To plot the graph of the function \( y = -3x - 1 \) from \( x = -2 \) to \( x = 2 \), let's follow a step-by-step approach to understand how the graph would look.
### Step-by-Step Solution:
1. Define the function:
The given function is:
[tex]\[
y = -3x - 1
\][/tex]
2. Determine the range of \( x \):
We need to plot the graph for \( x \) values ranging from \( -2 \) to \( 2 \).
3. Calculate corresponding \( y \)-values for specific \( x \)-values:
Here, we will calculate \( y \)-values for some key \( x \)-values within the given range.
- For \( x = -2 \):
[tex]\[
y = -3(-2) - 1 = 6 - 1 = 5
\][/tex]
- For \( x = -1 \):
[tex]\[
y = -3(-1) - 1 = 3 - 1 = 2
\][/tex]
- For \( x = 0 \):
[tex]\[
y = -3(0) - 1 = 0 - 1 = -1
\][/tex]
- For \( x = 1 \):
[tex]\[
y = -3(1) - 1 = -3 - 1 = -4
\][/tex]
- For \( x = 2 \):
[tex]\[
y = -3(2) - 1 = -6 - 1 = -7
\][/tex]
4. Plot the points:
We have the following points \((x, y)\):
[tex]\[
(-2, 5), (-1, 2), (0, -1), (1, -4), (2, -7)
\][/tex]
5. Draw the straight line:
Since the equation \( y = -3x - 1 \) is a linear equation, which means the graph will be a straight line passing through the points we have calculated above.
6. Graphical representation:
- Label the x-axis \( x \) and the y-axis \( y \).
- Plot the calculated points on the coordinate plane.
- Draw a straight line passing through these points to represent the equation.
Here is a rough sketch of how the graph should look:
```
7 | .
6 | .
5 | .
4 |
3 |
2 | .
1 |
0 | .
-1 | .
-2 |
-3 |
-4 | .
-5 |
-6 |
-7 | .
-8 |_________________________
-2 -1 0 1 2
```
Here's a summary of the coordinates plotted:
- \((-2, 5)\)
- \((-1, 2)\)
- \((0, -1)\)
- \((1, -4)\)
- \((2, -7)\)
Connect these points with a straight line, and you will have the graph of the function [tex]\( y = -3x - 1 \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 2 \)[/tex].
