To solve the problem, we need to plot the graphs of the equations \(y = 2x - 1\), \(y = 2x\), and \(y = 2x + 1\) from \(x = -2\) to \(x = 4\) and determine if the lines are parallel. Here’s a step-by-step solution:
### Step 1: Create a Table of Values for Each Equation
We will calculate the corresponding \(y\) values for each equation at selected \(x\) values.
#### For \(y = 2x - 1\):
- When \(x = -2\):
\[
y = 2(-2) - 1 = -4 - 1 = -5
\]
- When \(x = -1\):
\[
y = 2(-1) - 1 = -2 - 1 = -3
\]
- When \(x = 0\):
\[
y = 2(0) - 1 = 0 - 1 = -1
\]
- When \(x = 1\):
\[
y = 2(1) - 1 = 2 - 1 = 1
\]
- When \(x = 2\):
\[
y = 2(2) - 1 = 4 - 1 = 3
\]
- When \(x = 3\):
\[
y = 2(3) - 1 = 6 - 1 = 5
\]
- When \(x = 4\):
\[
y = 2(4) - 1 = 8 - 1 = 7
\]
#### For \(y = 2x\):
- When \(x = -2\):
\[
y = 2(-2) = -4
\]
- When \(x = -1\):
\[
y = 2(-1) = -2
\]
- When \(x = 0\):
\[
y = 2(0) = 0
\]
- When \(x = 1\):
\[
y = 2(1) = 2
\]
- When \(x = 2\):
\[
y = 2(2) = 4
\]
- When \(x = 3\):
\[
y = 2(3) = 6
\]
- When \(x = 4\):
\[
y = 2(4) = 8
\]
#### For \(y = 2x + 1\):
- When \(x = -2\):
\[
y = 2(-2) + 1 = -4 + 1 = -3
\]
- When \(x = -1\):
\[
y = 2(-1) + 1 = -2 + 1 = -1
\]
- When \(x = 0\):
\[
y = 2(0) + 1 = 0 + 1 = 1
\]
- When \(x = 1\):
\[
y = 2(1) + 1 = 2 + 1 = 3
\]
- When \(x = 2\):
\[
y = 2(2) + 1 = 4 + 1 = 5
\]
- When \(x = 3\):
\[
y = 2(3) + 1 = 6 + 1 = 7
\]
- When \(x = 4\):
\[
y = 2(4) + 1 = 8 + 1 = 9
\]
### Step 2: Summary of Points
Now we summarize the points for each equation:
- For \(y = 2x - 1\):
\[
(-2, -5), (-1, -3), (0, -1), (1, 1), (2, 3), (3, 5), (4, 7)
\]
- For \(y = 2x\):
\[
(-2, -4), (-1, -2), (0, 0), (1, 2), (2, 4), (3, 6), (4, 8)
\]
- For \(y = 2x + 1\):
\[
(-2, -3), (-1, -1), (0, 1), (1, 3), (2, 5), (3, 7), (4, 9)
\]
### Step 3: Plot the Points on Graph Paper
Using the points calculated, plot the points on graph paper for each equation and draw the lines through the points.
### Step 4: Determine if the Lines are Parallel
To check if the lines are parallel, we observe the slopes of the lines:
- The slope of \(y = 2x - 1\) is \(2\).
- The slope of \(y = 2x\) is \(2\).
- The slope of \(y = 2x + 1\) is \(2\).
Since all three lines have the same slope, they are parallel to each other.
### Conclusion
Yes, the graphs of the lines \(y = 2x - 1\), \(y = 2x\), and \(y = 2x + 1\) are parallel to each other.
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