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We have two linear functions, Function \( J \) and Function \( K \), and we are asked to determine which statement about them is true. Let's analyze both functions step by step.

### Function \( J \):
Given the equation:
\[
y = \frac{1}{2}x - 4
\]
- The slope (rate of change) of Function \( J \) is \( \frac{1}{2} \).
- The y-intercept is \( -4 \).

### Function \( K \):
For this, we have the table of values:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 6 \\
-1 & -8 \\
1 & -2 \\
2 & 0 \\
\hline
\end{array}
\]

To find the rate of change (slope) of Function \( K \), we use the slope formula between any two points:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's use two points: \( (-1, -8) \) and \( (1, -2) \):

\[
\text{slope} = \frac{-2 - (-8)}{1 - (-1)} = \frac{-2 + 8}{1 + 1} = \frac{6}{2} = 3
\]

Thus, the slope of Function \( K \) is \( 3 \).

To find the y-intercept of Function \( K \), we use the slope-intercept form \( y = mx + b \). Since we know the slope \( m = 3 \), we can substitute one of the points, say \( (1, -2) \), into the equation:

\[
-2 = 3(1) + b \implies -2 = 3 + b \implies b = -5
\]

So, the y-intercept of Function \( K \) is \( -5 \).

### Comparing the functions:
- The rate of change for Function \( J \) is \( \frac{1}{2} \), and for Function \( K \), it is \( 3 \).
- The y-intercept for Function \( J \) is \( -4 \), and for Function \( K \), it is \( -5 \).

### Analyzing the statements:
1. **The rate of change for Function \( J \) is equal to the rate of change for Function \( K \).**  
   False: The rate of change for \( J \) is \( \frac{1}{2} \), and for \( K \) it is \( 3 \).
   
2. **The y-intercept of Function \( J \) is greater than the y-intercept of Function \( K \).**  
   True: The y-intercept of \( J \) is \( -4 \), which is greater than \( -5 \) for \( K \).
   
3. **The y-intercept of Function \( J \) is equal to the y-intercept of Function \( K \).**  
   False: The y-intercepts are different (\( -4 \) and \( -5 \)).
   
4. **The rate of change for Function \( J \) is greater than the rate of change for Function \( K \).**  
   False: The rate of change for \( J \) is \( \frac{1}{2} \), which is less than \( 3 \) for \( K \).

### Correct answer: 
The second statement is true: **The y-intercept of Function \( J \) is greater than the y-intercept of Function \( K \).**

---

Would you like more details on any step? Here are some related questions for practice:

1. How do you find the slope of a line from two points?
2. What is the importance of the y-intercept in a linear equation?
3. How would the graph of Function \( J \) compare to the graph of Function \( K \)?
4. Can the y-intercept be used to determine where a line crosses the y-axis?
5. What is the difference between positive and negative slope?

**Tip:** Always ensure to compare both the slope and the y-intercept when analyzing two linear functions for differences.

Robert has two linear functions as shown below. Function J: y = 1/2x - 4. Function K is given in a table with points (-2, 6), (-1, -8), (1, -2), (2, 0). Which statement about the functions is true? (1) The rate of change for Function J is equal to the rate of change for Function K. (2) The y-intercept of Function J is greater than the y-intercept of Function K. (3) The y-intercept of Function J is equal to the y-intercept of Function K. (4) The rate of change for Function J is greater than the rate of change for Function K.

Learn how to compare linear functions by analyzing slope and y-intercept. Function J is represented as y = 1/2x - 4, while Function K is given by a set of points. This exercise teaches students to find slope and y-intercepts to compare the functions.

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