To draw the graph of the equation $y = 2x - 6$ for values of $x$ from $-2$ to $5$, we can follow these steps:

Step-by-Step Solution

1. Identify the Equation's Type:

   • The equation $y = 2x - 6$ is a linear equation in the slope-intercept form $y = mx + b$, where:
     • $m = 2$ (slope)
     • $b = -6$ (y-intercept)

2. Create a Table of Values:

   • Substitute values of $x$ from $-2$ to $5$ to find the corresponding $y$ values.

3. Plot the Points on the Grid:

   • For each pair $(x, y)$, plot the points $(-2, -10), (-1, -8), (0, -6), (1, -4), (2, -2), (3, 0), (4, 2), (5, 4)$.

4. Draw the Line:

   • Connect these points to form a straight line that represents $y = 2x - 6$.

Explanation of the Line

• Since the slope $m = 2$, the line rises by 2 units for every increase of 1 unit in $x$.
• The y-intercept $b = -6$ indicates that the line crosses the y-axis at $(0, -6)$.

Would you like me to go through any additional details, or do you have any questions?

Here are some related questions to extend your understanding:

1. How would the graph change if the slope were negative, say $y = -2x - 6$?
2. What would happen to the line if we changed the y-intercept to a positive number, like $y = 2x + 3$?
3. How can you determine the x-intercept from the equation $y = 2x - 6$?
4. How do you find the slope of a line if only given two points on the line?
5. How can you determine if two lines are parallel or perpendicular just by looking at their equations?

Tip: Remember, in linear equations, the slope $m$ controls the angle of the line, and the y-intercept $b$ determines where it crosses the y-axis.