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The equation you provided is \( y = 3x + 8 \), which is in slope-intercept form: \( y = mx + b \), where:

- \( m = 3 \) is the slope,
- \( b = 8 \) is the y-intercept (where the line crosses the y-axis).

### Steps to graph by hand:
1. **Start with the y-intercept**:
   - Plot the point \( (0, 8) \) on the y-axis because when \( x = 0 \), \( y = 8 \).

2. **Use the slope to find another point**:
   - The slope is 3, which means "rise over run." A slope of 3 can be written as \( \frac{3}{1} \), meaning for every 1 unit you move to the right (positive direction on the x-axis), move 3 units up (positive direction on the y-axis).
   - Starting from \( (0, 8) \), move 1 unit to the right (to \( x = 1 \)) and 3 units up (to \( y = 11 \)). Plot this point: \( (1, 11) \).

3. **Draw the line**:
   - Draw a straight line through the points \( (0, 8) \) and \( (1, 11) \).
   - You can also extend the line to cover more values by plotting additional points using the slope.

### Verifying using a graphing utility:
You can input the equation \( y = 3x + 8 \) into any graphing calculator or online graphing tool (such as Desmos) to verify that the line you plotted matches the graph generated by the utility.

Would you like a more detailed breakdown or any clarifications?

Here are 5 related questions for practice:
1. What would the graph look like if the slope were negative?
2. How would you graph \( y = 3x + 8 \) if you were given only the slope and a point not on the y-axis?
3. How do you find the slope if you are given two points?
4. How does changing the value of \( b \) affect the graph of a linear equation?
5. What is the equation of a line parallel to \( y = 3x + 8 \)?

**Tip**: The y-intercept helps you quickly plot the starting point of the line on the graph, and the slope tells you how to move from that point.

Graph the equation y = 3x + 8 by hand by plotting points. Verify your results using a graphing utility.

Learn how to graph the linear equation y = 3x + 8 by plotting points and using the slope-intercept form. This lesson covers the key concepts of graphing linear equations, including identifying the slope and y-intercept, and is suitable for students in Grades 7-10.

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