Question

i can identify the relationship between points on a coordinate plane. identify the quadrant where the point is located.

1. (-4, 10) quadrant:
2. (7, -2) quadrant:

problem solving
use the map for 3 & 4

1. the restaurants location has the same x - coordinate as the city hall but the opposite y - coordinate. in which quadrant is the restaurant located?
2. the grocery store has the same y - coordinate as the library but the opposite x - coordinate. the grocery store and the library are reflections of each other across which axis?

multiple choice

1. what are the coordinates of the point (-8, 5) if it is reflected across the x - axis?

a. (8, 5)
b. (8, -5)
c. (-8, 5)
d. (-8, -5)

Explanation:

Step1: Recall quadrant rules

In a coordinate - plane, in Quadrant I: \(x>0,y > 0\); in Quadrant II: \(x<0,y>0\); in Quadrant III: \(x < 0,y<0\); in Quadrant IV: \(x>0,y<0\).

Step2: Determine quadrant for (-4, 10)

For the point \((-4,10)\), \(x=-4<0\) and \(y = 10>0\), so it is in Quadrant II.

Step3: Determine quadrant for (7,-2)

For the point \((7,-2)\), \(x = 7>0\) and \(y=-2<0\), so it is in Quadrant IV.

Step4: Analyze restaurant location (assuming City - Hall location)

Without the actual location of City - Hall on the map, assume City - Hall has coordinates \((x,y)\). The restaurant has coordinates \((x, - y)\). If City - Hall is in Quadrant I \((x>0,y>0)\), the restaurant is in Quadrant IV \((x>0,-y<0)\); if in Quadrant II \((x<0,y>0)\), the restaurant is in Quadrant III \((x<0,-y<0)\); if in Quadrant III \((x<0,y<0)\), the restaurant is in Quadrant II \((x<0,-y>0)\); if in Quadrant IV \((x>0,y<0)\), the restaurant is in Quadrant I \((x>0,-y>0)\).

Step5: Analyze grocery - store and library relationship

If two points have the same \(y\) - coordinate and opposite \(x\) - coordinates, they are reflections of each other across the \(y\) - axis. For example, if a point is \((x,y)\) and the other is \((-x,y)\), the line \(x = 0\) (the \(y\) - axis) is the axis of symmetry.

Step6: Reflect point across x - axis

When a point \((x,y)\) is reflected across the \(x\) - axis, the \(x\) - coordinate remains the same and the \(y\) - coordinate changes its sign. For the point \((-8,5)\), when reflected across the \(x\) - axis, the new point is \((-8,-5)\).

Answer:

1. Quadrant II
2. Quadrant IV
3. Depends on City - Hall's quadrant
4. \(y\) - axis
5. D. \((-8,-5)\)