Graph â riangle ABCâ ext with vertices â extA(-4, 4), B(-1, 7), and C(-1, 4)â ext and its image after a 270° clockwise rotation about the origin. Then, reflect a rectangle with vertices at â extA(-3,

The vertices of $\triangle ABC$ are $A(-4, 4)$, $B(-1, 7)$, and $C(-1, 4)$.

A 270° clockwise rotation about the origin $(0,0)$ transforms a point $(x, y)$ to $(y, -x)$.

For vertex $A(-4, 4)$:
$A' = (4, -(-4)) = (4, 4)$

For vertex $B(-1, 7)$:
$B' = (7, -(-1)) = (7, 1)$

For vertex $C(-1, 4)$:
$C' = (4, -(-1)) = (4, 1)$

The vertices of the image $\triangle A'B'C'$ are $A'(4, 4)$, $B'(7, 1)$, and $C'(4, 1)$.

To graph, plot the original vertices $A(-4, 4)$, $B(-1, 7)$, $C(-1, 4)$ and connect them to form $\triangle ABC$.
Then, plot the image vertices $A'(4, 4)$, $B'(7, 1)$, $C'(4, 1)$ and connect them to form $\triangle A'B'C'$.

The coordinates of the image are:
$\boxed{A'(4, 4), B'(7, 1), C'(4, 1)}$

: A rectangle has vertices at $A(-3, 1)$, $B(-1, 1)$, $C(-1, 4)$, and $D(-3, 4)$. Reflect the rectangle across the line $y = 2$ and find the coordinates of the new vertices. Explain how the position of the shape changes after the reflection.

The vertices of the rectangle are $A(-3, 1)$, $B(-1, 1)$, $C(-1, 4)$, and $D(-3, 4)$.

A reflection across the horizontal line $y=k$ transforms a point $(x, y)$ to $(x, 2k - y)$.
Here, $k=2$, so the rule is $(x, y) \to (x, 2(2) - y) = (x, 4 - y)$.

For vertex $A(-3, 1)$:
$A' = (-3, 4 - 1) = (-3, 3)$

For vertex $B(-1, 1)$:
$B' = (-1, 4 - 1) = (-1, 3)$

For vertex $C(-1, 4)$:
$C' = (-1, 4 - 4) = (-1, 0)$

For vertex $D(-3, 4)$:
$D' = (-3, 4 - 4) = (-3, 0)$

The coordinates of the new vertices are:
$\boxed{A'(-3, 3), B'(-1, 3), C'(-1, 0), D'(-3, 0)}$

The reflection across the line $y=2$ flips the rectangle vertically. Points that were below the line $y=2$ are now above it, and points that were above the line $y=2$ are now below it. The x-coordinates of the vertices remain unchanged, while the y-coordinates are transformed such that their distance from the line $y=2$ is preserved but on the opposite side.

Reflection is an isometry, which means it preserves the size and shape of the figure. The rectangle remains a rectangle of the same dimensions (length and width) and its angles remain 90 degrees. Only its position and orientation in the coordinate plane change.