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

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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 $$y eq 0$$, the point lies **on the y-axis**. - If $$y = 0$$ and $$x eq 0$$, 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. To determine which quadrant a point $$(x, y)$$ is in, check the signs of $$x$$ and $$y$$: - **Quadrant I**: $$x > 0$$, $$y > 0$$ - **Quadrant II**: $$x < 0$$, $$y > 0$$ - **Quadrant III**: $$x < 0$$, $$y < 0$$ - **Quadrant IV**: $$x > 0$$, $$y < 0$$ Points on the axes or at the origin are not in any quadrant.

Describe how to determine which quadrant a given point lies in.

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