Question

A quadratic inequality opens downward, has a vertex at (0,5)
, passes through the point (1,2)
, and has a solid curve with shading below the curve.

Which of the following quadratic inequalities represents the description?

(1 point)
Responses

y≤−3x2+5
y is less than or equal to negative 3 x squared plus 5

y<−3x2+5
y is less than negative 3 x squared plus 5

y≥−3x2+5
y is greater than or equal to negative 3 x squared plus 5

y≤3x2+5

Answer 1

To find the correct quadratic inequality, we need to determine the equation of the quadratic function that fits the description. Since the quadratic opens downward and has a vertex at (0, 5), we can start with the vertex form of a quadratic equation:

\[ y = a(x - h)^2 + k \]
where (h, k) is the vertex.

Given the vertex (0, 5), we have:
\[ y = a(x - 0)^2 + 5 \]
or
\[ y = ax^2 + 5 \]

Since the quadratic opens downward, the coefficient \( a \) must be negative. We know that the parabola passes through the point (1, 2). To find the value of \( a \), we can substitute x = 1 and y = 2 into the equation:

\[ 2 = a(1)^2 + 5 \]
\[ 2 = a + 5 \]

Now, solving for \( a \):
\[ a = 2 - 5 \]
\[ a = -3 \]

Thus, the equation of the quadratic function is:
\[ y = -3x^2 + 5 \]

Since the shading is below the curve (which indicates we want the area where \( y \) is less than or equal to the quadratic function), our inequality will be:
\[ y \leq -3x^2 + 5 \]

Hence, the correct response is:

**y ≤ −3x² + 5**
(y is less than or equal to negative 3 x squared plus 5)