The vertical parabola that goes through [tex](x_{1}, y_{1}) = (-2, -3)[/tex], [tex](x_{2}, y_{2}) = (3, 12)[/tex] and [tex](x_{3}, y_{3}) = (5, 46)[/tex] is [tex]x^{2} + \frac{1}{2}\cdot D -\frac{1}{2}\cdot y -\frac{9}{2} = 0[/tex]
The general form of the equation for the vertical parabola is described below:
[tex]x^{2}+D\cdot x + E\cdot y + F = 0[/tex] (1)
Where [tex]D[/tex], [tex]E[/tex] and [tex]F[/tex] are coefficients of the parabola equation.
We need three distinct coplanar points to determine all coefficients.
If we know that [tex](x_{1}, y_{1}) = (-2, -3)[/tex], [tex](x_{2}, y_{2}) = (3, 12)[/tex] and [tex](x_{3}, y_{3}) = (5, 46)[/tex], then we get the following system of equations:
[tex]-2\cdot D -3\cdot E + F = - 4[/tex] (2)
[tex]3\cdot D + 12\cdot E + F = -9[/tex] (3)
[tex]5\cdot D + 46\cdot y + F = -25[/tex] (4)
The solution set of the system is: [tex]D = \frac{1}{2}[/tex], [tex]E = -\frac{1}{2}[/tex], [tex]F = - \frac{9}{2}[/tex]
The vertical parabola that goes through [tex](x_{1}, y_{1}) = (-2, -3)[/tex], [tex](x_{2}, y_{2}) = (3, 12)[/tex] and [tex](x_{3}, y_{3}) = (5, 46)[/tex] is [tex]x^{2} + \frac{1}{2}\cdot D -\frac{1}{2}\cdot y -\frac{9}{2} = 0[/tex].
We kindly invite to check this question on parabolae: https://brainly.com/question/21685473
