Answer
$[-3,9]$
Work Step by Step
We first write the equation in the form $y=c+a \cos [b(x-d)]$. Therefore, $y=3-6\sin (2x+\frac{\pi}{2})$ becomes $y=3-6\sin [2(x+\frac{\pi}{4})]$.
Comparing the equation to its general form, we find that $c=3$ and $a=-6$. Using $a=-6$, we had found previously in (b) that the amplitude is 6. Also, we know that $c$ represents the y-intercept and thus also the axis of the graph.
Using the axis and the amplitude, we can find the minimum and maximum values of y (which constitute the range of the function):
Minimum value$= 3-6=-3$
Minimum value$= 3+6=9$
Therefore, the range of the function is $[-3,9].$
