To find the quadratic inequality that meets the given criteria, we start from the vertex form of a quadratic function, which is given by:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. From your description, the vertex is \((2, 4)\), so we can rewrite this as:
\[ y = a(x - 2)^2 + 4 \]
Since the parabola opens downwards, the coefficient \(a\) will be negative.
Next, we know the parabola passes through the point \((1, 3)\). We can substitute \(x = 1\) and \(y = 3\) into the equation to solve for \(a\):
\[ 3 = a(1 - 2)^2 + 4 \]
Simplifying this:
\[ 3 = a(1)^2 + 4 \] \[ 3 = a + 4 \] \[ a = 3 - 4 = -1 \]
Now we can write the quadratic function:
\[ y = -1(x - 2)^2 + 4 \]
Expanding this:
\[ y = -1(x^2 - 4x + 4) + 4 \] \[ y = -x^2 + 4x - 4 + 4 \] \[ y = -x^2 + 4x \]
Now, since we have a dashed curve with shading above the curve, our inequality becomes:
\[ y < -x^2 + 4x \]
Thus, the quadratic inequality that represents the description given is:
\[ \boxed{y < -x^2 + 4x} \]