To determine which statement correctly describes the graph of the inequality [tex]\( y + 2 \geq -4(x - 3)^2 \)[/tex], let's proceed through the following steps:
1. Rewrite the inequality in the standard form:
[tex]\[
y + 2 \geq -4(x - 3)^2
\][/tex]
2. Isolate [tex]\( y \)[/tex]:
[tex]\[
y \geq -4(x - 3)^2 - 2
\][/tex]
3. Analyze the form of the quadratic inequality:
- The standard form for a parabola is [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex.
- Here, [tex]\(a = -4\)[/tex], [tex]\(h = 3\)[/tex], and [tex]\(k = -2\)[/tex].
- Since [tex]\(a\)[/tex] is negative, the parabola opens downward.
4. Identify the vertex:
[tex]\[
\text{Vertex:}(h, k) = (3, -2)
\][/tex]
5. Interpret the inequality [tex]\( y \geq -4(x - 3)^2 - 2 \)[/tex]:
- The inequality [tex]\( \geq \)[/tex] indicates the region where [tex]\( y \)[/tex] is greater than or equal to the parabola.
- Since the parabola opens downward and we are considering [tex]\( y \geq \)[/tex], the shaded region is above the parabola.
6. Merge the conclusions:
- The parabola opens downward.
- The shading is above the vertex.
Therefore, the statement that best describes the graph is:
[tex]\[
\boxed{\text{a parabola opening down, with shading above the vertex}}
\][/tex]
