Problem 17 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The points A(-4,1), B(-1,5), and C(0,-4) have been plotted correctly following the order of the coordinates for x-axis (horizontal movement) and y-axis (vertical movement).

Step 1: Understand the Coordinate System

The coordinate plane is defined by an x-axis (horizontal) and a y-axis (vertical) which divides the plane into four quadrants. Each point in the coordinate system is represented as an ordered pair defined as (x, y), where x is the abscissa (horizontal value) and y is the ordinate (vertical value). The origin (0,0) is the point where both axes intersect.

Step 2: Plot point A(-4,1)

This point is (-4,1). We start at the origin, move 4 units to the left on the x-axis (since x is -4) and then 1 unit up on the y-axis (since y is 1). This is where point A is plotted.

Step 3: Plot point B(-1,5)

This point is (-1,5). From the origin, move 1 unit to the left on the x-axis (since x is -1) and then 5 units up on the y-axis (since y is 5). This is where point B is plotted.

Step 4: Plot point C(0,-4)

This point is (0,-4). Starting from the origin, no move is needed on the x-axis (since x is 0) but then move 4 units down on the y-axis (since y is -4). This is where point C is plotted.

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in understanding the coordinate plane. An ordered pair consists of two elements written in a specific order, such as \(x, y\). This notation is crucial because it allows us to locate points on the coordinate plane.
The first element, \(x\), represents the horizontal position of the point, known as the abscissa. The second element, \(y\), represents the vertical position, known as the ordinate. Ordered pairs are always written in parentheses and separated by a comma.

• Example: For the ordered pair \( (-4, 1) \), \(x = -4\) and \(y = 1\).

Plotting Points

Plotting points on a coordinate plane involves using the ordered pairs \( (x, y) \) to determine the location of each point. It's like a treasure map where the X marks the spot!
First, start at the origin \( (0,0) \). Then move along the \(x\)-axis according to the x-value of the pair.
Next, move vertically to match the \(y\)-value.

• If the x-value is positive, move to the right. If it's negative, move to the left.
• If the y-value is positive, move up. If it's negative, move down.

This step-by-step method ensures accuracy when marking points on a graph.

Quadrants

The coordinate plane is divided into four sections called quadrants. These quadrants help us quickly identify the location of points in relation to the axes.

• Quadrant I: Positive x and positive y values \((+,+)\).
• Quadrant II: Negative x and positive y values \((-1,+)\).
• Quadrant III: Negative x and negative y values \((-1,-1)\).
• Quadrant IV: Positive x and negative y values \((+, -1)\).

Understanding which quadrant a point is in can help verify accuracy of plotting.

X-axis and Y-axis

The x-axis and y-axis are the two fundamental lines that define the coordinate plane. The x-axis runs horizontally, while the y-axis runs vertically.
Both axes intersect at a special point known as the origin, represented by the ordered pair \( (0,0) \).
The axes divide the plane into the four quadrants and provide reference lines to help locate points.

• Points on the x-axis will always have a y-value of 0, such as \( (3,0) \).
• Points on the y-axis have an x-value of 0, such as \( (0,-5) \).

These axes are essential for understanding how the coordinate system functions.