Given f(x) = -(x+2)^2 - 5, explain how the graph of the function changes at each transformation and give the coordinates of 3 points per phase of the graph.
You need to start with the identity f(x) = y, hence, substituting x + 2 for x, the graph f(x+2) = y is obtained by translating the graph f(x)=y to the left by 2 units.
Notice that f(x+2) is multiplied by -1, hence the graph of y = -f(x+2) is obtained by reflecting the graph of y=f(x+2) throuh the x axis.
Notice also that the graph of y = -f(x+2) - 5 is obtained by translating the graph of y=-f(x+2) down by 5 units.
Sketching the new graph obtained after a series of transformation were performed yields:
a= -1; h= -2; k= -5
(-) = reflects the graph on/ in the x-axis
1= since a<0, the graph opens down.
h & k translates the graph by the given units. That is it stretches the graph vertically or horizontally.
Therefore, h (-2) translates the graph by a factor of 2, and
K (-5) translates the graph 5 units down.