Describe how to determine which quadrant a given point \((x, y)\) lies in.

To determine which **quadrant** a given point \((x, y)\) lies in on the Cartesian coordinate system, you can follow these steps: ### Understanding the Cartesian Plane The Cartesian plane is divided into four regions called **quadrants** by the **x-axis** (horizontal axis) and the **y-axis** (vertical axis). The quadrants are numbered counterclockwise starting from the **upper right**: 1. **Quadrant I**: Both \(x\) and \(y\) are **positive**. 2. **Quadrant II**: \(x\) is **negative** and \(y\) is **positive**. 3. **Quadrant III**: Both \(x\) and \(y\) are **negative**. 4. **Quadrant IV**: \(x\) is **positive** and \(y\) is **negative**. Additionally, if a point lies **exactly on one of the axes** or at the **origin**, it is **not** considered to be in any quadrant. ### Step-by-Step Process 1. **Identify the Coordinates**: Start with the given point \((x, y)\). 2. **Determine the Signs of \(x\) and \(y\)**: - **Positive** (\(+\)) indicates the direction to the right (for \(x\)) or upward (for \(y\)). - **Negative** (\(-\)) indicates the direction to the left (for \(x\)) or downward (for \(y\)). 3. **Match the Signs to the Quadrant**: - **Quadrant I**: \(x > 0\) and \(y > 0\) - **Quadrant II**: \(x < 0\) and \(y > 0\) - **Quadrant III**: \(x < 0\) and \(y < 0\) - **Quadrant IV**: \(x > 0\) and \(y < 0\) 4. **Check for Special Cases**: - If \(x = 0\) and \(y \neq 0\), the point lies **on the y-axis**. - If \(y = 0\) and \(x \neq 0\), the point lies **on the x-axis**. - If \(x = 0\) and \(y = 0\), the point is at the **origin**. ### Examples 1. **Point (3, 4)**: - \(x = 3 > 0\) - \(y = 4 > 0\) - **Quadrant I** 2. **Point \((-2, 5)\)**: - \(x = -2 < 0\) - \(y = 5 > 0\) - **Quadrant II** 3. **Point \((-7, -3)\)**: - \(x = -7 < 0\) - \(y = -3 < 0\) - **Quadrant III** 4. **Point (6, -8)**: - \(x = 6 > 0\) - \(y = -8 < 0\) - **Quadrant IV** 5. **Point (0, 9)**: - Lies **on the y-axis** 6. **Point (5, 0)**: - Lies **on the x-axis** 7. **Point (0, 0)**: - Located at the **origin** ### Visual Representation Here's a simple diagram of the Cartesian plane with the quadrants labeled: ``` y | II | I | -------+------- x | III | IV | ``` - **Quadrant I** is in the upper right. - **Quadrant II** is in the upper left. - **Quadrant III** is in the lower left. - **Quadrant IV** is in the lower right. ### Summary By examining the **signs** of the \(x\) and \(y\) coordinates of a point, you can easily determine which quadrant the point resides in. Remember to also consider cases where the point lies directly on one of the axes or at the origin.