What is the vertex of y=5(x+3)^2-9 ?

Mar 4, 2018

The vertex coordinates are: $\left(- 3 , - 9\right)$

Explanation:

There are two ways to solve it:

For the equation $a {x}^{2} + b x + c = y$ :

The $x$-value of the vertex $= \frac{- b}{2 a}$

The $y$-value can be found out by solving the equation.

So now, we have to expand the equation we have to get it in quadratic form:

$5 {\left(x + 3\right)}^{2} - 9 = y$

$\to 5 \left(x + 3\right) \left(x + 3\right) - 9 = y$

$\to 5 \left({x}^{2} + 6 x + 9\right) - 9 = y$

$\to 5 {x}^{2} + 30 x + 45 - 9 = y$

$\to 5 {x}^{2} + 30 x + 36 = y$

Now, $a = 5$ and $b = 30$. (FYI, $c = 36$)

$\to \frac{- b}{2 a} = \frac{- \left(30\right)}{2 \left(5\right)}$

$\to \frac{- b}{2 a} = \frac{- 30}{10}$

$\to \frac{- b}{2 a} = - 3$

Thus, the $x$-value $= - 3$. Now, we substitute $- 3$ for $x$ to get the $y$ value of the vertex:

$5 {x}^{2} + 30 x + 36 = y$

becomes:

$5 {\left(- 3\right)}^{2} + 30 \left(- 3\right) + 36 = y$

$\to 45 + \left(- 90\right) + 36 = y$

$\to y = 81 - 90$

$\to y = - 9$

Thus, since $x = - 3$ and $y = - 9$, the vertex is:

$\left(- 3 , - 9\right)$

2) This is the easier way of doing it - by using the Vertex Formula:

In the equation $a {\left(x - h\right)}^{2} + k = y$, the vertex is $\left(h , k\right)$

We are already given an equation in the Vertex format, so it is easy to find out the Vertex coordinates:

$5 {\left(x + 3\right)}^{2} - 9 = y$

can be rewritten as:

$5 {\left(x - \left(- 3\right)\right)}^{2} - 9 = y$

Now we have it in the Vertex-form, where $h = - 3$, and $k = - 9$

So, the Vertex coordinates are:

$\left(h , k\right)$

$= \left(- 3 , - 9\right)$

Tip: you can change an equation in a quadratic form to a vertex form by completing the square. If you are not aware of this concept, search it up on the Internet or post a question on Socratic.