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 $$
.

1 answer

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:

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