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To solve the problem and determine the correct description of the graph of \( y \geq -x^2 + 8x - 2 \), let's follow these steps:

### 1. Rewrite the quadratic equation in vertex form:
The given quadratic equation is:
\[
y \geq -x^2 + 8x - 2
\]
We complete the square to rewrite this in vertex form.

1.1. Factor out the coefficient of \(-1\) from the first two terms:
\[
y \geq -(x^2 - 8x) - 2
\]

1.2. Complete the square inside the parentheses. Take half of the coefficient of \(x\), which is \(-8\), divide it by 2 to get \(-4\), and square it to get 16. Add and subtract 16 inside the parentheses:
\[
y \geq -(x^2 - 8x + 16 - 16) - 2
\]
\[
y \geq -(x - 4)^2 + 16 - 2
\]
\[
y \geq -(x - 4)^2 + 14
\]

So, the vertex form of the equation is:
\[
y \geq -(x - 4)^2 + 14
\]

### 2. Analyze the graph:
- **Vertex**: The vertex of the parabola is at \( (4, 14) \).
- **Direction**: Since the coefficient of \( (x - 4)^2 \) is negative, the parabola opens downward.
- **Line type**: The inequality is \( \geq \), so the boundary will be a **solid line**.
- **Shading**: Because the inequality is \( y \geq \), the shading will be **above the parabola**.

### 3. Compare with the answer choices:
The correct description should indicate:
- Vertex at \( (4, 14) \),
- The parabola opens down,
- The boundary is a solid line,
- Shading is above the parabola.

Thus, the correct answer is the first option:
**"The vertex is at (4, 14). The parabola is a solid line that opens down. Shading is above the parabola."**

Would you like further details or explanations? Here are 5 related questions to expand on this:

1. How do you graph a parabola given its vertex form?
2. What is the significance of completing the square in graphing quadratic functions?
3. How does the sign of the quadratic term affect the direction of the parabola?
4. What is the difference between solid and dotted boundary lines in graphing inequalities?
5. How do you determine where to shade when graphing inequalities?

**Tip**: When graphing quadratic inequalities, always pay attention to the inequality sign to determine the type of boundary line and where to shade.

Which best describes the graph of y ≥ -x² + 8x - 2?

This question involves graphing the quadratic inequality y ≥ -x² + 8x - 2 by completing the square, finding the vertex, and determining the shading area. Learn about key algebraic concepts such as graphing parabolas and vertex forms, suitable for students in Grades 9-11.

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