To graph the line given by the equation [tex]\( y = 2x - 6 \)[/tex], follow these steps:
1. Identify the Equation and Components:
- The given equation is [tex]\( y = 2x - 6 \)[/tex].
- The equation is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope (rate of change) of the line.
- [tex]\( b \)[/tex] is the y-intercept (the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]).
2. Determine the Slope and Y-Intercept:
- [tex]\( m = 2 \)[/tex] (slope), which means that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
- [tex]\( b = -6 \)[/tex] (y-intercept), which means the line crosses the y-axis at [tex]\( y = -6 \)[/tex].
3. Plot the Y-Intercept:
- Locate the point [tex]\( (0, -6) \)[/tex] on the graph and put a point there. This is the y-intercept.
4. Use the Slope to Find Another Point:
- Starting from the y-intercept [tex]\( (0, -6) \)[/tex], use the slope [tex]\( m = 2 \)[/tex] to find another point.
- Since the slope is [tex]\( 2 \)[/tex], move up 2 units in the y-direction and 1 unit in the x-direction.
- Doing this, from [tex]\( (0, -6) \)[/tex], we move to the point [tex]\( (1, -4) \)[/tex].
5. Plot the Second Point:
- Locate the point [tex]\( (1, -4) \)[/tex] on the graph and put a point there.
6. Draw the Line:
- Draw a straight line passing through the points [tex]\( (0, -6) \)[/tex] and [tex]\( (1, -4) \)[/tex].
- Extend this line in both directions and ensure it covers the entire graph.
7. Verify with More Points (Optional):
- You can verify the accuracy by choosing another [tex]\( x \)[/tex]-value, say [tex]\( x = 2 \)[/tex]:
- Plug [tex]\( x = 2 \)[/tex] into the equation [tex]\( y = 2(2) - 6 \)[/tex], which gives [tex]\( y = 4 - 6 = -2 \)[/tex].
- Plot the point [tex]\( (2, -2) \)[/tex] to ensure it lies on the line.
- Similarly, verify with a negative [tex]\( x \)[/tex]-value, say [tex]\( x = -1 \)[/tex]:
- Plug [tex]\( x = -1 \)[/tex] into the equation [tex]\( y = 2(-1) - 6 \)[/tex], which gives [tex]\( y = -2 - 6 = -8 \)[/tex].
- Plot the point [tex]\( (-1, -8) \)[/tex] to ensure it lies on the line.
Your graph should look like this:
```
y
|
10 | .
9 | .
8 | .
7 | .
6 |
5 | .
4 | .
3 | .
2 | .
1 | .
0 o--------------------------------------| x
-1 | .
-2 | .
-3 | .
-4 o (1, -4)
-5 |
-6 o (0, -6)
-7 |
-8 o (-1, -8)
-9 |
-10 |
```
This shows the line [tex]\( y = 2x - 6 \)[/tex] graphed on the coordinate plane. The line crosses the y-axis at [tex]\( y = -6 \)[/tex] and rises with a slope of [tex]\( 2 \)[/tex].
