To graph the line given by the equation [tex]\( y = -\frac{2}{3}x + 1 \)[/tex], follow these steps:
1. Identify the slope and y-intercept:
- The equation is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, the slope [tex]\( m \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex], and the y-intercept [tex]\( b \)[/tex] is 1.
2. Plot the y-intercept:
- Find the y-intercept on the graph, which is the point where the line crosses the y-axis.
- The y-intercept is at [tex]\( (0, 1) \)[/tex]. Plot the point [tex]\( (0, 1) \)[/tex] on the graph.
3. Use the slope to find another point:
- The slope [tex]\( -\frac{2}{3} \)[/tex] indicates that for every 3 units you move to the right along the x-axis, the line will move down 2 units along the y-axis (since the slope is negative).
- Starting from the y-intercept [tex]\( (0, 1) \)[/tex], move 3 units to the right (positive x direction) to the point [tex]\( (3, 1) \)[/tex].
- From [tex]\( (3, 1) \)[/tex], move down 2 units (negative y direction) to the point [tex]\( (3, -1) \)[/tex]. Plot the point [tex]\( (3, -1) \)[/tex].
4. Draw the line:
- Draw a straight line passing through the points [tex]\( (0, 1) \)[/tex] and [tex]\( (3, -1) \)[/tex].
- Extend the line in both directions, ensuring it passes through both points and continues infinitely in both directions.
By following these steps, you will have successfully graphed the line representing the equation [tex]\( y = -\frac{2}{3}x + 1 \)[/tex].
Here is how it would look on the graph:
```
y
|
4 | .
3 | .
2 |.
1 | . (0,1)
0 |----------------------------
-4 -3 -2 -1 0 1 2 3 4 x
-1 | (3,-1)
-2 |
-3 |
-4 |
|
```
The asterisks (*) represent the points [tex]\( (0, 1) \)[/tex] and [tex]\( (3, -1) \)[/tex] that you plotted, and the line should pass through these points.
