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

Aug 8, 2017

The graph of

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

The original function in this case is

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]}
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