Descartes, is a two-dimensional plane defined by a horizontal line known as the x-axis and a vertical line called the y-axis.

**Key Features:**

• The point where the x-axis and y-axis intersect is known as the origin, labeled as \((0,0)\).
• The plane is divided into four sections called quadrants, each identified by Roman numerals, starting with I in the top right and moving counter-clockwise.

Every point on the Cartesian plane is defined by an ordered pair of numbers that indicate its position relative to the origin.
Each axis is a number line that extends infinitely in positive and negative directions, helping to map out a grid where every position is unique and traceable.

**Using the Cartesian Plane:**

• It provides a visual way to represent algebraic equations and geometric figures.
• Allows for the plotting of points, graphs, and even lines or curves.

Plotting Points on a Cartesian Plane

Plotting points involves "placing" them in their correct positions based on their ordered pairs.
Here is how you can plot a point like \((x, y)\):

• Start at the origin \((0,0)\).
• Move horizontally to the x-coordinate value (left if negative, right if positive).
• From this new spot, move vertically to the y-coordinate value (down if negative, up if positive).
• Mark the spot and label it according to the problem, such as 'A', 'B' or 'C'.

In the original exercise, points are expressed as:

• A(-1,-2) means we go 1 unit left and 2 units down.
• B(-4,5) means shifting 4 units left and 5 units up.
• C(0,2) involves no movement on the x-axis, only 2 units up on the y-axis.

The accuracy of plotted points helps in forming graphs and understanding geometric relationships on the Cartesian plane.