To determine the interval over which the graph of the function [tex]\( f(x) = 2(x+3)^2 + 2 \)[/tex] is decreasing, let's analyze the function step-by-step.
1. Form of the Function:
The given function is in the vertex form of a parabola:
[tex]\[
f(x) = a(x - h)^2 + k
\][/tex]
Here, [tex]\( a = 2 \)[/tex], [tex]\( h = -3 \)[/tex], and [tex]\( k = 2 \)[/tex], so we can rewrite the function as:
[tex]\[
f(x) = 2(x + 3)^2 + 2
\][/tex]
2. Vertex of the Parabola:
The vertex of the parabola given by [tex]\(f(x) = 2(x + 3)^2 + 2\)[/tex] is at the point [tex]\((-3, 2)\)[/tex].
3. Direction of the Parabola:
The coefficient [tex]\( a = 2 \)[/tex] is positive, which tells us that the parabola opens upwards. This means the graph decreases to the left of the vertex and increases to the right.
4. Determine the Interval:
For parabolas that open upwards, the function decreases as we move leftward from the vertex. Therefore, the function is decreasing in the interval that includes all x-values less than the x-coordinate of the vertex [tex]\(-3\)[/tex].
5. Correct Interval:
The x-coordinates for which the function [tex]\( f(x) \)[/tex] is decreasing is:
[tex]\[
(-\infty, -3)
\][/tex]
So, the interval over which the graph of [tex]\( f(x) = 2(x+3)^2 + 2 \)[/tex] is decreasing is:
[tex]\[
(-\infty, -3)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{(-\infty, -3)}
\][/tex]
