Answer:
[tex]y = 2\, (x - 9)^{2} - 1[/tex].
Step-by-step explanation:
If the vertex of a parabola is [tex](h,\, k)[/tex], there would exist a constant [tex]a[/tex] ([tex]a \ne 0[/tex]) such that the following would be an equation for this parabola:
[tex]y = a\, (x - h)^{2} + k[/tex].
The equation above is in the vertex form.
In this question, the vertex of this parabola is [tex](9,\, -1)[/tex]. (Such that [tex]h = 9[/tex] and [tex]k = -1[/tex].) Therefore, the equation of this parabola in the vertex form would be:
[tex]\text{$y = a\, (x - 9)^{2} + (-1)$ for some $a \ne 0$}[/tex].
It was also given that the point [tex](7,\, 7)[/tex] is on the parabola. In other words, if [tex]x = 7[/tex] and [tex]y = 7[/tex], the equation of this parabola ([tex]\text{$y = a\, (x - 9)^{2} + (-1)$}[/tex]) should be satisfied. In other words:
[tex]7= a\, (7 - 9)^{2} + (-1)[/tex].
Solve this equation for [tex]a[/tex].
[tex]a = 2[/tex].
Therefore, the equation of this parabola in vertex form would be[tex]y = 2\, (x - 9) + (-1)[/tex], which is equivalently [tex]y = 2\, (x - 9)^{2} - 1[/tex].
