Sure, we need to graph the line given by the equation [tex]\( y = -3x - 1 \)[/tex]. Here is a detailed, step-by-step explanation of how to do this:
### Step 1: Identify the slope and y-intercept
The equation [tex]\( y = -3x - 1 \)[/tex] is in slope-intercept form, [tex]\( y = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope,
- [tex]\( b \)[/tex] is the y-intercept.
For this equation:
- The slope [tex]\( m \)[/tex] is [tex]\(-3\)[/tex],
- The y-intercept [tex]\( b \)[/tex] is [tex]\(-1\)[/tex].
### Step 2: Plot the y-intercept
Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. For [tex]\( b = -1 \)[/tex], this point is [tex]\((0, -1)\)[/tex].
### Step 3: Use the slope to find another point
The slope tells us how to move from one point to another along the line. A slope of [tex]\(-3\)[/tex] means that for every 1 unit we move to the right (in the positive x-direction), we move 3 units down (in the negative y-direction).
Starting from the intercept [tex]\((0, -1)\)[/tex]:
- Move 1 unit to the right to [tex]\((1, -1)\)[/tex],
- Move 3 units down to [tex]\((1, -4)\)[/tex].
Now we have another point on the line: [tex]\((1, -4)\)[/tex].
### Step 4: Plot the second point
Plot the point [tex]\((1, -4)\)[/tex] on the graph.
### Step 5: Draw the line
Draw a straight line through the points [tex]\((0, -1)\)[/tex] and [tex]\((1, -4)\)[/tex]. This line should extend in both directions beyond these points.
### Step 6: Label the graph
Make sure to label the axes (x and y) and the equation of the line [tex]\( y = -3x - 1 \)[/tex] on the graph. You can also draw a small arrow on each end of the line to indicate that it extends infinitely.
### Example Graph
```
|
4 | .
3 |
2 |
1 |
|
-1.5 |
-2.5 +.-----------------+
| 0 0 1 2 3
|
```
On a coordinate system, with the x-axis and y-axis intersecting at (0,0), the line y = -3x - 1 crosses the y-axis at (0, -1) and extends in both directions.
By plotting these points and drawing the line through them, we have successfully graphed the equation [tex]\( y = -3x - 1 \)[/tex].
