`f (x) =ax ^(2)+3x+5`
The function f is defined above, and `f (3) =-4.` If a is a constant, what is the value of `f (2)` ?

Options

A. `-2`
B. 3
C. 5
D. 19

Explanation

To solve the problem step by step, we will start with the function given and the condition provided.

### Step 1: Write down the function and the condition
The function is given as:
\[ f(x) = ax^2 + 3x + 5 \]
We are also given that:
\[ f(3) = -4 \]

### Step 2: Substitute \( x = 3 \) into the function
We substitute \( x = 3 \) into the function to find the value of \( a \):
\[ f(3) = a(3^2) + 3(3) + 5 \]
This simplifies to:
\[ f(3) = a(9) + 9 + 5 \]
So we have:
\[ f(3) = 9a + 14 \]

### Step 3: Set the equation equal to -4
Since we know that \( f(3) = -4 \), we can set up the equation:
\[ 9a + 14 = -4 \]

### Step 4: Solve for \( a \)
Now, we will solve for \( a \):
1. Subtract 14 from both sides:
\[ 9a = -4 - 14 \]
\[ 9a = -18 \]
2. Divide both sides by 9:
\[ a = \frac{-18}{9} \]
\[ a = -2 \]

### Step 5: Substitute \( a \) back into the function
Now that we have the value of \( a \), we can substitute it back into the function:
\[ f(x) = -2x^2 + 3x + 5 \]

### Step 6: Find \( f(2) \)
Now we need to find \( f(2) \):
\[ f(2) = -2(2^2) + 3(2) + 5 \]
This simplifies to:
\[ f(2) = -2(4) + 6 + 5 \]
\[ f(2) = -8 + 6 + 5 \]
\[ f(2) = -8 + 11 \]
\[ f(2) = 3 \]

### Final Answer
Thus, the value of \( f(2) \) is:
\[ \boxed{3} \]