Question

i can describe the effect of transformations by using coordinates.

1. point h is located at (-7, 5). where is h after a 270° clockwise rotation? (5, 7)
2. point r is located at (8, 0). where is r after a translation 9 units left and 6 units down?
3. point l is located at (3, -6). where is l after a reflection over the y - axis? (3, - 6)
4. point b is located at (4, - 16). where is b after a dilation with a scale factor of 3/2? (3, - 12)
5. point w is located at (6, 2). where is w after a 180° counterclockwise rotation? (-6, -2)
6. point q is located at (-9, 10). where is q after a translation 4 units right and 11 units down?

i can describe the effect of transformations by using coordinates.
match each of the descriptions with the correct algebraic representation of the transformation.

1. reflection over the x - axis c
2. rotation 90° counterclockwise e
3. translation 2 left and 2 up f
4. dilation; scale factor 2 b
5. rotation 180° clockwise a
6. translation 2 right and 2 down d
7. reflection over the y - axis g

a (-x, -y)
b (2x, 2y)
c (x, -y)
d (x + 2, y - 2)
e (-y, x)
f (x - 2, y + 2)
g (-x, y)
h (y, -x)
i can describe the effect of transformations by using coordinates.

1. point j(-3, -7) was rotated to j(7, -3). how many degrees was the rotation? clockwise 270°
2. point c(12, 5) was reflected to c(-12, 5). what was the line of reflection? y - axis
3. point m(2, 4) was dilated to m(9, 18). what was the scale factor, and was the dilation an enlargement or reduction? enlargement
4. point z(-5, -6) was translated to z(-8, 4). describe the direction and distance of the translation.

Explanation:

Step1: Recall transformation rules

For a point $(x,y)$:

• Reflection over the $x -$axis gives $(x,-y)$.
• $90^{\circ}$ counter - clockwise rotation gives $(-y,x)$.
• Translation $a$ units left and $b$ units up gives $(x - a,y + b)$.
• Dilation with scale factor $k$ gives $(kx,ky)$.
• $180^{\circ}$ clockwise or counter - clockwise rotation gives $(-x,-y)$.
• Translation $a$ units right and $b$ units down gives $(x + a,y - b)$.
• Reflection over the $y -$axis gives $(-x,y)$.

Step2: Solve each part

Part 1:

1. Given point $H(-7,5)$, for a $270^{\circ}$ clockwise (same as $90^{\circ}$ counter - clockwise) rotation, using the rule $(x,y)\to(-y,x)$, we have $(-7,5)\to(5,7)$.
2. Given point $R(8,0)$, for a translation 9 units left and 6 units down, using the rule $(x,y)\to(x - 9,y - 6)$, we get $(8 - 9,0 - 6)=(-1,-6)$.
3. Given point $L(-3,6)$, for a reflection over the $y -$axis, using the rule $(x,y)\to(-x,y)$, we get $(3,6)$.
4. Given point $B(4,-16)$, for a dilation with scale factor $\frac{3}{2}$, using the rule $(x,y)\to(kx,ky)$ where $k = \frac{3}{2}$, we have $(4\times\frac{3}{2},-16\times\frac{3}{2})=(6,-24)$.
5. Given point $W(6,2)$, for a $180^{\circ}$ counter - clockwise rotation, using the rule $(x,y)\to(-x,-y)$, we get $(-6,-2)$.
6. Given point $Q(-9,10)$, for a translation 4 units right and 11 units down, using the rule $(x,y)\to(x + 4,y - 11)$, we have $(-9+4,10 - 11)=(-5,-1)$.

Part 2:

1. Reflection over the $x -$axis: $(x,y)\to(x,-y)$ (matches C).
2. $90^{\circ}$ counter - clockwise rotation: $(x,y)\to(-y,x)$ (matches E).
3. Translation 2 left and 2 up: $(x,y)\to(x - 2,y + 2)$ (matches F).
4. Dilation with scale factor 2: $(x,y)\to(2x,2y)$ (matches B).
5. $180^{\circ}$ clockwise rotation: $(x,y)\to(-x,-y)$ (matches A).
6. Translation 2 right and 2 down: $(x,y)\to(x + 2,y - 2)$ (matches D).
7. Reflection over the $y -$axis: $(x,y)\to(-x,y)$ (matches G).

Part 3:

1. Given $J(-3,-7)$ rotated to $J'(7,-3)$. A $90^{\circ}$ clockwise rotation rule is $(x,y)\to(y,-x)$. Here $(-3,-7)\to(-7,3)$ is not correct. A $270^{\circ}$ clockwise rotation rule is $(x,y)\to(-y,x)$, and $(-3,-7)\to(7,-3)$. So the rotation is $270^{\circ}$ clockwise.
2. Given $C(12,5)$ reflected to $C'(-12,5)$. The line of reflection is the $y -$axis since for reflection over the $y -$axis, $(x,y)\to(-x,y)$.
3. Given $M(2,4)$ dilated to $M'(9,18)$. The scale factor $k=\frac{9}{2}= 4.5$ (for $x$ - coordinate) and $\frac{18}{4}=4.5$ (for $y$ - coordinate). Since $k = 4.5>1$, it is an enlargement.
4. Given $Z(-5,-6)$ translated to $Z'(-8,4)$. The change in $x$ is $-8-(-5)=-3$ (3 units left) and the change in $y$ is $4-(-6)=10$ (10 units up).

Answer:

1. $R'(-1,-6)$
2. $L'(3,6)$
3. $B'(6,-24)$
4. $Q'(-5,-1)$
5. Matching: 11 - C, 12 - E, 13 - F, 14 - B, 15 - A, 16 - D, 17 - G
6. $270^{\circ}$ clockwise
7. $y$ - axis
8. Scale factor $4.5$, enlargement
9. 3 units left and 10 units up