The graph of linear function k passes through the points (−7, 0)

To analyze the statements regarding the linear function \( k \) that passes through the points \( (-7, 0) \) and \( (1, 8) \), we'll follow these steps:

1. **Calculate the slope** of the line using the formula \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\) where \((x_1, y_1) = (-7, 0)\) and \((x_2, y_2) = (1, 8)\).

\[
\text{slope} = \frac{8 - 0}{1 - (-7)} = \frac{8}{1 + 7} = \frac{8}{8} = 1
\]

2. **Find the y-intercept** using the point-slope form or slope-intercept form \( y = mx + b \). Here, we can use one of the points, say \((-7, 0)\):

\[
0 = 1 \cdot (-7) + b \implies 0 = -7 + b \implies b = 7
\]

Hence, the equation of the line is:
\[
y = 1x + 7 \quad \text{or} \quad y = x + 7
\]

Now let's analyze the statements:

A. **The slope of the graph of k is −4 over 3**.
- This statement is **false** since we calculated the slope to be 1.

B. **The graph of k passes through the point (−1, −8)**.
- To check this, substitute \( x = -1 \) into the equation \( y = x + 7 \):
\[
y = -1 + 7 = 6 \quad \text{(not -8)}
\]
- So this statement is **false**.

C. **The zero of k is 7**.
- The zero of the function is the x-value where \( y = 0 \):
\[
0 = x + 7 \implies x = -7
\]
- Thus, this statement is **false**.

D. **The x-intercept of the graph of k is −7**.
- The x-intercept (where \( y = 0 \)) is already found to be \( x = -7 \) as shown above.
- So this statement is **true**.

Therefore, the correct answer is:

**D. The x-intercept of the graph of k is −7.**