Δabc has coordinates of a (–6, –7), b (4, –2), and c (0, 7). find the coordinates of its image after a dilation centered at the origin with a scale factor of 2.

The rule for dilation (scale factor k) about the origin is given by:
(x,y)->(kx,ky)
This means that when k=2,
(x,y)->(2x,2y) after dilation.
A(-6,-7)->A'(-12,-14)
B(4,-2)->B'(8,-4)
C(0,7)->C'(0,14)

A dilation with a scale factor k about the origin can be described using the following notation:

(x,y)→(kx,ky).

In your case, k=2, then

A (–6, –7)→A'(2·(-6),2·(-7))=A'(-12,-14);B (4, –2)→B'(2·4,2·(-2))=B'(8,-4);C (0, 7)→C'(2·0,2·7)=C'(0,14).

A'(-12,-14), B'(8,-4), C'(0,14).

What do you mean by that?