Since the graph passes through (0, 4), has a line of symmetry at x = 2.5 and is a smiley face curve, it means that the x-coordinate of the graph’s turning point is 2.5.

We can write the equation of our curve as

y = a (x + h)^2 + k

The given equation is y = x^2 + bx + c, so we can safely assume that a = 1.

y = (x + h)^2 + k

Since the line of symmetry is x = 2.5, then “x - 2.5 = 0”, so h = -2.5.

y = (x - 2.5)^2 + k

Since the graph passes (0, 4),

4 = (0 - 2.5)^2 + k
4 = 6.25 + k
-2.25 = k

So, our equation of the curve is

y = (x - 2.5)^2 - 2.25
y = x^2 - 2.5x - 2.5x + 6.25 - 2.25
y = x^2 - 5x + 4

By direct comparison, b = -5 and c = 4.

y-coordinate of turning point (x = 2.5)
= (2.5 - 2.5)^2 - 2.25
= 0 - 2.25
= -2.25

Thank you !! ^