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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.

Compare the graphs of the following functions and determine their rate of change: f(x) = 2x - 3, g(x) = x + 3, h(x) = 3x + 1, j(x) = 1/2x - 4.

This exercise focuses on graphing four linear functions: f(x) = 2x - 3, g(x) = x + 3, h(x) = 3x + 1, and j(x) = 1/2x - 4. Learn how to plot these functions and compare their slopes to understand their rates of change. Suitable for Grades 8-10.

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