Question

which figure shows $\triangle mnp$ reflected over the $y$-axis to form $\triangle mnp$?

Explanation:

Step1: Recall reflection over y - axis rule

The rule for reflecting a point \((x,y)\) over the \(y\) - axis is \((x,y)\to(-x,y)\). This means that the \(y\) - coordinate of the point remains the same, and the \(x\) - coordinate changes its sign.

Step2: Analyze each figure

• First figure (top - left): Let's assume the coordinates of \(M\), \(N\), \(P\) in \(\triangle MNP\). For example, if \(N\) is at \((2,1)\), \(P\) is at \((4,1)\), \(M\) is at \((3,5)\) (approximate from the grid). After reflection over \(y\) - axis, \(N'\) should be at \((- 2,1)\), \(P'\) at \((-4,1)\), \(M'\) at \((-3,5)\). Looking at the first figure, \(N'\) is at \((-4,1)\), \(P'\) at \((-2,1)\), \(M'\) at \((-3,5)\) which matches the reflection rule (since original \(N\) could be \((4,1)\), \(P\) \((2,1)\), \(M\) \((3,5)\) and after reflection \(x\) - coordinates are negated).
• Second figure (top - right): The reflection here seems to be over the origin or some other line, not the \(y\) - axis, because the \(y\) - coordinates are changing (e.g., \(P\) is above the \(x\) - axis and \(P'\) is below), so it does not follow the \(y\) - axis reflection rule.
• Third figure (bottom - left): The \(y\) - coordinates of the reflected points are negative, which means it is a reflection over the \(x\) - axis (rule \((x,y)\to(x, - y)\)) rather than the \(y\) - axis.
• Fourth figure (bottom - right): The \(x\) - coordinates are not negated in a way that follows the \(y\) - axis reflection. The points seem to be shifted horizontally without the correct sign change for \(x\) - coordinates.

Answer:

The first figure (top - left) shows \(\triangle MNP\) reflected over the \(y\) - axis to form \(\triangle M'N'P'\). (If we consider the options as radio buttons, the first option (top - left) is the correct one.)