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)
1 answer