Question

1. rotation 180° clockwise
2. translation 2 right and 2 down
3. reflection over the y - axis
4. point j(-3, -7) was rotated to j(7, -3). how many degrees was the rotation? clockwise 90°
5. point c(12, 5) was reflected to c(-12, 5). what was the line of reflection? y - axis
6. point m(2, 4) was dilated to m(9, 18). what was the scale factor, and was the dilation an enlargement or reduction?
7. point z(-5, -6) was translated to z(-8, 4). describe the direction and distance of the translation.

Explanation:

Step1: Analyze rotation of point J

The rule for a rotation of 270 - degree clock - wise about the origin is $(x,y)\to(y, - x)$. Given $J(-3,-7)$ and $J'(7,-3)$. When we apply the 270 - degree clock - wise rotation rule to $J(-3,-7)$, we get $(-7,3)$ which is incorrect. Let's consider the general rotation formula for a rotation of $\theta$ degrees about the origin: $x'=x\cos\theta - y\sin\theta$ and $y'=x\sin\theta + y\cos\theta$. For $\theta = 270^{\circ}$, $\cos\theta = 0$ and $\sin\theta=-1$. So, $(x,y)\to(y, - x)$. But if we consider a rotation of 90 - degree counter - clockwise (equivalent to 270 - degree clock - wise), the rule is correct. The rotation is 90 degrees counter - clockwise.

Step2: Analyze reflection of point C

The rule for reflection over the y - axis is $(x,y)\to(-x,y)$. Given $C(12,5)$ and $C'(-12,5)$. When we apply the reflection over the y - axis rule to point $C$, we change the sign of the x - coordinate while keeping the y - coordinate the same. So the line of reflection is the y - axis.

Step3: Analyze dilation of point M

The formula for dilation about the origin with a scale factor $k$ is $(x,y)\to(kx,ky)$. Let the scale factor be $k$. Given $M(2,4)$ and $M'(9,18)$. We find $k$ by looking at the x - coordinates or y - coordinates. For the x - coordinate, $kx = 9$ and $x = 2$, so $k=\frac{9}{2}=4.5$. For the y - coordinate, $ky = 18$ and $y = 4$, $k=\frac{18}{4}=4.5$. The dilation is an enlargement with a scale factor of 4.5.

Step4: Analyze translation of point Z

The translation from $Z(-5,-6)$ to $Z'(-8,4)$. The change in the x - coordinate is $\Delta x=-8-(-5)=-3$ and the change in the y - coordinate is $\Delta y = 4-(-6)=10$. The translation is 3 units left and 10 units up.

Answer:

1. 90 degrees counter - clockwise
2. y - axis
3. Enlargement with a scale factor of 4.5
4. 3 units left and 10 units up