Question

a line segment has endpoints at (-1,4) and (4,1). which reflection will produce an image with endpoints at (-4,1) and (-1,-4)? a reflection of the line segment across the x - axis a reflection of the line segment across the y - axis a reflection of the line segment across the line y = x a reflection of the line segment across the line y = -x

Explanation:

Step1: Recall reflection rules

• Reflection across the \(x\) - axis: \((x,y)\to(x, - y)\)
• Reflection across the \(y\) - axis: \((x,y)\to(-x,y)\)
• Reflection across the line \(y = x\): \((x,y)\to(y,x)\)
• Reflection across the line \(y=-x\): \((x,y)\to(-y,-x)\)

Step2: Analyze each endpoint transformation

• For the point \((-1,4)\):
• If we first reflect \((-1,4)\) across the \(y\) - axis: \((-1,4)\to(1,4)\), then this is not helpful.
• If we reflect \((-1,4)\) across the \(x\) - axis: \((-1,4)\to(-1,-4)\)
• For the point \((4,1)\):
• If we reflect \((4,1)\) across the \(y\) - axis: \((4,1)\to(-4,1)\)

Answer:

a reflection of the line segment across the \(x\) - axis and then across the \(y\) - axis. But if we consider the composition of reflections, first reflecting across the \(y\) - axis \((x,y)\to(-x,y)\) for \((4,1)\to(-4,1)\) and then reflecting \((-1,4)\) across the \(x\) - axis \((x,y)\to(x, - y)\) gives \((-1,-4)\). However, if we check the options, the key is to note that \((4,1)\to(-4,1)\) (reflection across \(y\) - axis: \(x\) changes sign, \(y\) remains the same) and \((-1,4)\to(-1,-4)\) (reflection across \(x\) - axis: \(y\) changes sign, \(x\) remains the same). But if we consider the order of operations for the options given, we can also think of it as two separate reflections. But if we check the options, we can verify by applying the rules:

• For a reflection across the \(y\) - axis: \((x,y)\to(-x,y)\). When we apply it to \((4,1)\), we get \((-4,1)\)
• For a reflection across the \(x\) - axis: \((x,y)\to(x, - y)\). When we apply it to \((-1,4)\), we get \((-1,-4)\)

So the answer is a reflection of the line segment across the \(x\) - axis and then across the \(y\) - axis. But looking at the options, if we consider the two - step process (since the options are single reflections, but actually it's a composition). However, if we check each option:

• Option 1: Reflection across \(x\) - axis: \((-1,4)\to(-1,-4)\) and \((4,1)\to(4, - 1)\) (incorrect)
• Option 2: Reflection across \(y\) - axis: \((-1,4)\to(1,4)\) and \((4,1)\to(-4,1)\) (incorrect)
• Option 3: Reflection across \(y = x\): \((-1,4)\to(4,-1)\) and \((4,1)\to(1,4)\) (incorrect)
• Option 4: Reflection across \(y=-x\): \((-1,4)\to(-4,1)\) and \((4,1)\to(-1,-4)\) (correct)

So the answer is a reflection of the line segment across the line \(y =-x\)