Point \(A\) is to be graphed in a quadrant, not on an axis, of the standard \((x, y)\) coordinate plane below. graph can't copy If the \(x\) -coordinate and the \(y\) -coordinate of point \(A\) are to have opposite signs, then point \(A\) must be located in: A. Quadrant II only: B. Quadrant IV only. C. Quadrant I or III only. D. Quadrant I or IV only. E. Quadrant II or IV only.

The Cartesian coordinate system is a foundation of algebra and geometry, named after the French mathematician René Descartes. It's a method to visualize relationships between two or more variables in a plane. In a two-dimensional space, this system uses two perpendicular axes: the horizontal axis (often labeled as the x-axis) and the vertical axis (labeled as the y-axis).

The point where these axes intersect is called the origin, usually denoted as the point (0, 0). Every point in the plane can be described by a pair of numbers, known as coordinates. The first number in the pair is the x-coordinate and it reflects the point's horizontal position, while the second number, the y-coordinate, reflects its vertical position.

• Points to the right of the origin have positive x-coordinates.
• Points to the left have negative x-coordinates.
• Points above the origin have positive y-coordinates.
• Points below have negative y-coordinates.

The two axes divide the plane into four areas called quadrants, which are numbered counterclockwise starting from the upper right quadrant.

Graphing points on the Cartesian coordinate system is a fundamental skill for plotting geometric figures, functions, and various other mathematical representations. To graph a point, you need a pair of coordinates: (x, y). Start at the origin, move horizontally to the x-coordinate value, and then vertically to the y-coordinate value. Where these two movements intersect is the location of the point on the plane.

Remember, moving right indicates positive x-values and left indicates negative. Likewise, moving up signifies positive y-values, while going down indicates negative ones. For example, to graph the point (3, -2), move three units to the right of the origin and two units down. This simple process of graphing points is the building block for more complex operations such as plotting lines and curves.

Quadrant analysis is crucial for understanding the positioning and relationships of points on the Cartesian coordinate system. With four quadrants, each represents a unique combination of positive and negative coordinates.

• Quadrant I has both coordinates positive (x, y).
• Quadrant II has a negative x-coordinate and a positive y-coordinate (-x, y).
• Quadrant III has both coordinates negative (-x, -y).
• Quadrant IV has a positive x-coordinate and a negative y-coordinate (x, -y).

This framework allows us to categorize and make sense of points and their movements across the plane. For instance, if a point is moving from Quadrant I to II, we know the x-value is becoming negative while the y-value remains positive.

Points with opposite signs coordinates lie in Quadrants II and IV. In these quadrants, one coordinate is positive while the other is negative, which means they're on opposite sides of one axis but the same side of the other.

Opposite signs are important to recognize because they indicate specific behaviors or conditions in mathematical problems and real-world contexts. For example, if a point has a positive x-coordinate and a negative y-coordinate, such as (5, -3), it is located in Quadrant IV. Conversely, a point with a negative x-coordinate and a positive y-coordinate, like (-3, 2), would be in Quadrant II. Recognizing and understanding these relationships between coordinates and their signs can help you effectively navigate problems involving the coordinate plane.