To graph the linear equation \( y = 3x + 1 \), follow these steps:
Step 1: Identify the slope and y-intercept
The equation is in the slope-intercept form \( y = mx + b \), where:
- \( m \) is the slope
- \( b \) is the y-intercept
For the equation \( y = 3x + 1 \):
- The slope \( m = 3 \) (this means for every 1 unit you move to the right on the x-axis, you move 3 units up on the y-axis).
- The y-intercept \( b = 1 \) (this means the line crosses the y-axis at the point (0, 1)).
Step 2: Plot the y-intercept
Start by plotting the y-intercept (0, 1) on the graph.
Step 3: Use the slope to find another point
From the y-intercept, use the slope to find another point:
- Starting at (0, 1), move 1 unit to the right (increase x by 1), and then move 3 units up (increase y by 3).
- This gives you the point (1, 4).
Step 4: Plot the second point
Plot the point (1, 4) on the graph.
Step 5: Draw the line
Draw a straight line through the points (0, 1) and (1, 4). Extend the line in both directions.
Step 6: Add arrows
Add arrows at both ends of the line to indicate that it continues indefinitely.
Optional: Create a table of values
If you want to verify your graph or add more points, you can create a table of values by choosing different values of \( x \) and calculating \( y \):
| \( x \) | \( y = 3x + 1 \) | |---------|------------------| | -1 | -2 | | 0 | 1 | | 1 | 4 | | 2 | 7 |
Plot these points as well, ensuring they align with the line you’ve drawn.
Conclusion
You should now have a graph of the equation \( y = 3x + 1 \), demonstrating a straight line with a slope of 3 and a y-intercept at (0, 1).
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