Answer :
To graph the line with the equation [tex]\( y = -\frac{2}{3}x + 4 \)[/tex], we need to follow these steps:
### Step 1: Identify the slope and y-intercept
- Slope (m): The slope of the line is [tex]\(-\frac{2}{3}\)[/tex]. This means the line decreases by 2 units in the y-direction for every 3 units it moves in the x-direction.
- y-intercept (b): The y-intercept is 4. This is where the line crosses the y-axis, at the point (0, 4).
### Step 2: Plot the y-intercept
Start by plotting the y-intercept on the graph. The y-intercept is at the point (0, 4).
### Step 3: Use the slope to find other points
Starting from the y-intercept (0, 4), use the slope to determine additional points on the line:
- If you move 3 units to the right on the x-axis, move 2 units down on the y-axis (because the slope is [tex]\(-\frac{2}{3}\)[/tex]).
- If you move 3 units to the left on the x-axis, move 2 units up on the y-axis.
Here are some specific points calculated by applying the slope from the y-intercept:
1. Moving 3 units to the right and 2 units down:
[tex]\[ (0 + 3, 4 - 2) = (3, 2) \][/tex]
2. Moving another 3 units to the right and another 2 units down:
[tex]\[ (3 + 3, 2 - 2) = (6, 0) \][/tex]
3. Moving 3 units to the left and 2 units up:
[tex]\[ (0 - 3, 4 + 2) = (-3, 6) \][/tex]
4. Moving another 3 units to the left and another 2 units up:
[tex]\[ (-3 - 3, 6 + 2) = (-6, 8) \][/tex]
### Step 4: Plot the calculated points
Now we plot all the points calculated:
- (0, 4) — the y-intercept
- (3, 2)
- (6, 0)
- (-3, 6)
- (-6, 8)
### Step 5: Draw the line
Finally, draw a straight line through all the plotted points. Extend the line in both directions.
### Additional Points for Accuracy
To enhance the accuracy of our graph, consider additional points:
For [tex]\( x = -10 \)[/tex]:
[tex]\[ y = -\frac{2}{3}(-10) + 4 = 10.666666666666666 \][/tex]
For [tex]\( x = -9 \)[/tex]:
[tex]\[ y = -\frac{2}{3}(-9) + 4 = 10.0 \][/tex]
Continue similarly for all values of [tex]\( x \)[/tex] from -10 to 10:
- [tex]\( x = -8: y = 9.333333333333332 \)[/tex]
- [tex]\( x = -7: y = 8.666666666666666 \)[/tex]
- [tex]\( x = -6: y = 8.0 \)[/tex]
- [tex]\( x = -5: y = 7.333333333333333 \)[/tex]
- [tex]\( x = -4: y = 6.666666666666666 \)[/tex]
- [tex]\( x = -3: y = 6.0 \)[/tex]
- [tex]\( x = -2: y = 5.333333333333333 \)[/tex]
- [tex]\( x = -1: y = 4.666666666666667 \)[/tex]
- [tex]\( x = 0: y = 4.0 \)[/tex]
- [tex]\( x = 1: y = 3.3333333333333335 \)[/tex]
- [tex]\( x = 2: y = 2.666666666666667 \)[/tex]
- [tex]\( x = 3: y = 2.0 \)[/tex]
- [tex]\( x = 4: y = 1.3333333333333335 \)[/tex]
- [tex]\( x = 5: y = 0.666666666666667 \)[/tex]
- [tex]\( x = 6: y = 0.0 \)[/tex]
- [tex]\( x = 7: y = -0.6666666666666661 \)[/tex]
- [tex]\( x = 8: y = -1.333333333333333 \)[/tex]
- [tex]\( x = 9: y = -2.0 \)[/tex]
- [tex]\( x = 10: y = -2.666666666666666 \)[/tex]
By verifying and plotting these points, we can see the true path of the line.
And that's how you graph the line [tex]\( y = -\frac{2}{3}x + 4 \)[/tex].
### Step 1: Identify the slope and y-intercept
- Slope (m): The slope of the line is [tex]\(-\frac{2}{3}\)[/tex]. This means the line decreases by 2 units in the y-direction for every 3 units it moves in the x-direction.
- y-intercept (b): The y-intercept is 4. This is where the line crosses the y-axis, at the point (0, 4).
### Step 2: Plot the y-intercept
Start by plotting the y-intercept on the graph. The y-intercept is at the point (0, 4).
### Step 3: Use the slope to find other points
Starting from the y-intercept (0, 4), use the slope to determine additional points on the line:
- If you move 3 units to the right on the x-axis, move 2 units down on the y-axis (because the slope is [tex]\(-\frac{2}{3}\)[/tex]).
- If you move 3 units to the left on the x-axis, move 2 units up on the y-axis.
Here are some specific points calculated by applying the slope from the y-intercept:
1. Moving 3 units to the right and 2 units down:
[tex]\[ (0 + 3, 4 - 2) = (3, 2) \][/tex]
2. Moving another 3 units to the right and another 2 units down:
[tex]\[ (3 + 3, 2 - 2) = (6, 0) \][/tex]
3. Moving 3 units to the left and 2 units up:
[tex]\[ (0 - 3, 4 + 2) = (-3, 6) \][/tex]
4. Moving another 3 units to the left and another 2 units up:
[tex]\[ (-3 - 3, 6 + 2) = (-6, 8) \][/tex]
### Step 4: Plot the calculated points
Now we plot all the points calculated:
- (0, 4) — the y-intercept
- (3, 2)
- (6, 0)
- (-3, 6)
- (-6, 8)
### Step 5: Draw the line
Finally, draw a straight line through all the plotted points. Extend the line in both directions.
### Additional Points for Accuracy
To enhance the accuracy of our graph, consider additional points:
For [tex]\( x = -10 \)[/tex]:
[tex]\[ y = -\frac{2}{3}(-10) + 4 = 10.666666666666666 \][/tex]
For [tex]\( x = -9 \)[/tex]:
[tex]\[ y = -\frac{2}{3}(-9) + 4 = 10.0 \][/tex]
Continue similarly for all values of [tex]\( x \)[/tex] from -10 to 10:
- [tex]\( x = -8: y = 9.333333333333332 \)[/tex]
- [tex]\( x = -7: y = 8.666666666666666 \)[/tex]
- [tex]\( x = -6: y = 8.0 \)[/tex]
- [tex]\( x = -5: y = 7.333333333333333 \)[/tex]
- [tex]\( x = -4: y = 6.666666666666666 \)[/tex]
- [tex]\( x = -3: y = 6.0 \)[/tex]
- [tex]\( x = -2: y = 5.333333333333333 \)[/tex]
- [tex]\( x = -1: y = 4.666666666666667 \)[/tex]
- [tex]\( x = 0: y = 4.0 \)[/tex]
- [tex]\( x = 1: y = 3.3333333333333335 \)[/tex]
- [tex]\( x = 2: y = 2.666666666666667 \)[/tex]
- [tex]\( x = 3: y = 2.0 \)[/tex]
- [tex]\( x = 4: y = 1.3333333333333335 \)[/tex]
- [tex]\( x = 5: y = 0.666666666666667 \)[/tex]
- [tex]\( x = 6: y = 0.0 \)[/tex]
- [tex]\( x = 7: y = -0.6666666666666661 \)[/tex]
- [tex]\( x = 8: y = -1.333333333333333 \)[/tex]
- [tex]\( x = 9: y = -2.0 \)[/tex]
- [tex]\( x = 10: y = -2.666666666666666 \)[/tex]
By verifying and plotting these points, we can see the true path of the line.
And that's how you graph the line [tex]\( y = -\frac{2}{3}x + 4 \)[/tex].
