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 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**.