Problem 21 - Algebra 1

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$(6,17)$$

Step-by-Step Solution

Answer

The point (6, 17) lies in Quadrant I.

Step 1: Identify the coordinates

Here, the given point is (6, 17). This means the x-coordinate is 6 and the y-coordinate is 17.

Step 2: Determine the quadrant

By definition, if both x and y are positive then the point lies in the first quadrant. Here, both x and y are positive, so the point (6, 17) lies in Quadrant I.

Key Concepts

Understanding the Coordinate Plane

The coordinate plane, commonly referred to as the Cartesian plane, is a two-dimensional surface on which we can plot points, lines, and curves. It is divided into two perpendicular lines: the horizontal line known as the x-axis and the vertical line called the y-axis. These axes intersect at a point known as the origin, which has coordinates (0,0).

From the origin, you can locate any point on this plane by moving along the x and y-axes. The position of a point is indicated by a pair of numbers, \(x, y\), known as coordinates. The x-value tells us how far along the horizontal axis we should move from the origin, and the y-value indicates the vertical displacement from the origin.

Dividing the Plane into Quadrants

The coordinate plane is divided into four sections, called quadrants, which are numbered counterclockwise starting from the top right quadrant. Each quadrant comprises different signs for the x and y coordinates.

• Quadrant I (\(QI\)) is located in the top right section where both x and y coordinates are positive.
• Quadrant II (\(QII\)) is positioned in the top left section, where x is negative and y is positive.
• Quadrant III (\(QIII\)) can be found on the bottom left section where both x and y coordinates are negative.
• Finally, Quadrant IV (\(QIV\)) is in the bottom right section, where x is positive and y is negative.

Understanding which quadrant a point lies in becomes straightforward once you know the signs of its x and y coordinates.

Positive Coordinates and Their Quadrant

When we have a point with positive x and y coordinates, such as \( (6,17) \), we can easily determine that it lies in Quadrant I. This is because, as previously mentioned, Quadrant I is defined by both coordinates being positive.

A point like \( (6,17) \) tells us that from the origin, we move 6 units to the right along the x-axis and then 17 units up along the y-axis. By following these directions, we end up in the upper right section of the plane, thereby confirming the location of the point in Quadrant I. Recognizing the signs of a point's coordinates is a fundamental step in understanding graphing and the layout of the coordinate plane.