`f(x)=ax^(2)+bx+c, agt0`
The coordinates of the lowest point on the graph of the function defined by the equation above is `(3, 2)`. If `f(-1)=p`, then which of the following represents the value of p?

Options

A. `f(-5)`
B. `f(-4)`
C. `f(6)`
D. `f(7)`

Answer

To solve the problem, we need to find the value of \( p \) where \( p = f(-1) \) for the quadratic function \( f(x) = ax^2 + bx + c \) with the vertex (lowest point) at the coordinates \( (3, 2) \).

### Step-by-Step Solution:

1. **Identify the vertex form of the quadratic function**:
Since the vertex of the parabola is given as \( (3, 2) \), we can express the function in vertex form:
\[ f(x) = a(x - 3)^2 + 2 \]
Here, \( a > 0 \) ensures that the parabola opens upwards.

2. **Calculate \( f(-1) \)**:
We need to find \( f(-1) \):
\[ f(-1) = a(-1 - 3)^2 + 2 \]
Simplifying this:
\[ f(-1) = a(-4)^2 + 2 = 16a + 2 \]

3. **Use symmetry of the parabola**:
The vertex \( (3, 2) \) implies that the parabola is symmetric about the line \( x = 3 \). Therefore, the point \( (-1, f(-1)) \) is symmetric to the point \( (7, f(7)) \) across the line \( x = 3 \). Thus, we have:
\[ f(-1) = f(7) \]

4. **Calculate \( f(7) \)**:
Now, we can also find \( f(7) \):
\[ f(7) = a(7 - 3)^2 + 2 \]
Simplifying this:
\[ f(7) = a(4)^2 + 2 = 16a + 2 \]

5. **Conclusion**:
Since \( f(-1) = f(7) \), we have:
\[ p = f(-1) = 16a + 2 \]

### Final Answer:
The value of \( p \) can be represented as:
\[ p = 16a + 2 \]