Question

what is the x - coordinate of the image when ( m(4,-1) ) is reflected in ( y = 3 ).
a) 7
b) 4
c) - 1
d) - 4

Explanation:

Step1: Understand reflection over a horizontal line

When reflecting a point \((x,y)\) over a horizontal line \(y = k\), the \(x -\)coordinate remains the same. The formula for the \(y -\)coordinate of the reflected point is \(y'=2k - y\). But we are interested in the \(x -\)coordinate.
For a reflection over a horizontal line \(y = c\) (in this case \(c = 3\)), the rule for a point \((x,y)\) is that the \(x -\)value of the point and its reflection is invariant. Mathematically, if we have a point \(P(x,y)\) and we reflect it over the line \(y = c\), the reflected point \(P'(x',y')\) has \(x'=x\).

Step2: Apply the rule to the given point

We are given the point \(M(4,-1)\) and the line \(y = 3\). Using the rule for reflection over a horizontal line \(y = c\) (where \(c = 3\)) for the \(x -\)coordinate. Since the rule \(x'=x\) (because reflection over a horizontal line does not change the \(x -\)value of the point), when \(x = 4\), the \(x -\)coordinate of the reflected point is also \(x'=4\).

Answer:

B. 4