Over what interval is the graph of f(x) =-(x+8)^2-1 decreasing?

Accepted Solution

Answer:Step-by-step explanation:If we used calculus this would be pretty simple, but I'm going to assume this is not a calc class, so I will answer with algebra.You're going to need to know your graph transformations.  And the base graph of x^2.  When does x^2 decrease?  Well, it reaches its lowest point at  (0,0), so everything before that it is decreasing.  then after that it is increasing, so keep that in mind, especially the point 0,0.Now, there are three transformations here.  -1*x^2 (x+8)^2 and x^2-1, so let's see what each of those do.-1*x^2 or just -x^2 flips the graph over the x axis.  So now, instead of the u shape x62 usually has it now has an upside down u shape, which means the increasing and decreasing flipped.  So now it increases from - infinity to 0, and then decreases from 0 to infinity.   It reaches it's highest point at (0,0) still though since it was not moved up, down, left or right.(x+8)^2 moves the graph to the LEFT 8 times.  I stress left because it's moving int he negative direction on the x axis even though we are adding x  You need to keep this in mind.  Anyway, that peak point (0,0) is moved left by 8 now, so that's (-8,0).  this means the graph increases from - infinity to 8 now, and then decreases from 8 to infinity.This final transformation is just adding 1.  x^2-1 moves the graph down 1.  Unlike the +8 from the last one this is outside of the parenthesis.  if this was (x-1)^2 it would move the graph to the right by 1.  Also note being outside those parenthesis moves it in the corresponding direction, down on the y axis.  Anyway, x^2-1 would move the tip (or the vertex) down to (0,-1), but we moved our vertex to (-8,0) so now it gets moved to (-8,1).Now the vertex is a maximum, which means the graph increases until it gets to the vertex, and then decreases.  Since the 1 only moved it down the interval left and right doesn't change for increasing/ decreasing.  So it still decreases from -8 to infinity.Again, let me know if you didn't get something.