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Chapter 3: Problem 7

Fill in the blanks. a. To plot \((-5,4),\) we start at the ___and move 5 units to the ___ and then move 4 units___. b. To plot \(\left(6,-\frac{3}{2}\right),\) we start at the___ and move 6 units to the___ and then move \(\frac{3}{2}\) units___.

Short Answer

Expert verified
a. origin, left, up; b. origin, right, down.

Step by step solution

01

Understanding the Coordinate System

When plotting a point on a Cartesian coordinate system, we start at the origin, which is the point \(0, 0\). The first number in a coordinate pair, called the x-coordinate, tells us how far to move horizontally from the origin. The second number, the y-coordinate, tells us how far to move vertically.
02

Plotting the First Coordinate (-5,4)

For the point \(-5, 4\), start at the origin. The x-coordinate is \(-5\), meaning we will move 5 units to the left since negative x-values indicate a leftward movement. After that, the y-coordinate \(4\) tells us to move 4 units up since positive y-values indicate upward movement.
03

Filling the Blanks for (-5,4)

In the sentence given, after moving 5 units based on the x-coordinate, we should have moved to the 'left'. After moving based on the y-coordinate, we moved 'up'. Thus, the blanks can be filled as: \( \text{origin}, \text{left}, \text{up} \).
04

Plotting the Second Coordinate (6,-3/2)

For the point \(6, -\frac{3}{2}\), start at the origin. The x-coordinate is \(6\), meaning we move 6 units to the right since positive x-values indicate rightward movement. The y-coordinate \(-\frac{3}{2}\) indicates a downward movement of \(\frac{3}{2}\) units because negative y-values indicate downwards.
05

Filling the Blanks for (6,-3/2)

In the sentence given, after moving 6 units based on the x-coordinate, we should have moved to the 'right'. After moving based on the y-coordinate, we moved 'down'. Thus, the blanks can be filled as: \( \text{origin}, \text{right}, \text{down} \).

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Key Concepts

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

Understanding the X-Coordinate
In the Cartesian coordinate system, each point is defined by two numbers: the x-coordinate and the y-coordinate. The x-coordinate is the first number in the pair \(x, y\), and it dictates the horizontal movement on the coordinate plane from the origin, where the axes meet at \(0, 0\).
  • If the x-coordinate is positive like in the point \(6, -\frac{3}{2}\), it indicates a movement to the right.
  • If the x-coordinate is negative as seen in the point \(-5, 4\), it indicates a movement to the left.
It is essential to always begin from the origin when plotting any point. This starting point ensures that you are converting the numerical x-value into physical steps left or right on your graph. By understanding the x-coordinate, you ensure that each point finds its horizontal position accurately on the grid.
Decoding the Y-Coordinate
While the x-coordinate moves you horizontally, the y-coordinate tells you how to move vertically on the Cartesian coordinate system. The y-coordinate is the second number in the coordinate pair \(x, y\).
  • A positive y-coordinate, such as the \(4\) in the point \(-5, 4\), directs you to go up from the current horizontal position.
  • A negative y-coordinate, like \(-\frac{3}{2}\) in the point \(6, -\frac{3}{2}\), commands a downward move.
Always remember that the y-coordinate modifies your location after the x-movement has been established. This sequential correction 鈥� first x, then y 鈥� ensures that you land precisely at the intended point.
Origin in Coordinate System
The origin serves as the heart of the Cartesian coordinate system. At the point \(0, 0\), it acts as a universal reference starting point. Every plotting of coordinates stems from this set location. The origin doesn't imply any real direction on its own but emphasizes a neutral starting frame.When plotting, \(0, 0\) is not just a beginning strategy but a crucial anchor:
  • Starting at \(0, 0\) prevents cumulative errors because movement directions and distances are always constant from this point.
  • The origin is symmetrical, meaning both the x-axis and y-axis cross through it. This symmetry helps in visual alignment and check accuracy in plotting.
By treating the origin as the primary anchor for all subsequent moves, each x and y movement retains its directed integrity, ensuring precise positional results on the graph.

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