Question

which graph shows $\triangle abc$ and its reflection across the $y$-axis, $\triangle abc$?

choose the correct answer below.

\bigcirc a.\quad\includegraphicsscale=0.3{graph1.png}\quad\bigcirc b.\quad\includegraphicsscale=0.3{graph2.png}\quad\bigcirc c.\quad\includegraphicsscale=0.3{graph3.png}

Explanation:

To determine which graph shows \(\triangle ABC\) and its reflection across the \(y\)-axis, we use the rule for reflecting a point \((x, y)\) across the \(y\)-axis: the new coordinates become \((-x, y)\). This means the \(x\)-coordinate changes sign, but the \(y\)-coordinate remains the same.

Analyzing each option:

• Option A: The triangles appear to be reflected across the \(x\)-axis (since the \(y\)-coordinates change sign, not the \(x\)-coordinates). So this is not a reflection over the \(y\)-axis.
• Option B: The triangles are on the same side of the \(y\)-axis (both have positive \(x\)-coordinates or both negative), so this is not a reflection over the \(y\)-axis (which should flip the \(x\)-coordinate sign).
• Option C: For each vertex of \(\triangle ABC\), the \(x\)-coordinate is negated (e.g., a point \((a, b)\) on the left of the \(y\)-axis reflects to \((-a, b)\) on the right, or vice versa), while the \(y\)-coordinate stays the same. This matches the reflection rule over the \(y\)-axis.

Answer:

C