Answer :
We are given an inequality \( y \geq -x^2 + 8x - 2 \). Let's analyze the components step-by-step to determine the graph's characteristics.
### Step 1: Determine the Vertex
The given inequality represents a quadratic equation. To find the vertex of the quadratic equation \( y = -x^2 + 8x - 2 \), we use the vertex form of a quadratic equation, which is \( x = -\frac{b}{2a} \) where \( a = -1 \) and \( b = 8 \).
[tex]\[ x = -\frac{8}{2(-1)} = -\frac{8}{-2} = 4 \][/tex]
Next, we substitute \( x = 4 \) back into the equation to find the \( y \)-coordinate of the vertex.
[tex]\[ y = -(4)^2 + 8(4) - 2 \][/tex]
[tex]\[ y = -16 + 32 - 2 \][/tex]
[tex]\[ y = 14 \][/tex]
So, the vertex is at \( (4, 14) \).
### Step 2: Determine the Direction of the Parabola
The coefficient \( a \) in the quadratic equation \( y = -x^2 + 8x - 2 \) is negative (\( a = -1 \)). Therefore, the parabola opens downwards.
### Step 3: Determine the Type of Line and Shading
Since the inequality is \( y \geq -x^2 + 8x - 2 \):
- The line is solid because the inequality is \( \geq \) (greater than or equal to).
- The shading is above the parabola because we are considering \( y \) values that are greater than or equal to the values on the parabola.
### Conclusion
Based on the analysis, the correct description of the graph of \( y \geq -x^2 + 8x - 2 \) is:
- The vertex is at \( (4, 14) \)
- The parabola is a solid line that opens down
- The shading is above the parabola
Thus, the correct choice is:
The vertex is at [tex]\( (4, 14) \)[/tex]. The parabola is a solid line that opens down. Shading is above the parabola.
### Step 1: Determine the Vertex
The given inequality represents a quadratic equation. To find the vertex of the quadratic equation \( y = -x^2 + 8x - 2 \), we use the vertex form of a quadratic equation, which is \( x = -\frac{b}{2a} \) where \( a = -1 \) and \( b = 8 \).
[tex]\[ x = -\frac{8}{2(-1)} = -\frac{8}{-2} = 4 \][/tex]
Next, we substitute \( x = 4 \) back into the equation to find the \( y \)-coordinate of the vertex.
[tex]\[ y = -(4)^2 + 8(4) - 2 \][/tex]
[tex]\[ y = -16 + 32 - 2 \][/tex]
[tex]\[ y = 14 \][/tex]
So, the vertex is at \( (4, 14) \).
### Step 2: Determine the Direction of the Parabola
The coefficient \( a \) in the quadratic equation \( y = -x^2 + 8x - 2 \) is negative (\( a = -1 \)). Therefore, the parabola opens downwards.
### Step 3: Determine the Type of Line and Shading
Since the inequality is \( y \geq -x^2 + 8x - 2 \):
- The line is solid because the inequality is \( \geq \) (greater than or equal to).
- The shading is above the parabola because we are considering \( y \) values that are greater than or equal to the values on the parabola.
### Conclusion
Based on the analysis, the correct description of the graph of \( y \geq -x^2 + 8x - 2 \) is:
- The vertex is at \( (4, 14) \)
- The parabola is a solid line that opens down
- The shading is above the parabola
Thus, the correct choice is:
The vertex is at [tex]\( (4, 14) \)[/tex]. The parabola is a solid line that opens down. Shading is above the parabola.
