A linear function, f, has a slope of -2, f(1)=2 and f(2)=q. find q.

Options

A. 0
B. `(3)/(2)`
C. `5/2`
D. 3

Explanation

To solve the problem, we need to find the value of \( q \) for the linear function \( f \) given that it has a slope of -2, \( f(1) = 2 \), and we need to find \( f(2) = q \).

### Step-by-Step Solution:

1. **Write the equation of the linear function**:
The general form of a linear function is:
\[ f(x) = mx + c \]
where \( m \) is the slope and \( c \) is the y-intercept. Given that the slope \( m = -2 \), we can write:
\[ f(x) = -2x + c \]

2. **Use the given point to find \( c \)**:
We know that \( f(1) = 2 \). Substitute \( x = 1 \) into the function:
\[ f(1) = -2(1) + c = 2 \]
Simplifying this gives:
\[ -2 + c = 2 \]
To find \( c \), add 2 to both sides:
\[ c = 2 + 2 = 4 \]

3. **Write the complete function**:
Now that we have \( c \), we can write the complete function:
\[ f(x) = -2x + 4 \]

4. **Find \( f(2) \)**:
Now we need to find \( f(2) \) to determine \( q \):
\[ f(2) = -2(2) + 4 \]
Simplifying this gives:
\[ f(2) = -4 + 4 = 0 \]

5. **Conclusion**:
Therefore, \( q = f(2) = 0 \).

### Final Answer:
\[ q = 0 \]