To graph the line with the equation [tex]\( y = 2x - 2 \)[/tex], follow these steps:
1. Understand the Equation:
- The equation [tex]\( y = 2x - 2 \)[/tex] is in slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, the slope [tex]\( m = 2 \)[/tex] and the y-intercept [tex]\( b = -2 \)[/tex].
2. Determine Key Points:
- By manipulating [tex]\( x \)[/tex] values, we can calculate corresponding [tex]\( y \)[/tex] values to determine points on the line.
3. Generate Points:
- Let’s generate 100 points from [tex]\( x = -10 \)[/tex] to [tex]\( x = 10 \)[/tex] and compute the corresponding [tex]\( y \)[/tex] values using the equation [tex]\( y = 2x - 2 \)[/tex].
4. Plot Points:
- You can now plot the points on the graph. Below is a table of some selected points from the list of 100 points:
| [tex]\( x \)[/tex] | [tex]\( y \)[/tex] |
|---------------|----------------|
| -10 | -22 |
| -9.8 | -21.6 |
| -9.6 | -21.2 |
| -9.4 | -20.8 |
| -9.2 | -20.4 |
| -9.0 | -20 |
| ... | ... |
| -0.2 | -2.4 |
| 0 | -2 |
| 0.2 | -1.6 |
| ... | ... |
| 9.6 | 17.2 |
| 9.8 | 17.6 |
| 10 | 18 |
Here are specific points to get you started:
- When [tex]\( x = -10 \)[/tex], [tex]\( y = 2(-10) - 2 = -22 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2(0) - 2 = -2 \)[/tex]
- When [tex]\( x = 10 \)[/tex], [tex]\( y = 2(10) - 2 = 18 \)[/tex]
5. Sketch the Line:
- Start by plotting the y-intercept [tex]\((0, -2)\)[/tex].
- Next, plot a few more points, such as [tex]\((-10, -22)\)[/tex] and [tex]\((10, 18)\)[/tex].
- Use a ruler to draw a straight line through these points.
6. Graph Features:
- Note that the line crosses the y-axis at -2.
- The positive slope [tex]\(m = 2\)[/tex] means the line rises to the right. For every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
By following these steps, you can successfully plot the line defined by the equation [tex]\( y = 2x - 2 \)[/tex] on a graph.
