Question

A quadratic inequality opens upward, has a vertex at (−1,−1)
, passes through the point (0,1)
, and has a solid curve with shading below the curve.

Write the quadratic inequality that represents the description.

(1 point)
The quadratic inequality that represents the description is $$
.

Answer 1

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:

\[
2(x + 1)^2 - 1 \leq 0
\]