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.

Question

Which of the following quadratic inequalities represents the description?

(1 point)
Responses

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

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

Answer 1

To find the correct quadratic inequality representing the given conditions, let's analyze the provided information.

1. The quadratic opens downward: This indicates that the coefficient of \(x^2\) in the quadratic expression is negative.

2. Vertex at (0, 5): The quadratic can be expressed in the vertex form as: \[ y = a(x - 0)^2 + 5 = ax^2 + 5 \] where \(a < 0\) since it opens downward.

3. Passes through the point (1, 2): We can use this point to find the value of \(a\): \[ 2 = a(1^2) + 5 \] \[ 2 = a + 5 \] \[ a = 2 - 5 = -3 \] Thus, the quadratic function is: \[ y = -3x^2 + 5 \]

4. Shading below the curve: Since the shading is below the curve, the correct inequality would be: \[ y \leq -3x^2 + 5 \]

Now, let's match this with the provided options:

• \(y \leq 3x^2 + 5\) (not valid as the parabola opens upward)
• \(y \leq -3x^2 + 5\) (correct)
• \(y < -3x^2 + 5\) (not valid as we require solid shading below the curve)
• \(y \geq -3x^2 + 5\) (not valid as this indicates shading above the curve)

Thus, the correct response is: y ≤ −3x² + 5.