Question

A quadratic inequality opens downward, has a vertex at (2,4)
, passes through the point (1,3)
, and has a dashed curve with shading above the curve.

Write the quadratic inequality that represents the description.

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

Answer 1

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:

\[
\boxed{y < -x^2 + 4x}
\]