What is the equation of the parabola that has a vertex at  (7, 9)  and passes through point  (0, 2) ?

Jul 15, 2016

$y = - \frac{1}{7} {\left(x - 7\right)}^{2} + 9$

Explanation:

This problem requires that we understand how a function can be shifted around and stretched to meet particular parameters. In this case, our basic function is $y = {x}^{2}$. This describes a parabola which has its vertex at $\left(0 , 0\right)$. However we can expand it as:

$y = a {\left(x + b\right)}^{2} + c$

In the most basic situation:
$a = 1$
$b = c = 0$

But by altering these constants, we can control the shape and position of our parabola. We will start with the vertex. Since we know it needs to be at $\left(7 , 9\right)$ we need to shift the default parabola to the right by $7$ and up by $9$. That means manipulating the $b$ and $c$ parameters:

Obviously $c = 9$ because that will mean all $y$ values will increase by $9$. But less obviously, $b = - 7$. This is because when we add a factor to the $x$ term, the shift will be opposite that factor. We can see that here:
$x + b = 0$
$x = - b$

When we add $b$ to $x$, we move the vertex to $- b$ in the $x$ direction.

So our parabola so far is:
$y = a {\left(x - 7\right)}^{2} + 9$

But we need to stretch it to pass through point $\left(0 , 2\right)$. This is as simple as plugging in those values:
$2 = a {\left(- 7\right)}^{2} + 9$
$2 = 49 a + 9$
$- 7 = 49 a$
$a = - \frac{1}{7}$

That means our parabola will have this equation:
$y = - \frac{1}{7} {\left(x - 7\right)}^{2} + 9$