Question

Plot each point in the xy-plane. State which quadrant or on what coordinate axis each point lies. (a) \(A=(-3,2)\) (d) \(D=(6,5)\) (b) \(B=(6,0)\) (e) \(E=(0,-3)\) (c) \(C=(-2,-2)\) (f) \(F=(6,-3)\)

Short answer

A: Quadrant II, D: Quadrant I, B: x-axis, E: y-axis, C: Quadrant III, F: Quadrant IV.

Step-by-step solution

Identify Quadrants and Axes

The xy-plane is divided into four quadrants: - Quadrant I: Both x and y are positive - Quadrant II: x is negative and y is positive- Quadrant III: Both x and y are negative- Quadrant IV: x is positive and y is negative Points on the x-axis have y = 0 and points on the y-axis have x = 0.

Examine Point A

For point A=(-3,2), the x-value is -3 and the y-value is 2. Since x is negative and y is positive, point A lies in Quadrant II.

Examine Point D

For point D=(6,5), both the x-value and the y-value are positive. Therefore, point D lies in Quadrant I.

Examine Point B

For point B=(6,0), the x-value is 6 and the y-value is 0. Since y is 0, point B lies on the x-axis.

Examine Point E

For point E=(0,-3), the x-value is 0 and the y-value is -3. Since x is 0, point E lies on the y-axis.

Examine Point C

For point C=(-2,-2), both the x-value and the y-value are negative. Therefore, point C lies in Quadrant III.

Examine Point F

For point F=(6,-3), the x-value is positive and the y-value is negative. Therefore, point F lies in Quadrant IV.

Key concepts

Quadrants

The xy-plane (or Cartesian plane) is conveniently split into four specific areas called quadrants. These quadrants help to identify where a point lies based on its coordinates. The quadrants are numbered I, II, III, and IV and follow a counter-clockwise direction starting from the upper right.
The characteristics of each quadrant are:

• Quadrant I: Both x (horizontal value) and y (vertical value) are positive.
• Quadrant II: x is negative and y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive and y is negative.

Additionally, points that lie exactly on the axes (not in any quadrant) have either x=0 or y=0.

xy-plane

The xy-plane is a two-dimensional plane where any point can be defined using two numbers called coordinates.
It gets its name from the two perpendicular lines that cross each other at the origin (0,0) - the x-axis (horizontal line) and the y-axis (vertical line). This creates a grid where every location can be pinpointed using an ordered pair \(x, y\).

• First coordinate \(x\) represents the horizontal position.
• Second coordinate \(y\) specifies the vertical position.

Each point on this plane holds a significant meaning depending on its position relative to these axes.

Coordinate Axes

The x-axis and y-axis are the two number lines that form the backbone of the coordinate plane. When these two axes intersect at right angles, they split the plane into the four quadrants mentioned earlier.

• The x-axis runs horizontally and is used to measure values to the left (negative) or right (positive) of the origin.
• The y-axis runs vertically and measures values above (positive) or below (negative) the origin.

Points that lie directly on the axes are important:

• If a point lies on the x-axis, its y-value will be 0 (e.g., (3, 0)).
• If a point lies on the y-axis, its x-value will be 0 (e.g., (0, -2)).

Understanding the relationship between points and these axes is crucial for plotting.

Plotting Points

Plotting points on the coordinate plane involves locating their exact positions using their x and y coordinates.
Here’s a simple guide to help you plot points accurately:

• Start at the origin (0,0).
• Move horizontally along the x-axis according to the x-value.
• From there, move vertically based on the y-value.

For example, to plot the point \(3, -2\), you would:

• Start at the origin.
• Move 3 units to the right.
• Then move 2 units down.

Repeat this process for each point given in your exercise. Use a pencil and draw a small dot or circle at each point’s location for clarity.
This methodical approach will help ensure you place points in the correct quadrant or on the appropriate axis.