Question

1. a circle is graphed on a coordinate grid with its center point at (5, 4). the circle be reflected across the x - axis 8.10c

which rule describes the center of the new circle after the reflection?
a ((x,y)\to(-x,y))
b ((x,y)\to(x,-y))
c ((x,y)\to(-x,-y))
d ((x,y)\to(y,-x))

Explanation:

Step1: Recall reflection over x - axis rule

When a point \((x,y)\) is reflected across the \(x\) - axis, the \(x\) - coordinate remains the same and the \(y\) - coordinate changes its sign. The transformation rule is \((x,y)\to(x, - y)\).

Step2: Analyze the center point

The center of the original circle is \((5,4)\). After reflecting across the \(x\) - axis, the new center should be \((5,- 4)\), which follows the rule \((x,y)\to(x, - y)\).

Step3: Match with the given options

• Option A: \((x,y)\to(-x,y)\) is the rule for reflection over the \(y\) - axis, so A is incorrect.
• Option B: \((x,y)\to(x, - y)\) matches the reflection over \(x\) - axis rule, so B is correct.
• Option C: \((x,y)\to(-x,-y)\) is the rule for reflection over the origin, so C is incorrect.
• Option D: \((x,y)\to(y, - x)\) is not a standard reflection rule for \(x\) - axis, so D is incorrect.

Answer:

B. \((x,y)\to(x, - y)\)