y = x^2 - 3x - 5
y = (x^2 - 3x + 9/4) - 5 - 9/4 (complete the square)
y = (x - 3/2)^2 - 29/4)
vertex is (h,k) or (3/2, -29/4)
axis of symentry is x = 3/2.
since the y is negative there are roots on the x-axis.
set y = 0 to find them
0 = (-3/2)^2 - 29/4; there are two roots one on each side of 3/2.
9/4 - 29/4 or 9/4 + 29/4
-20/4 or 38/4
(-5, 0) or (9 1/2, 0)
You have three points now these two and the vertex.
y = x^2 + 2
vertex is (0, 2)
axis of symentry is the y-axis.
it does not cross the x-axis because it is above it.
set y = 0
0 = x^2 + 2
x^2 = -2
x = +or- sqrt(-2) no real solutions.