`g(x)=ax^(2)+24`
For the function g defined above, a is a constant and `g(4) = 8`. What is the value of `g(−4)` ?

Options

A. 8
B. 0
C. `-1`
D. `-8`

Answer

To solve the problem step by step, we will follow these steps:

1. **Write down the function and the given condition**:
The function is given as \( g(x) = ax^2 + 24 \) and we know that \( g(4) = 8 \).

2. **Substitute \( x = 4 \) into the function**:
We will substitute \( x = 4 \) into the function to find the value of \( a \):
\[ g(4) = a(4^2) + 24 = 8 \]
This simplifies to:
\[ g(4) = 16a + 24 = 8 \]

3. **Solve for \( a \)**:
Now, we will isolate \( a \):
\[ 16a + 24 = 8 \]
Subtract 24 from both sides:
\[ 16a = 8 - 24 \]
\[ 16a = -16 \]
Now, divide both sides by 16:
\[ a = -1 \]

4. **Substitute \( a \) back into the function**:
Now that we have \( a = -1 \), we can rewrite the function:
\[ g(x) = -1 \cdot x^2 + 24 \]

5. **Find \( g(-4) \)**:
Now we need to find \( g(-4) \):
\[ g(-4) = -1 \cdot (-4)^2 + 24 \]
This simplifies to:
\[ g(-4) = -1 \cdot 16 + 24 \]
\[ g(-4) = -16 + 24 \]
\[ g(-4) = 8 \]

6. **Conclusion**:
The value of \( g(-4) \) is \( 8 \).