Answer:
(See explanation for further details)
Step-by-step explanation:
The standard equation of the parabola is:
[tex]y + 4 = C \cdot (x-k)^{2}[/tex]
The formula is now expanded into a the form of a second-order polynomial:
[tex]y + 4 = C\cdot x^{2} -2\cdot C\cdot k \cdot x +C\cdot k^{2}[/tex]
[tex]y = C\cdot x^{2} - (2\cdot C \cdot k) \cdot x + (C\cdot k^{2}-4)[/tex]
The general equation of the second-order polynomial is:
[tex]x = \frac{2\cdot C \cdot k \pm \sqrt{4\cdot C^{2}\cdot k^{2}-4\cdot C\cdot (C\cdot k^{2}-4)}}{2\cdot C}[/tex]
[tex]x = k \pm \frac{\sqrt{C^{2}\cdot k^{2}-C^{2}\cdot k^{2}+4\cdot C}}{C}[/tex]
[tex]x = k \pm 2\cdot \frac{\sqrt{C}}{C}[/tex]
[tex]x = k \pm \frac{2}{\sqrt{C}}[/tex]
The equations to be solved are presented herein:
[tex]-3 = k -\frac{2}{\sqrt{C}}[/tex]
[tex]5 = k + \frac{2}{\sqrt{C}}[/tex]
Now, the solution of the system is:
[tex]-3 +\frac{2}{\sqrt{C}} = 5 -\frac{2}{\sqrt{C}}[/tex]
[tex]\frac{4}{\sqrt{C}} = 8[/tex]
[tex]\sqrt{C} = \frac{1}{2}[/tex]
[tex]C = \frac{1}{4}[/tex]
[tex]k = 5 - \frac{2}{\sqrt{\frac{1}{4} }}[/tex]
[tex]k = 1[/tex]
The equation of the parabola is:
[tex]y = \frac{1}{4}\cdot (x-1)^{2} -4[/tex]
Lastly, the graphic of the function is included as attachment.
