To solve the given problem, we need to thoroughly analyze the inequality [tex]\( y \geq -x^2 + 8x - 2 \)[/tex].
First, let's rewrite it in a more familiar form, [tex]\( y = ax^2 + bx + c \)[/tex], where:
[tex]\[ a = -1 \][/tex]
[tex]\[ b = 8 \][/tex]
[tex]\[ c = -2 \][/tex]
### Step 1: Finding the Vertex
The vertex of a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the vertex formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{8}{2(-1)} \][/tex]
[tex]\[ x = -\frac{8}{-2} \][/tex]
[tex]\[ x = 4 \][/tex]
Next, to find the y-coordinate of the vertex, substitute [tex]\( x = 4 \)[/tex] back into the equation [tex]\( y = ax^2 + bx + c \)[/tex]:
[tex]\[ y = - (4)^2 + 8(4) - 2 \][/tex]
[tex]\[ y = -16 + 32 - 2 \][/tex]
[tex]\[ y = 14 \][/tex]
Thus, the vertex is [tex]\( (4, 14) \)[/tex].
### Step 2: Analyzing the Parabola's Direction
The coefficient [tex]\( a \)[/tex] (which is -1) determines the direction the parabola opens. Since [tex]\( a < 0 \)[/tex], the parabola opens downward.
### Step 3: Line Type and Shading
Given the inequality [tex]\( y \geq -x^2 + 8x - 2 \)[/tex]:
- The inequality symbol [tex]\( \geq \)[/tex] indicates that the line is included in the solution — which means it is a solid line.
- Since the inequality is [tex]\( \geq \)[/tex], the shaded region will be above the parabola.
### Conclusion
From these steps, we conclude the following description best fits the graph of [tex]\( y \geq -x^2 + 8x - 2 \)[/tex]:
- The vertex is at [tex]\( (4, 14) \)[/tex].
- The parabola is a solid line that opens down.
- Shading is above the parabola.
Thus, the correct answer is:
The vertex is at [tex]\( (4, 14) \)[/tex]. The parabola is a solid line that opens down. Shading is above the parabola.
