Question

for exercise 11-13, answer the questions. 11. error analysis a classmate says that when you rotate any point, (x, y), 180° about the origin, the image is (-x, y)? how would you respond to them?

Explanation:

To determine the correct image of a point \((x, y)\) after a \(180^{\circ}\) rotation about the origin, we use the rotation rule for \(180^{\circ}\) rotations. The rule for rotating a point \((x, y)\) \(180^{\circ}\) about the origin is \((x, y)\to(-x, -y)\).

Let's verify this with an example. Take the point \((2, 3)\). Rotating it \(180^{\circ}\) about the origin should map it to \((-2, -3)\), not \((-2, 3)\) as the classmate suggested. Another example: the point \((-1, 4)\) should rotate to \((1, -4)\), not \((1, 4)\).

The classmate made a mistake in the \(y\)-coordinate transformation. The correct transformation for a \(180^{\circ}\) rotation about the origin changes both the \(x\)- and \(y\)-coordinates to their opposites, not just the \(x\)-coordinate.

Answer:

The classmate is incorrect. The rule for rotating a point \((x, y)\) \(180^{\circ}\) about the origin is \((x, y)\to(-x, -y)\), not \((-x, y)\). This is because a \(180^{\circ}\) rotation about the origin reflects the point across both the \(x\)-axis and \(y\)-axis (or equivalently, through the origin), which changes both the \(x\)- and \(y\)-coordinates to their additive inverses. For example, rotating \((2, 3)\) \(180^{\circ}\) about the origin gives \((-2, -3)\), not \((-2, 3)\).