To graph the linear equation $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept, follow these steps:

1. Identify the Slope and Y-Intercept

• Slope (m): This represents the steepness and direction of the line. If $m$ is positive, the line slopes upward from left to right. If $m$ is negative, the line slopes downward from left to right.
• Y-Intercept (b): This is the point where the line crosses the y-axis. This will be the starting point for graphing the line.

2. Plot the Y-Intercept (b)

• Start by marking the point $(0, b)$ on the y-axis. This is your first point on the graph.

3. Use the Slope to Find Another Point

• The slope $m$ is often written as a fraction $\frac{rise}{run}$.
• From the y-intercept, move vertically (up or down) by the "rise" and horizontally (right or left) by the "run".
• For example, if $m = \frac{3}{2}$, go up 3 units and right 2 units from the y-intercept to find your second point.

4. Draw the Line

• Connect the y-intercept and the second point you found by drawing a straight line through them.
• Extend the line across the graph in both directions.

5. Add Arrowheads and Label the Line (Optional)

• Add arrowheads on both ends of the line to indicate it extends infinitely.
• Label the line with its equation $y = mx + b$ if needed.

Would you like more details on any of these steps, or have any other questions?

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Further Questions

1. How does changing the slope $m$ affect the steepness of the line?
2. What happens to the graph if the y-intercept $b$ is zero?
3. How would you graph a line if the slope $m$ is negative?
4. How does the slope-intercept form compare to the standard form of a linear equation?
5. How can you find the equation of a line given two points?

Tip

Remember: if $m = 0$, the line is horizontal. If $b = 0$, the line passes through the origin.