y=(x-h)^2 + k
0=(4-h)^2 +k
2.25 = (3-h)^2 + k
2 equations, 2 unknowns, solve for (h,k) the vertex
then expand and rewrite the equation to solve for the constant term = y intercept
what you initially have with (4,0) is the x intercept
there are an infinite number of equations for parabolas through the two given points
but the above is for the most basic standard upward opening parabola though those 2 points
0=(4-h)^2 + k = 16-8h+h^2 + k
2.25= (3-h)^2 = 9-6h+h^2 + k
subtract
2.25 =-7 +2h
2h = 9.25
h = 4.625
0=(4-4.625)^2 + k
k=-(4-4.625)^2=-(-.625)^2 =-.378225
y=(x-4.625)^2 -0.378225
y=x^2 -9.25x + 4.625^2-0.378225
y=x^2 -9.25x + 21.0124
y intercept =the constant term = 21.0124 = the point (0, 21.0124)
y intercept = about 21
(no guarantees there's no arithmetic mistake above, as the calculations get a little tedious, but if you plot the points, the vertex and intercept, and sketch a parabola through them, it looks about right. It's the general method for finding a basic simple parabola through two given points)
Or, solve for the other x intercept
if h was calculated correctly earlier as 4.625, then the other x intercept is symmetric about the vertical line x=4.625. the other x intercept = 4+2(.625) = 5 1/4 = 5.25
then the parabola equation is
y=(x-4)(x-5.25) where the two x intercepts are 4 and 5.25, the points (4,0) and (5.25,0)
but that leads to
y=x^2 -42x + 18.75 which is different from the equation earlier reached so, there's a likely arithmetic error somewhere. although the y intercept is roughly the same, 18.75 compared to about 21
