The image contains a set of functions and asks to graph them to compare their rates of change. Let's break it down step by step.

Functions to graph:

1. $f(x) = 2x - 3$
2. $g(x) = x + 3$
3. $h(x) = 3x + 1$
4. $j(x) = \frac{1}{2}x - 4$

Steps to solve:

1. Graphing each function:

   • For each linear function, recall that the general form is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

2. Determine the slope (rate of change) for each function:

   • $f(x) = 2x - 3$: slope $m = 2$
   • $g(x) = x + 3$: slope $m = 1$
   • $h(x) = 3x + 1$: slope $m = 3$
   • $j(x) = \frac{1}{2}x - 4$: slope $m = \frac{1}{2}$

3. Plot points using the y-intercept and slope:

   • For each equation, the y-intercept is the constant term, and you can use the slope to plot the next point.

Step-by-step details for plotting:

1. $f(x) = 2x - 3$

• Slope $m = 2$, y-intercept $-3$.
• Plot: (0, -3), then move up 2 units and right 1 unit to find the next point.

2. $g(x) = x + 3$

• Slope $m = 1$, y-intercept $3$.
• Plot: (0, 3), then move up 1 unit and right 1 unit.

3. $h(x) = 3x + 1$

• Slope $m = 3$, y-intercept $1$.
• Plot: (0, 1), then move up 3 units and right 1 unit.

4. $j(x) = \frac{1}{2}x - 4$

• Slope $m = \frac{1}{2}$, y-intercept $-4$.
• Plot: (0, -4), then move up 1 unit and right 2 units.

Comparing rates of change:

• The function with the greatest slope will have the steepest line.
  • $h(x)$ has the steepest slope ($m = 3$).
  • $j(x)$ has the least steep slope ($m = \frac{1}{2}$).

Would you like more details on graphing or solving these equations? Here are some related questions to extend your understanding:

1. What happens to the graph of a line when the slope is negative?
2. How do changes in the y-intercept affect the position of the graph?
3. What would the graph look like if the slope was zero?
4. How can you determine the slope from a graph directly?
5. What is the difference between the steepness of the graphs based on their slopes?

Tip: When graphing a line, always start from the y-intercept and use the slope to find additional points.