Question

4 a. what does the sign of the x-coordinate of a point tell you about the location of the point in the coordinate plane?
b. what does the absolute value of the x-coordinate tell you?
5 reflect what does it mean when the x-coordinate of an ordered pair is negative? what does it mean when the y-coordinate is negative?

Explanation:

Part 4a

The coordinate plane has a vertical y - axis and a horizontal x - axis, which intersect at the origin \((0,0)\) and divide the plane into four quadrants. The sign of the \(x\) - coordinate of a point determines which side of the \(y\) - axis the point lies on. If the \(x\) - coordinate is positive (\(x>0\)), the point is to the right of the \(y\) - axis (in Quadrants I or IV). If the \(x\) - coordinate is negative (\(x < 0\)), the point is to the left of the \(y\) - axis (in Quadrants II or III). If the \(x\) - coordinate is \(0\), the point lies on the \(y\) - axis.

The absolute value of a number \(|x|\) represents the distance of the number \(x\) from \(0\) on the number line. In the context of the \(x\) - coordinate of a point \((x,y)\) in the coordinate plane, the absolute value of the \(x\) - coordinate, \(|x|\), tells us the horizontal distance of the point from the \(y\) - axis. For example, if a point has an \(x\) - coordinate of \(3\) or \(- 3\), \(|3|=| - 3| = 3\), which means the point is \(3\) units away from the \(y\) - axis (either \(3\) units to the right if \(x = 3\) or \(3\) units to the left if \(x=-3\)).

In an ordered pair \((x,y)\) that represents a point in the coordinate plane:

• When the \(x\) - coordinate is negative: The coordinate plane is divided into four quadrants by the \(x\) - axis and \(y\) - axis. The \(y\) - axis is the vertical line \(x = 0\). A negative \(x\) - coordinate (\(x<0\)) means that the point is to the left of the \(y\) - axis. Depending on the sign of the \(y\) - coordinate, the point will be in either Quadrant II (if \(y>0\)) or Quadrant III (if \(y < 0\)). For example, the point \((-2,3)\) is in Quadrant II (left of \(y\) - axis, above \(x\) - axis) and the point \((-2,-3)\) is in Quadrant III (left of \(y\) - axis, below \(x\) - axis).
• When the \(y\) - coordinate is negative: The \(x\) - axis is the horizontal line \(y = 0\). A negative \(y\) - coordinate (\(y < 0\)) means that the point is below the \(x\) - axis. Depending on the sign of the \(x\) - coordinate, the point will be in either Quadrant III (if \(x < 0\)) or Quadrant IV (if \(x>0\)). For example, the point \((-2,-3)\) is in Quadrant III (left of \(y\) - axis, below \(x\) - axis) and the point \((2,-3)\) is in Quadrant IV (right of \(y\) - axis, below \(x\) - axis).

Answer:

The sign of the \(x\) - coordinate tells us whether the point is to the left (negative \(x\) - coordinate), to the right (positive \(x\) - coordinate) of the \(y\) - axis, or on the \(y\) - axis (\(x = 0\)). If \(x>0\), the point is to the right of the \(y\) - axis; if \(x < 0\), the point is to the left of the \(y\) - axis; if \(x = 0\), the point is on the \(y\) - axis.

Part 4b