The problem provided involves a square $BCDE$ with vertices $B(-3, -4)$, $C(0, -3)$, $D(1, -6)$, and $E(-2, -7)$. The task is to apply a dilation to the square using a scale factor of $k = 4$, with the center of dilation at $(0, -5)$.

Steps to Solve the Dilation:

1. Dilation Formula: The dilation formula for any point $(x, y)$ with respect to a center of dilation $(x_c, y_c)$ and scale factor $k$ is: $(x', y') = (x_c + k(x - x_c), y_c + k(y - y_c))$ where $(x', y')$ are the new coordinates after dilation.

2. Center of Dilation: $(0, -5)$ We'll apply this formula for each vertex of the square.

3. Dilation for each vertex:

   • For $B(-3, -4)$: $x' = 0 + 4(-3 - 0) = -12, \quad y' = -5 + 4(-4 - (-5)) = -5 + 4(1) = -1$ So, the new coordinates of $B'$ are $(-12, -1)$.

   • For $C(0, -3)$: $x' = 0 + 4(0 - 0) = 0, \quad y' = -5 + 4(-3 - (-5)) = -5 + 4(2) = 3$ So, the new coordinates of $C'$ are $(0, 3)$.

   • For $D(1, -6)$: $x' = 0 + 4(1 - 0) = 4, \quad y' = -5 + 4(-6 - (-5)) = -5 + 4(-1) = -9$ So, the new coordinates of $D'$ are $(4, -9)$.

   • For $E(-2, -7)$: $x' = 0 + 4(-2 - 0) = -8, \quad y' = -5 + 4(-7 - (-5)) = -5 + 4(-2) = -13$ So, the new coordinates of $E'$ are $(-8, -13)$.

New Coordinates after Dilation:

• $B'(-12, -1)$
• $C'(0, 3)$
• $D'(4, -9)$
• $E'(-8, -13)$

These are the vertices of the square after applying the dilation.

Would you like any further details or clarifications?

Additional Questions:

1. How do you find the center of dilation if not given directly?
2. What happens to the shape of the figure when you apply a dilation with a scale factor less than 1?
3. Can dilation change the orientation of a figure? Why or why not?
4. What is the effect of a negative scale factor in dilation?
5. How would you apply dilation if the center is outside the coordinate plane?

Tip: Always double-check your calculations, especially when applying a scale factor, as small errors can significantly change the shape or size.