Problem 24 - Algebra 1

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$(3,9)$$

Step-by-Step Solution

Answer

The point (3,9) is in Quadrant I.

Step 1: Identify the coordinates

Identify the coordinates of the point. Here, the point is (3,9). So, x=3 and y=9.

Step 2: Determine the quadrant based on the signs of the coordinates

In the Cartesian coordinate system, the sign of the coordinates determines the quadrant of the point. Given that x=3 (positive) and y=9 (positive), we can ascertain that the point (3,9) is in quadrant I. Quadrant I consists of points with both x and y coordinates positive.

Key Concepts

Cartesian coordinate system

The Cartesian coordinate system is a fundamental framework for understanding the position of points in a two-dimensional space. It consists of two perpendicular axes: the horizontal axis (usually called the x-axis) and the vertical axis (y-axis). These axes intersect at a point known as the origin, labeled as (0,0).

Each point in this system is defined by a pair of numbers, known as coordinates, that indicate its position relative to these axes. The first number, called the x-coordinate, tells us how far to move horizontally from the origin, while the second number, the y-coordinate, specifies the vertical displacement. For example, the coordinates (3,9) mean that we move 3 units to the right and 9 units upwards from the origin.

Quadrants of the coordinate plane

Dividing the Cartesian coordinate system with the x and y axes results in four distinct areas, or quadrants, each with their own characteristic sign patterns for coordinates. Starting from the upper right and moving counter-clockwise, these quadrants are numbered I through IV.

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

To determine which quadrant a point belongs to, one must simply look at the signs of the x and y coordinates. As in our example, the point (3,9), with both coordinates positive, falls within Quadrant I.

Positive coordinates

Coordinates can be positive or negative, corresponding to directions on the Cartesian plane. Positive coordinates make understanding the position of points more straightforward when both values are above zero.

In Quadrant I, where both x and y coordinates are positive, points represent positions that are to the right and above the origin. This is typically the default quadrant for many mathematical functions and geometrical shapes because it reflects the natural way we think about space. A point like (3,9) indicates a location that is 3 units to the right and 9 units up from the origin, ensuring it lies in Quadrant I.