1) Perform the composition of transformations for the following points.
1st: \((x, y) → (x + 1, y – 2)\) then 2nd: reflect across the \(x\)-axis
a) point \(A(3, 4)\rightarrow A’\)________ \(\rightarrow A’’\)________
b) point \(B(1, 5)\rightarrow B’\)________ \(\rightarrow B’’\)________
2) Perform the composition of transformations for the following \(\overline{JK}\)
1st: a\(180^{\circ}\) rotation about \((0, 0)\) followed by 2nd: a reflection across the \(y\)-axis
a) \(J(-7, -6)\rightarrow J’\)________ \(\rightarrow J’’\)________
b) \(K(-4, 0)\rightarrow K’\)________ \(\rightarrow K’’\)________
1st: \((x, y) → (x + 1, y – 2)\) then 2nd: reflect across the \(x\)-axis
a) point \(A(3, 4)\rightarrow A’\)________ \(\rightarrow A’’\)________
b) point \(B(1, 5)\rightarrow B’\)________ \(\rightarrow B’’\)________
2) Perform the composition of transformations for the following \(\overline{JK}\)
1st: a\(180^{\circ}\) rotation about \((0, 0)\) followed by 2nd: a reflection across the \(y\)-axis
a) \(J(-7, -6)\rightarrow J’\)________ \(\rightarrow J’’\)________
b) \(K(-4, 0)\rightarrow K’\)________ \(\rightarrow K’’\)________
3) Move point \(P\) under the composition of transformations for each problem below and match the result with the correct point \((X, Y\) or \(Z)\)
a) 1st: reflect point \(P\) across line \(t\), then 2nd: translate point \(P’\) by the rule \((x, y)\rightarrow (x + 1, y – 3)\). b) 1st: translate point \(P\) by the rule \((x, y)\rightarrow (x, y – 2)\), then 2nd: reflect \(P’\) across line \(t\). c)1st: rotate point \(P\) \(90^{\circ}\) counter-clockwise about point \(Q\), then 2nd: translate point \(P’\) by the rule \((x, y)\rightarrow (x + 5, y – 1)\) |
4) Write a single transformation that is the same as the following composition of transformations:
1st: \((x, y)\rightarrow (x – 3, y + 4)\) then 2nd: \((x, y)\rightarrow (x – 5, y – 1)\)
1st: \((x, y)\rightarrow (x – 3, y + 4)\) then 2nd: \((x, y)\rightarrow (x – 5, y – 1)\)
7) Which transformation is a glide reflection of point \(P\)? Is it choice “a” or choice “b”? Explain.
a: 1st: translate point \(P\) by the rule \((x, y)\rightarrow (x + 2, y + 4)\), then 2nd: reflect \(P’\) across line \(t\). b: 1st: reflect point \(P\) across line \(t\), then 2nd: translate point \(P’\) by the rule \((x, y) → (x + 0, y + 3)\) |
8) Which composition of transformations has \(Q’’\) as its image point? Is it choice “a” or choice “b”? Explain.
a: 1st: translate \(Q\) by the rule \((x, y) \rightarrow (x – 1, y + 2)\), then 2nd: reflect \(Q’\) across line m. b: 1st: reflect point \(Q\) across line \(m\), then 2nd: translate point \(Q’\) by the rule \((x, y)\rightarrow (x – 1, y + 2)\). |
9) Perform the following composition of transformations on \(\triangle ABC\).
1st: rotate counter-clockwise \(270^{\circ}\) by the origin, then
2nd: translate by the rule \((x, y)\rightarrow (x – 3, y – 2)\), then
3rd: reflect in the line \(y = -x\).
\(A(3, 7)\rightarrow A’\) _________ \(\rightarrow A''\)________ \(\rightarrow A'''\)________
\(B(1, 3) \rightarrow B’\) _________ \(\rightarrow B''\)________ \(\rightarrow B'''\)________
\(C(2, 6) \rightarrow C’\) _________ \(\rightarrow C''\)________ \(\rightarrow C'''\)________
1st: rotate counter-clockwise \(270^{\circ}\) by the origin, then
2nd: translate by the rule \((x, y)\rightarrow (x – 3, y – 2)\), then
3rd: reflect in the line \(y = -x\).
\(A(3, 7)\rightarrow A’\) _________ \(\rightarrow A''\)________ \(\rightarrow A'''\)________
\(B(1, 3) \rightarrow B’\) _________ \(\rightarrow B''\)________ \(\rightarrow B'''\)________
\(C(2, 6) \rightarrow C’\) _________ \(\rightarrow C''\)________ \(\rightarrow C'''\)________
14) Given Line \(m\parallel n\). Reflect \(\triangle JOE\) across line \(m\), then reflect \(\triangle J'O'E'\) across line n.
a) How far is it from \(J\) to \(J’’\)? b) How far is it from \(O\) to \( O’’\)? c) How far is it from \(E\) to \( E’’\)? d) How far is it between the parallel lines \(m\) and \(n\)? |
16) Line \(e\) and line \(f\) are placed so that they intersect at \(45^{\circ}\) angles. Line \(e\) has a slope of \(1\) (like the line \(y = x\)) and line \(f\) is vertical.
1st: reflect \(\triangle ABC\) across line \(e\). 2nd: reflect \(\triangle A’B’C’\) across line \(f\). Plot \(\triangle A’’B’’C’’\) Verify that \(\triangle ABC\) can be mapped to \(A’’B’’C’’\) by a single transformation. Correctly describe that transformation. |
18) Notice the graph \(y = x^2\). Perform the following composition of transformation on the graph:
1st: reflect across the \(x\)-axis, then 2nd: translate by the rule \((x, y)\rightarrow (x + 2, y – 3)\). Name two points on the graph in the final image other than the image of the vertex. BONUS: Determine the equation of the graph of the final image in vertex form. |
19) Create a composition of transformations for the figure.
21) Move \(\overleftrightarrow{DE}\) so that
a) \(\overleftrightarrow{DE}\parallel\overleftrightarrow{SM}\)
b) \(\overleftrightarrow{DE}\perp\overleftrightarrow{SM}\)
c) What are the linear equations that you created?
a) \(\overleftrightarrow{DE}\parallel\overleftrightarrow{SM}\)
b) \(\overleftrightarrow{DE}\perp\overleftrightarrow{SM}\)
c) What are the linear equations that you created?