1) Find the coordinates of \((8, –3)\) when it is reflected over the \(x\)-axis, then rotated \(90^{\circ}\).
2) The point \(Q(-3, 2)\) is rotated \(90^{\circ}\) around the origin to point \(Q'\). Point \(Q'\) is then reflected in the line \(y = x\) to point \(Q''\). Find the coordinates of point \(Q''\).
3) Graph \(\overline{F''G''}\) after a composition of the transformations in the order they are listed. Then perform the transformations in reverse order. Does the order affect the final image of \(\overline{F''G''}\)?
\(F(-5, 2), G(-2, 4)\)
Translation: \((x, y)\rightarrow(x + 3, y – 8)\)
Reflection: in the \(x\)-axis
4) The composite of three reflections, across the \(x\)-axis, across the line \(y = x\), and then across the line \(y = 2x\), is equivalent to a single reflection across some line \(y = mx\). Find the value of \(m\).
5) Graph the following compositions. List which figures are isometries.
a) \(A(1, 3), B(3, 3), C(6, 1), D(1, 1)\)
Rotation: \(270^{\circ}\) clockwise about the origin
Translation: \((x, y)\rightarrow (2x - 3, 2y)\)
Reflection: in the line \(y = 2\)
b) \(A(-2, 0), B(1, -1), C(-3, -3)\)
Rotation: \(180^{\circ}\) about the origin
Translation: \((x, y)\rightarrow (3x, 3y + 5)\)
Reflection: in the line \(y = -x\)
6) Describe a composition of transformations that maps \(ABCD\) onto \(A’’B’’C’’D’’\).
2) The point \(Q(-3, 2)\) is rotated \(90^{\circ}\) around the origin to point \(Q'\). Point \(Q'\) is then reflected in the line \(y = x\) to point \(Q''\). Find the coordinates of point \(Q''\).
3) Graph \(\overline{F''G''}\) after a composition of the transformations in the order they are listed. Then perform the transformations in reverse order. Does the order affect the final image of \(\overline{F''G''}\)?
\(F(-5, 2), G(-2, 4)\)
Translation: \((x, y)\rightarrow(x + 3, y – 8)\)
Reflection: in the \(x\)-axis
4) The composite of three reflections, across the \(x\)-axis, across the line \(y = x\), and then across the line \(y = 2x\), is equivalent to a single reflection across some line \(y = mx\). Find the value of \(m\).
5) Graph the following compositions. List which figures are isometries.
a) \(A(1, 3), B(3, 3), C(6, 1), D(1, 1)\)
Rotation: \(270^{\circ}\) clockwise about the origin
Translation: \((x, y)\rightarrow (2x - 3, 2y)\)
Reflection: in the line \(y = 2\)
b) \(A(-2, 0), B(1, -1), C(-3, -3)\)
Rotation: \(180^{\circ}\) about the origin
Translation: \((x, y)\rightarrow (3x, 3y + 5)\)
Reflection: in the line \(y = -x\)
6) Describe a composition of transformations that maps \(ABCD\) onto \(A’’B’’C’’D’’\).
7-11) Graph and describe a composition of transformations that maps \(\triangle ABC\) onto \(\triangle DEF\). You can use the app below to graph the triangles by entering the coordinates into the tables.
7) \(A(5, 5), B(1, 3), C(2, 1)\) onto \(D(0, -5), E(-4, -3), F(-3, -1)\)
8) \(A(-2, 2), B(2, 8), C(6, 2)\) onto \(D(-4, -1), E(0, -7), F(4, -1)\)
9) \(A(2, 6), B(2, 3), C(7, 3)\) onto \(D(3, -2), E(0, -2), F(0, -7)\)
10) \(A(2, -1), B(4, 1), C(1, 2)\) onto \(D(-2, -4), E(-4, -2), F(-1, -1)\)
11) \(A(-3, -4), B(-5, -1), C(-1, 1)\) onto \(D(1, 4), E(-1, 1), F(3, -1)\)
7) \(A(5, 5), B(1, 3), C(2, 1)\) onto \(D(0, -5), E(-4, -3), F(-3, -1)\)
8) \(A(-2, 2), B(2, 8), C(6, 2)\) onto \(D(-4, -1), E(0, -7), F(4, -1)\)
9) \(A(2, 6), B(2, 3), C(7, 3)\) onto \(D(3, -2), E(0, -2), F(0, -7)\)
10) \(A(2, -1), B(4, 1), C(1, 2)\) onto \(D(-2, -4), E(-4, -2), F(-1, -1)\)
11) \(A(-3, -4), B(-5, -1), C(-1, 1)\) onto \(D(1, 4), E(-1, 1), F(3, -1)\)