To write the quadratic inequality based on the provided information, let's start by establishing the quadratic function based on the vertex and the point it passes through.
A quadratic function can be expressed in vertex form as:
\[ f(x) = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. Given the vertex is \((-1, -1)\), we have:
\[ f(x) = a(x + 1)^2 - 1 \]
Next, we know that the parabola passes through the point \((0, 1)\). We can use this point to find the value of \(a\):
\[ 1 = a(0 + 1)^2 - 1 \]
\[ 1 = a(1) - 1 \]
\[ 1 + 1 = a \]
\[ 2 = a \]
Now, we can write the quadratic function:
\[ f(x) = 2(x + 1)^2 - 1 \]
Now, we want to write the corresponding inequality. Since the problem states that the shading is below the curve, we will write:
\[ f(x) \leq 0 \]
Thus, substituting \(f(x)\) in the inequality, we get:
\[ 2(x + 1)^2 - 1 \leq 0 \]
To conclude, the quadratic inequality that represents the description is:
\[ 2(x + 1)^2 - 1 \leq 0 \]