Certainly! Let's graph the line given by the equation [tex]\( y = -x - 3 \)[/tex].
Here is a step-by-step guide:
### Step 1: Understand the Equation
The equation of the line is given in the slope-intercept form [tex]\( y = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( b \)[/tex] is the y-intercept (the point where the line crosses the y-axis).
For the equation [tex]\( y = -x - 3 \)[/tex]:
- The slope [tex]\( m \)[/tex] is -1.
- The y-intercept [tex]\( b \)[/tex] is -3.
### Step 2: Plot the y-intercept
Find the y-intercept of the line on the graph. This is the point where the line crosses the y-axis (i.e., when [tex]\( x = 0 \)[/tex]).
For our equation:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -3 \)[/tex].
So, plot the point (0, -3) on the graph.
### Step 3: Use the Slope to Find Another Point
The slope of -1 means that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 1 unit.
Starting from the y-intercept (0, -3):
- Increase [tex]\( x \)[/tex] by 1 unit (thus, [tex]\( x = 1 \)[/tex]).
- Decrease [tex]\( y \)[/tex] by 1 unit (thus, [tex]\( y = -4 \)[/tex]).
So, the point (1, -4) is another point on the line. Plot this point on the graph.
### Step 4: Draw the Line
Now, draw a straight line through the points (0, -3) and (1, -4). Extend this line in both directions to cover the graph.
### Step 5: Label the Graph
Don't forget to label your axes and the line itself. Here is a summary of the points you have identified:
- y-intercept at (0, -3)
- Another point at (1, -4)
Your graph should now look like this:
[tex]\[
\begin{array}{c|cr}
x & y = -x - 3 \\ \hline
0 & -3 \\
1 & -4
\end{array}
\][/tex]
### Visual Representation:
```
y
^
|
|
4|
3|
2|
1|
0|-----------------------------> x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-1|
-2|
-3|(0, -3)
-4| (1, -4)
|
```
This is the graph of the line [tex]\( y = -x - 3 \)[/tex].
### Recap:
- The y-intercept is at [tex]\( (0, -3) \)[/tex].
- Using the slope of -1, the line passes through points such as [tex]\( (1, -4) \)[/tex].
- Draw a line through these points and extend it in both directions.
That's the complete graph for the equation [tex]\( y = -x - 3 \)[/tex].
