Sure, let's graph the line [tex]\( y = -x - 4 \)[/tex] for the given range of [tex]\( x \)[/tex] values from -6 to 0. Here are the steps for constructing the graph:
1. Identify the Equation and Determine Key Points:
The equation provided is [tex]\( y = -x - 4 \)[/tex]. This is a linear equation, hence forms a straight line. To graph this, we need several pairs of [tex]\( (x, y) \)[/tex] points.
2. Calculate the [tex]\( y \)[/tex] Values for Given [tex]\( x \)[/tex] Range:
We need to compute [tex]\( y \)[/tex] for various [tex]\( x \)[/tex] values within the range [tex]\([-6, 0]\)[/tex]:
- For [tex]\( x = -6 \)[/tex]: [tex]\( y = -(-6) - 4 = 6 - 4 = 2 \)[/tex]
- For [tex]\( x = -5.94 \)[/tex]: [tex]\( y = -(-5.94) - 4 \approx 1.94 \)[/tex]
- For [tex]\( x = -5.88 \)[/tex]: [tex]\( y = -(-5.88) - 4 \approx 1.88 \)[/tex]
- Continue similar calculations for multiple [tex]\( x \)[/tex] values up to [tex]\( x = 0 \)[/tex].
3. List of Calculated Points:
Here’s a list of calculated points for convenience (we will use a subset of the values for clarity):
[tex]\(
\begin{align*}
& (-6, 2), \\
& (-5.94, 1.94), \\
& (-5.88, 1.88), \\
& \vdots \\
& (-1, -3), \\
& (-0.5, -3.5), \\
& (0, -4).
\end{align*}
\)[/tex]
4. Plot the Points on a Graph:
- Draw a coordinate grid with [tex]\( x \)[/tex]-axis (horizontal) ranging from -6 to 0 and [tex]\( y \)[/tex]-axis (vertical) from -4 to 2.
- Plot the points listed above:
- [tex]\( (-6, 2) \)[/tex]
- [tex]\( (-5.94, 1.94) \)[/tex]
- [tex]\( (-5.88, 1.88) \)[/tex]
- [tex]\( \vdots \)[/tex]
- [tex]\( (-1, -3) \)[/tex]
- [tex]\( (-0.5, -3.5) \)[/tex]
- [tex]\( (0, -4) \)[/tex]
5. Draw the Line:
- After all points are plotted, connect them with a straight line. Since this is a linear function, the points will form a straight line.
6. Label the Graph:
- Label the x-axis and y-axis appropriately.
- Add the equation of the line [tex]\( y = -x - 4 \)[/tex] as a title or label near the line.
Here’s how your line should appear when you plot these points and connect them:
```plaintext
y
|
2 | (-6, 2)
1 |
0 |
-1 |
-2 |
-3 |
-4 | (0, -4)
|
---------------------------------- x
-6 -5 -4 -3 -2 -1 0
```
Therefore, the graph of the equation [tex]\( y = -x - 4 \)[/tex] is a straight line that starts from point [tex]\((-6, 2)\)[/tex] and goes down to the point [tex]\((0, -4)\)[/tex].
