Question

The table represents some points on the graph of linear function k.

x –3 2 5 11
k(x) –130 0 78 234

Which function represents k?
A.k(x)=−2(26x−1)

B.k(x)=−26(2x−1)

C.k(x)=13(x−2)

D.k(x)=26(x−2)

Answer 1

To determine which of the given functions represents \(k(x)\), we need to analyze the points provided in the table and see how they correspond to the linear functions in the options.

The points from the table are:

1. \( (-3, -130) \)
2. \( (2, 0) \)
3. \( (5, 78) \)
4. \( (11, 234) \)

### Step 1: Find the slope of the line.

The slope \( m \) of a linear function can be calculated using any two points from the table. Let's use the points \( (2, 0) \) and \( (5, 78) \):

\[
m = \frac{k(5) - k(2)}{5 - 2} = \frac{78 - 0}{5 - 2} = \frac{78}{3} = 26
\]

### Step 2: Use the slope-intercept form.

Now that we know the slope \( m = 26 \), we can use point-slope form to express the equation of the line. Let's use the point \( (2, 0) \):

\[
k(x) = m(x - 2) + k(2)
\]
Since \( k(2) = 0 \):

\[
k(x) = 26(x - 2)
\]

### Step 3: Verify against the options.

Looking at the provided function options, we see:

- **A.** \( k(x) = -2(26x - 1) \)
- **B.** \( k(x) = -26(2x - 1) \)
- **C.** \( k(x) = 13(x - 2) \)
- **D.** \( k(x) = 26(x - 2) \)

The derived equation \( k(x) = 26(x - 2) \) matches option **D**.

### Conclusion:

Thus, the function that represents \( k \) is **D: \( k(x) = 26(x - 2) \)**.