Question

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

Answer 1

The graph of `y=-(x+2)^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(x)` 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(x)=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 `color(blue)2` being added directly to the `x`, so it is `f(x+color(blue)2)`, making it `y=(x+color(blue)2)^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 `color(red)-` outside of the `f(x+2)`, making it `y=color(red)-(x+color(blue)2)^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 `color(green)2` being subtracted to the whole function, so `color(red)-f(x+color(blue)2)-color(green)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]}