The table provides several points on the graph of the linear function $k(x)$. The points given are:

• $(-3, -130)$
• $(2, 0)$
• $(5, 78)$
• $(11, 234)$

To determine which function represents $k(x)$, we need to calculate the slope and find the equation of the line using the points provided.

Step 1: Find the slope of the line

We can use the slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's use the points $(2, 0)$ and $(5, 78)$:

$m = \frac{78 - 0}{5 - 2} = \frac{78}{3} = 26$

So the slope $m = 26$.

Step 2: Find the equation of the line

Using the slope-point form of the equation of a line, $y - y_1 = m(x - x_1)$, and the point $(2, 0)$:

$y - 0 = 26(x - 2)$

$k(x) = 26(x - 2)$

This matches option D.

Thus, the correct function is:

$\boxed{D. \, k(x) = 26(x - 2)}$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How do you verify that this equation satisfies all the points in the table?
2. How would the equation change if the slope was negative?
3. What is the significance of the slope in a linear equation?
4. How can you find the y-intercept from the equation of a line?
5. Can you rewrite the equation in slope-intercept form?

Tip: In a linear equation, the slope indicates how much $y$ changes for each unit change in $x$.