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Sequences of rigid motions

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51319110
The line \(s\) is given by \(s(x) = \frac{2}{3}x + 5\). 1. Reflect \(s\) across the y-axis. Call the image \(t\) and write its equation. 2. Reflect \(t\) across the x-axis. Call the new image \(u\) and write its equation. 3. Compare \(s\) and \(u\). What single rigid motion maps \(s\) directly to \(u\)?

Hints

- Apply the two reflections in order and track which coordinate sign changes each time. - Compare the original equation with the final equation. - Which single transformation changes both \(x\) and \(y\) to their opposites?

Solution

1. Reflecting across the y-axis replaces \(x\) with \(-x\), so \(t(x) = -\frac{2}{3}x + 5\). 2. Reflecting across the x-axis multiplies all outputs by \(-1\), so \(u(x) = \frac{2}{3}x - 5\). 3. Reflecting across both coordinate axes is equivalent to a \(180^\circ\) rotation about the origin. That rotation maps \(s\) directly to \(u\).

Answer

1. \(t(x) = -\frac{2}{3}x + 5\) 2. \(u(x) = \frac{2}{3}x - 5\) 3. A \(180^\circ\) rotation about the origin
53216810
The graph shows quadrilateral \(ABCD\) and its image \(A'B'C'D'\). a) Read the coordinates of the vertices of both quadrilaterals from the graph. b) Which rule maps the coordinates \((x, y)\) of a point on \(ABCD\) to the coordinates \((x', y')\) of the corresponding point on \(A'B'C'D'\)? - **Rule I:** \((x, y)\to(x-5, y-3)\) - **Rule II:** \((x, y)\to(-x, y-3)\) - **Rule III:** \((x, y)\to(-x, -y)\) - **Rule IV:** \((x, y)\to(x-3, -y)\) c) Describe two basic transformations that can be performed in sequence to map \(ABCD\) onto \(A'B'C'D'\).
Figure for problem 532168

Hints

- Read each original vertex and its corresponding image vertex carefully. - Compare the \(x\)-coordinates in each corresponding pair. - Compare the \(y\)-coordinates and determine the constant change. - Interpret a sign change in the \(x\)-coordinate as a geometric transformation. - Interpret adding or subtracting a constant from a coordinate as a translation.

Solution

1. The vertices are \(A(1, 2)\), \(B(4, 1)\), \(C(5, 4)\), and \(D(2, 5)\). The image vertices are \(A'(-1, -1)\), \(B'(-4, -2)\), \(C'(-5, 1)\), and \(D'(-2, 2)\). 2. For each corresponding pair, the \(x\)-coordinate changes sign and the \(y\)-coordinate decreases by \(3\). Therefore, Rule II is correct: \((x, y)\to(-x, y-3)\). 3. Reflect \(ABCD\) across the \(y\)-axis, then translate the result \(3\) units down.

Answer

a) \(A(1, 2)\), \(B(4, 1)\), \(C(5, 4)\), \(D(2, 5)\); \(A'(-1, -1)\), \(B'(-4, -2)\), \(C'(-5, 1)\), \(D'(-2, 2)\). b) Rule II: \((x, y)\to(-x, y-3)\). c) Reflect across the \(y\)-axis, then translate \(3\) units down.

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