Question

What is the equation written in vertex form of a parabola with a vertex of (9, -1) that passes through (7. 7)?

Answer 1

Answer:

$y = 2\, (x - 9)^{2} - 1$.

Step-by-step explanation:

If the vertex of a parabola is $(h,\, k)$, there would exist a constant $a$ ($a \ne 0$) such that the following would be an equation for this parabola:

$y = a\, (x - h)^{2} + k$.

The equation above is in the vertex form.

In this question, the vertex of this parabola is $(9,\, -1)$. (Such that $h = 9$ and $k = -1$.) Therefore, the equation of this parabola in the vertex form would be:

$\text{ y = a\, (x - 9)^{2} + (-1) for some a \ne 0 }$.

It was also given that the point $(7,\, 7)$ is on the parabola. In other words, if $x = 7$ and $y = 7$, the equation of this parabola ($y = a\, (x - 9)^{2} + (-1)$) should be satisfied. In other words:

$7= a\, (7 - 9)^{2} + (-1)$.

Solve this equation for $a$.

$a = 2$.

Therefore, the equation of this parabola in vertex form would be$y = 2\, (x - 9) + (-1)$, which is equivalently $y = 2\, (x - 9)^{2} - 1$.