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

1 Answer
Aug 8, 2017

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.


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]}