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Chapter 4: Problem 43

Graph each ordered pair. $$\left(\frac{7}{2}, 1\right)$$

Short Answer

Expert verified
To graph the ordered pair \(\left(\frac{7}{2}, 1\right)\), set up the Cartesian plane and move \(\frac{7}{2}\) units to the right from the origin along the x-axis. Then, move \(1\) unit upwards parallel to the y-axis. Mark and label the point \(\left(\frac{7}{2}, 1\right)\) on the plane. Finally, verify that the point is in the correct location relative to the x and y axes.

Step by step solution

01

Identify the x and y coordinates

The given ordered pair is \(\left(\frac{7}{2}, 1\right)\), so the x-coordinate is \(\frac{7}{2}\) and the y-coordinate is \(1\).
02

Set up the Cartesian plane

Draw the Cartesian plane, which consists of two perpendicular axes. The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. The point where they intersect is called the origin, and it has the coordinates \((0, 0)\).
03

Locate the x-coordinate

The x-coordinate of the point is \(\frac{7}{2}\). Since it is positive, move \(\frac{7}{2}\) units to the right from the origin along the x-axis.
04

Locate the y-coordinate

The y-coordinate of the point is \(1\). Since it is positive, move \(1\) unit upwards from the point found in step 3, parallel to the y-axis.
05

Mark the point

Mark the point on the Cartesian plane where the x and y coordinates align. In this case, the point will be \(\left(\frac{7}{2}, 1\right)\). Label the point with its coordinates so that it is clear which ordered pair it represents.
06

Check your work

Verify that the point is indeed located at \(\left(\frac{7}{2}, 1\right)\) on the plane by double-checking the position relative to the x and y axes. If the point is at the correct location, the exercise is complete.

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Key Concepts

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

Cartesian Plane
The Cartesian plane is a fundamental concept in graphing and mathematics. It's a two-dimensional surface defined by two perpendicular number lines.
These number lines are known as the axes. The horizontal line is the x-axis, and the vertical line is the y-axis.
They meet at the origin, which has the coordinates (0, 0).
  • X-axis: Runs horizontally from left to right.
  • Y-axis: Runs vertically from bottom to top.
  • Origin: The intersection point of both axes, starting point of measurements.
The Cartesian plane allows us to plot points, graph lines, and display geometric shapes. It's essential for visualizing mathematical relationships.
Understanding the X-Coordinate
The x-coordinate is the first number in an ordered pair and determines the horizontal position of a point on the Cartesian plane.
In our example, the x-coordinate is \( \frac{7}{2} \). This tells us to move \( \frac{7}{2} \) units along the x-axis.
  • Positive Values: Move right from the origin.
  • Negative Values: Move left from the origin.
The x-coordinate is essential for accurately locating a point's horizontal placement on the graph.
Understanding the Y-Coordinate
The y-coordinate is the second number in an ordered pair. It determines the vertical position of a point.
In our given example, the y-coordinate is \( 1 \). This means we move 1 unit up from the x-coordinate position.
  • Positive Values: Move upwards from the x-axis.
  • Negative Values: Move downwards from the x-axis.
The y-coordinate is crucial for fixing a point's vertical position on the Cartesian plane, ensuring accurate plotting.
Ordered Pair Representation
An ordered pair represents a specific point on the Cartesian plane. It is written in the form \( (x, y) \), where x indicates the x-coordinate, and y indicates the y-coordinate.
For our graphing example, the ordered pair is \( \left( \frac{7}{2}, 1 \right) \).
  • First Value: Represents the x-coordinate (horizontal position).
  • Second Value: Represents the y-coordinate (vertical position).
Ordered pair representation is essential in graphing, as it allows us to pinpoint exact locations and interpret spatial relationships between points.

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