To determine the interval over which the graph of the function [tex]\( f(x) = 2(x + 3)^2 + 2 \)[/tex] is decreasing, we need to analyze the properties of the function.
1. Understanding the Function's Form:
The given function is in vertex form of a parabola, [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola. Comparing the given function [tex]\( f(x) = 2(x + 3)^2 + 2 \)[/tex] to the vertex form:
- Here, [tex]\( a = 2 \)[/tex], [tex]\( h = -3 \)[/tex], and [tex]\( k = 2 \)[/tex].
- The vertex of the parabola is therefore at [tex]\( (-3, 2) \)[/tex].
2. Direction of the Parabola:
- Since the coefficient [tex]\( a = 2 \)[/tex] is positive, the parabola opens upwards.
3. Behavior of the Function Around the Vertex:
- For parabolas that open upwards, they decrease to the left of the vertex and increase to the right of the vertex.
- Thus, the function [tex]\( f(x) \)[/tex] will be decreasing to the left of the vertex (i.e., for [tex]\( x < -3 \)[/tex]).
4. Determining the Interval:
- Since the vertex [tex]\( x = -3 \)[/tex] is the turning point where the function changes from decreasing to increasing, the interval over which the function is decreasing is [tex]\( x < -3 \)[/tex].
5. Conclusion:
The interval over which the function [tex]\( f(x) = 2(x + 3)^2 + 2 \)[/tex] is decreasing is [tex]\((-∞, -3)\)[/tex].
Thus, the correct answer is:
[tex]\[
(-\infty, -3)
\][/tex]
