# How do you sketch the graph of y=-(x+2)^2-2 and describe the transformation?

Aug 8, 2017

The graph of $y = - {\left(x + 2\right)}^{2} - 2$ is:

graph{-(x+2)^2-2 [-10, 10, -5, 5]}

Its transformation is a reflection over the x-axis, a translation of 2 units left and a translation of 2 units down.

#### Explanation:

Have a look at the following summary for transformation rules of graphs:

Transformations are called transformations because they start off with the "original" or "standard" function $f \left(x\right)$ and then move/transform it to a different point based on a variety of things being added to the function or multiplied to it.

The original function in this case is $f \left(x\right) = {x}^{2}$. Let's graph this first to see how the translations affect it:

graph{x^2 [-10, 10, -5, 5]}
We notice that it has 3 transformations happening to it:

1. There is a $\textcolor{b l u e}{2}$ being added directly to the $x$, so it is $f \left(x + \textcolor{b l u e}{2}\right)$, making it $y = {\left(x + \textcolor{b l u e}{2}\right)}^{2}$ --> this means that there will be a horizontal translation left of 2 units. In the graph, we take the original function and shift it left 2 units:
graph{(x+2)^2 [-10, 10, -5, 5]}
2. There is a negative sign $\textcolor{red}{-}$ outside of the $f \left(x + 2\right)$, making it $y = \textcolor{red}{-} {\left(x + \textcolor{b l u e}{2}\right)}^{2}$ --> this means that there will be a reflection over the x-axis. In the graph, we take this shifted function and "flip" it over the x-axis:
graph{-(x+2)^2 [-10, 10, -5, 5]}
3. Finally, there is a $\textcolor{g r e e n}{2}$ being subtracted to the whole function, so $\textcolor{red}{-} f \left(x + \textcolor{b l u e}{2}\right) - \textcolor{g r e e n}{2}$. In the graph, this means that the shifted function needs to be shifted two units down:
graph{-(x+2)^2-2 [-10, 10, -5, 5]}