The graph of the linear function f has intercepts at (c,0) and (0,d) in the xy-plane. If `2c= d and d ne 0,` which of the following in true about the slope of the graph of f ?

Options

A. It is positive.
B. It is negative.
C. It equals zero.
D. It is underfined.

Explanation

To solve the problem, we need to analyze the given information about the linear function \( f \) and its intercepts.

### Step-by-step Solution:

1. **Identify the Intercepts**:
The function \( f \) has intercepts at points \( (c, 0) \) and \( (0, d) \). Here, \( c \) is the x-intercept and \( d \) is the y-intercept.

2. **Use the Given Relationship**:
We are given that \( 2c = d \). This implies that the y-intercept \( d \) is twice the x-intercept \( c \).

3. **Determine the Signs of \( c \) and \( d \)**:
Since \( d \neq 0 \), we can analyze two cases:
- **Case 1**: If \( d > 0 \), then \( c \) must also be positive because \( d = 2c \) implies \( c = \frac{d}{2} > 0 \).
- **Case 2**: If \( d < 0 \), then \( c \) must also be negative because \( d = 2c \) implies \( c = \frac{d}{2} < 0 \).

4. **Graph the Function**:
- In **Case 1** (both \( c \) and \( d \) are positive), the line will be in the first quadrant.
- In **Case 2** (both \( c \) and \( d \) are negative), the line will be in the third quadrant.

5. **Calculate the Slope**:
The slope \( m \) of the line can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For the intercepts \( (c, 0) \) and \( (0, d) \):
\[ m = \frac{d - 0}{0 - c} = \frac{d}{-c} = -\frac{d}{c} \]

6. **Substituting \( d \)**:
From the relationship \( d = 2c \), we substitute \( d \) into the slope formula:
\[ m = -\frac{2c}{c} = -2 \]

7. **Conclusion**:
The slope of the graph of \( f \) is \( -2 \), which is negative.

### Final Answer:
The slope of the graph of \( f \) is negative.