Question Drag the blue movable points to produce a graph that correctly represents the inequality \( y \leq-x^{2}-2 x-6 \). You may select one of the five possible shading options by sliding the black point up or down the panel on the left side of the graphing window.

To graph the inequality \( y \leq -x^2 - 2x - 6 \), we need to follow these steps: 1. **Identify the boundary line**: The inequality \( y \leq -x^2 - 2x - 6 \) has a boundary line given by the equation \( y = -x^2 - 2x - 6 \). This is a quadratic equation, and its graph will be a parabola that opens downwards. 2. **Find the vertex**: The vertex of the parabola can be found using the formula for the vertex of a quadratic equation \( y = ax^2 + bx + c \), where the x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, \( a = -1 \) and \( b = -2 \). \[ x = -\frac{-2}{2 \cdot -1} = 1 \] Now, substitute \( x = 1 \) back into the equation to find the y-coordinate of the vertex: \[ y = -1^2 - 2(1) - 6 = -1 - 2 - 6 = -9 \] So, the vertex is at the point \( (1, -9) \). 3. **Determine the intercepts**: - **Y-intercept**: Set \( x = 0 \): \[ y = -0^2 - 2(0) - 6 = -6 \] So, the y-intercept is \( (0, -6) \). - **X-intercepts**: Set \( y = 0 \): \[ 0 = -x^2 - 2x - 6 \] Rearranging gives: \[ x^2 + 2x + 6 = 0 \] The discriminant \( b^2 - 4ac = 2^2 - 4(1)(6) = 4 - 24 = -20 \) is negative, indicating there are no real x-intercepts. 4. **Graph the parabola**: Plot the vertex \( (1, -9) \) and the y-intercept \( (0, -6) \). Since the parabola opens downwards, draw the curve accordingly. 5. **Shading the region**: Since the inequality is \( y \leq -x^2 - 2x - 6 \), we will shade the region below the parabola. 6. **Final adjustments**: Ensure that the boundary line is solid (indicating that points on the line are included in the solution set). Now, you can drag the blue movable points to create the graph of the parabola and shade the appropriate region below it.