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Chapter 8: Problem 201

Plot the points. \(\left(-1,-\frac{\pi}{2}\right)\)

Short Answer

Expert verified
The point \((-1, -\frac{\pi}{2})\) is located to the left of the origin and downward on the coordinate plane.

Step by step solution

01

Identify the Coordinate Components

The given point is \((-1, -\frac{\pi}{2})\). The first component \(-1\) represents the x-coordinate, and the second component \(-\frac{\pi}{2}\) represents the y-coordinate.
02

Set Up the Coordinate Plane

To plot the point, first draw a standard coordinate plane with a horizontal x-axis and a vertical y-axis. Mark a central origin point where the two axes intersect.
03

Locate the x-coordinate on the x-axis

Move left from the origin along the x-axis to -1. Since the x-coordinate is negative, you move left from the origin.
04

Locate the y-coordinate on the y-axis

From the position you identified for \(x = -1\), move vertically downwards because the y-coordinate \(-\frac{\pi}{2}\) is negative. Approximately place this point since \(-\frac{\pi}{2}\) (approximately -1.57) will be below the x-axis.
05

Plot the Point

Mark the intersection of the movements from steps 3 and 4. Ensure the mark is clear and distinguishable as your point \((-1, -\frac{\pi}{2})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plotting Points
Plotting points is a fundamental concept in mathematics that combines art and precision. It involves placing a point on a grid or graph to represent its coordinates. When starting, recognize that every point is defined by an ordered pair \((x, y)\). These values determine a unique location on the graph.
  • To plot a point, begin at the origin, where the x-axis and y-axis intersect. This point is \(0,0\).
  • Using the x-coordinate, move horizontally along the x-axis.
  • Next, use the y-coordinate to move vertically from your intended x-coordinate.
  • The final spot where these two moves meet is your plotted point.
Make sure to use small, easily visible marks when plotting. Understanding how to transform numeric values into visual representations can greatly aid in learning graph-related concepts, giving insight into distance, direction, and position.
Cartesian Plane
The Cartesian Plane, also known as the Coordinate Plane, provides a visual framework for plotting points. Named after French mathematician Ren茅 Descartes, this plane is defined by two perpendicular number lines that divide the space into four quadrants.
  • The x-axis runs horizontally, and the y-axis runs vertically.
  • The point where they intersect is called the origin, labeled as \(0,0\).
  • This plane is infinite in size but often displayed on a grid to simplify plotting.
  • Each quadrant is a specific region: I (top-right), II (top-left), III (bottom-left), IV (bottom-right).
Understanding the Cartesian Plane helps organize and understand spatial relationships between different points. It is a crucial foundation for many advanced topics in algebra and calculus.
Coordinates Identification
When you see coordinates, you are provided with a specific method to locate points in space. Identifying coordinates is key to efficiently plotting points and interpreting graphs.
  • The first value in the pair, or the x-coordinate, tells you how far along the horizontal axis to go.
  • The second value, or the y-coordinate, tells you how far up or down the vertical axis to move.
  • If the x-coordinate is negative, move left from the origin; if it is positive, move right.
  • If the y-coordinate is negative, move downward; if it is positive, move upward.
Always write coordinates in parentheses with a comma separating the values, like \((x, y)\). This system means every point has a unique location, enabling precise communication of its position on the graph.

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