Explanation:
y = -(x - 5)² - 7
this has only negative y-values.
because a square is always positive, the negative of it is always negative, and -7 of something already negative is negative.
plus the fact that the factor of x² is negative (-1), so it opens downwards.
so, this is green.
y = (x + 5)² - 7
when x = -5, y = -7, the vertex is (-5, -7). there is no extra stretching or narrowing factor, it opens up as wide as the others.
so, it is brown.
y = (x - 5)² + 7
this has only positive y-values (as an square is always positive, and +7 to something positive is also positive), the vertex is at x = 5 : (5, 7)
it is also the mirrored image (× -1) of the first (green) case : y = -(x - 5)² - 7 across the x-axis.
so, it is blue.
y = (x + 5)² + 7
this has only positive y-values (as an square is always positive, and +7 to something positive is also positive), the vertex is at x = -5 : (-5, 7)
so, it is turquoise.
y = 6(x + 5)² - 7
remember the case above : y = (x + 5)² - 7.
when x = -5, y = -7, the vertex is (-5, -7). but here is an extra stretching or narrowing factor (6), it opens up much narrower than the others.
so, it is gray.
y = (x - 5)² - 7
when x = 5, y = -7, the vertex is (5, -7). there is no extra stretching or narrowing factor, it opens up as wide as the others.
so, it is purple.