The graph of a quadratic function f intersects the x-axis at `x=-2 and x=6`. If `f(8)=f(p)`, which could be the value of p?

Options

A. `-6`
B. `-4`
C. `-2`
D. `0`

Answer

To solve the problem, we need to analyze the properties of the quadratic function given that it intersects the x-axis at \(x = -2\) and \(x = 6\).

### Step-by-Step Solution:

1. **Identify the Roots of the Quadratic Function:**
The quadratic function \(f(x)\) can be expressed in factored form based on its roots:
\[ f(x) = a(x + 2)(x - 6) \]
where \(a\) is a non-zero constant.

2. **Determine the Vertex of the Quadratic Function:**
The vertex of a quadratic function in the form \(f(x) = a(x - r_1)(x - r_2)\) is located at the midpoint of the roots. The roots here are \(-2\) and \(6\).
\[ \text{Vertex } x = \frac{-2 + 6}{2} = \frac{4}{2} = 2 \]
Thus, the vertex is at \(x = 2\).

3. **Evaluate \(f(8)\):**
To find \(f(8)\), we need to substitute \(x = 8\) into the function:
\[ f(8) = a(8 + 2)(8 - 6) = a(10)(2) = 20a \]

4. **Find the Symmetric Point:**
Since the graph of the quadratic function is symmetric about the vertex, the value of \(p\) that satisfies \(f(8) = f(p)\) must be equidistant from the vertex \(x = 2\). The distance from \(8\) to \(2\) is:
\[ 8 - 2 = 6 \]
Therefore, to find \(p\), we subtract this distance from the vertex:
\[ p = 2 - 6 = -4 \]

5. **Conclusion:**
Thus, the value of \(p\) such that \(f(8) = f(p)\) is:
\[ p = -4 \]

### Final Answer:
The value of \(p\) could be \(-4\).