Question

composite transformation practice
perform each composite transformation according to the list of directions below. write the coordinates of the transformed image after each step and dont forget to complete the coordinate connection.

1. reflect δhij across the line y = -x, then reflect hij across the x - axis to create hij, then rotate hij 90 clockwise about the origin to create hij.
2. translate lm n down 4 and left 5 units, then reflect lmn across the y - axis to create lmn. finally, rotate lmn 270 counter - clockwise about the origin to create lmn.

h( , ) h( , ) h( , ) h( , )
i( , ) i( , ) i( , ) i( , )
j( , ) j( , ) j( , ) j( , )
coordinate connection
(x,y)→( , )→( , )→( , )
l( , ) l( , ) l( , ) l( , )
m( , ) m( , ) m( , ) m( , )
n( , ) n( , ) n( , ) n( , )
coordinate connection
(x,y)→( , )→( , )→( , )

1. reflect quad across the x - axis to create quad. rotate quad 180° about the origin to create quad. translate quad right 3 and down 9 to create quad.

q( , ) q( , ) q( , ) q( , )
u( , ) u( , ) u( , ) u( , )
a( , ) a( , ) a( , ) a( , )
d( , ) d( , ) d( , ) d( , )
coordinate connection
(x,y)→( , )→( , )→( , )

Explanation:

Step1: Recall transformation rules

Reflection across $y = -x$: $(x,y)\to(-y,-x)$; Reflection across $x - axis$: $(x,y)\to(x, - y)$; Rotation 90 - clockwise about origin: $(x,y)\to(y,-x)$; Translation left $a$ units and down $b$ units: $(x,y)\to(x - a,y - b)$; Reflection across $y - axis$: $(x,y)\to(-x,y)$; Rotation 270 - counter - clockwise about origin (equivalent to 90 - clockwise) : $(x,y)\to(y,-x)$; Reflection across $x - axis$: $(x,y)\to(x,-y)$; Rotation 180 - about origin: $(x,y)\to(-x,-y)$; Translation right $a$ units and down $b$ units: $(x,y)\to(x + a,y - b)$.

Step2: For the first shape (with vertices H, I, J)

Let's assume $H(x_1,y_1)$, $I(x_2,y_2)$, $J(x_3,y_3)$.

• Reflection across $y=-x$: $H(x_1,y_1)\to H'(-y_1,-x_1)$
• Reflection across $x - axis$: $H'(-y_1,-x_1)\to H''(-y_1,x_1)$
• Rotation 90 - clockwise about origin: $H''(-y_1,x_1)\to H'''(x_1,y_1)$

Repeat these steps for $I$ and $J$.

Step3: For the second shape (with vertices L, M, N)

Let $L(x_4,y_4)$, $M(x_5,y_5)$, $N(x_6,y_6)$.

• Translation down 4 and left 5: $(x,y)\to(x - 5,y - 4)$, so $L(x_4,y_4)\to L'(x_4 - 5,y_4 - 4)$
• Reflection across $y - axis$: $L'(x_4 - 5,y_4 - 4)\to L''(5 - x_4,y_4 - 4)$
• Rotation 270 - counter - clockwise about origin: $L''(5 - x_4,y_4 - 4)\to L'''(y_4 - 4,x_4 - 5)$

Repeat for $M$ and $N$.

Step4: For the third shape (with vertices Q, U, A, D)

Let $Q(x_7,y_7)$, $U(x_8,y_8)$, $A(x_9,y_9)$, $D(x_{10},y_{10})$.

• Reflection across $x - axis$: $(x,y)\to(x,-y)$, so $Q(x_7,y_7)\to Q'(x_7,-y_7)$
• Rotation 180 - about origin: $Q'(x_7,-y_7)\to Q''(-x_7,y_7)$
• Translation right 3 and down 9: $Q''(-x_7,y_7)\to Q'''(-x_7 + 3,y_7-9)$

Repeat for $U$, $A$ and $D$.

Since the original coordinates of the vertices (H, I, J, L, M, N, Q, U, A, D) are not given in the problem - statement, we can only provide the general transformation rules for the coordinate connection:

• For the first set of transformations: $(x,y)\to(-y,-x)\to(-y,x)\to(x,y)$
• For the second set of transformations: $(x,y)\to(x - 5,y - 4)\to(5 - x,y - 4)\to(y - 4,x - 5)$
• For the third set of transformations: $(x,y)\to(x,-y)\to(-x,y)\to(-x + 3,y - 9)$

Answer:

Without the original coordinates of the vertices of the shapes (H, I, J, L, M, N, Q, U, A, D), we can only give the general coordinate - connection rules:
For the first shape: $(x,y)\to(-y,-x)\to(-y,x)\to(x,y)$
For the second shape: $(x,y)\to(x - 5,y - 4)\to(5 - x,y - 4)\to(y - 4,x - 5)$
For the third shape: $(x,y)\to(x,-y)\to(-x,y)\to(-x + 3,y - 9)$