Certainly! To graph the line represented by the equation [tex]\( y = -\frac{2}{3} x + 1 \)[/tex], follow these steps:
### Step 1: Understand the Equation
This is a linear equation in 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]\( y = -\frac{2}{3} x + 1 \)[/tex]:
- The slope ([tex]\( m \)[/tex]) is [tex]\( -\frac{2}{3} \)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is 1.
### Step 2: Plot the Y-Intercept
Start by plotting the y-intercept on the graph.
- The y-intercept is 1, so plot the point (0, 1) on the coordinate plane.
### Step 3: Use the Slope to Find Another Point
The slope [tex]\( -\frac{2}{3} \)[/tex] means that for every 3 units you move to the right (positive direction on the x-axis), you move 2 units down (negative direction on the y-axis).
From the point (0, 1):
- Move 3 units to the right to (3, 1).
- From (3, 1), move 2 units down to (3, -1).
This gives you another point (3, -1) on the line. Plot this point on the graph as well.
### Step 4: Draw the Line
Using a ruler, draw a straight line through the points (0, 1) and (3, -1). Extend the line further in both directions, and add arrows at both ends to indicate that it continues infinitely.
### Step 5: Verify Additional Points (Optional)
You can verify by finding more points on the line if needed. For instance, using the same slope:
- From (0, 1), you can also move 3 units to the left (negative direction on the x-axis) and 2 units up (positive direction on the y-axis).
From (0, 1):
- Move 3 units to the left to (-3, 1).
- From (-3, 1), move 2 units up to (-3, 3).
Plot the point (-3, 3) and ensure your drawn line passes through it as well for consistency.
### Graph Summary
- The line passes through (0, 1) and (3, -1).
- It extends infinitely in both directions.
- The slope is -2/3, indicating a downward slant from left to right.
- The y-intercept is 1, the point where the line crosses the y-axis.
Your final graph should look something like this:
```
y
↑
2 | (-3, 3)
1 | (0, 1)
0 |------------------------------------> x
-1 | (3, -1)
-2 |
```
The line through these points illustrates the equation [tex]\( y = -\frac{2}{3} x + 1 \)[/tex].
