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To graph the function [tex]\( f(x) = -\frac{3}{5}x + 2 \)[/tex], follow these steps:
1. Identify the type of function:
The given function is a linear equation in the form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Determine the slope and y-intercept:
- Slope ([tex]\( m \)[/tex]) = [tex]\(-\frac{3}{5}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]) = 2
3. Plot the y-intercept:
The y-intercept is the point where the graph crosses the y-axis. For this function, it is the point [tex]\( (0, 2) \)[/tex]. Plot this point on the graph.
4. Use the slope to find another point:
The slope [tex]\(-\frac{3}{5}\)[/tex] means that for every increase of 5 units in the x-direction, the function decreases by 3 units in the y-direction.
Starting from the y-intercept [tex]\( (0, 2) \)[/tex]:
- Move 5 units to the right (positive x-direction) and then 3 units down (negative y-direction).
- This gives the point [tex]\( (5, 2 - 3) = (5, -1) \)[/tex].
5. Plot the second point:
Plot the point [tex]\( (5, -1) \)[/tex] on the graph.
6. Draw the line:
Draw a straight line through the points [tex]\( (0, 2) \)[/tex] and [tex]\( (5, -1) \)[/tex]. This line represents the function [tex]\( f(x) = -\frac{3}{5}x + 2 \)[/tex].
7. Check additional points (optional, for accuracy):
If desired, you can calculate and plot additional points:
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ f(-5) = -\frac{3}{5}(-5) + 2 = 3 + 2 = 5 \][/tex]
Point: [tex]\( (-5, 5) \)[/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[ f(10) = -\frac{3}{5}(10) + 2 = -6 + 2 = -4 \][/tex]
Point: [tex]\( (10, -4) \)[/tex]
Plot these additional points to ensure the line is correctly drawn.
### Summary:
For further clarity, here is the visual representation:
1. Y-intercept point: [tex]\( (0, 2) \)[/tex]
2. Another calculated point using the slope: [tex]\( (5, -1) \)[/tex]
3. Optional additional points: [tex]\( (-5, 5) \)[/tex] and [tex]\( (10, -4) \)[/tex]
Finally, draw a straight line through these points on graph paper or a graphing tool. This line is the graph of the function [tex]\( f(x) = -\frac{3}{5}x + 2 \)[/tex].
### Graph:
You can create this graph on graph paper or use a graphing tool by plotting the points [tex]\( (0, 2) \)[/tex], [tex]\( (5, -1) \)[/tex], and additional points as described, then drawing a straight line through them.
1. Identify the type of function:
The given function is a linear equation in the form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Determine the slope and y-intercept:
- Slope ([tex]\( m \)[/tex]) = [tex]\(-\frac{3}{5}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]) = 2
3. Plot the y-intercept:
The y-intercept is the point where the graph crosses the y-axis. For this function, it is the point [tex]\( (0, 2) \)[/tex]. Plot this point on the graph.
4. Use the slope to find another point:
The slope [tex]\(-\frac{3}{5}\)[/tex] means that for every increase of 5 units in the x-direction, the function decreases by 3 units in the y-direction.
Starting from the y-intercept [tex]\( (0, 2) \)[/tex]:
- Move 5 units to the right (positive x-direction) and then 3 units down (negative y-direction).
- This gives the point [tex]\( (5, 2 - 3) = (5, -1) \)[/tex].
5. Plot the second point:
Plot the point [tex]\( (5, -1) \)[/tex] on the graph.
6. Draw the line:
Draw a straight line through the points [tex]\( (0, 2) \)[/tex] and [tex]\( (5, -1) \)[/tex]. This line represents the function [tex]\( f(x) = -\frac{3}{5}x + 2 \)[/tex].
7. Check additional points (optional, for accuracy):
If desired, you can calculate and plot additional points:
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ f(-5) = -\frac{3}{5}(-5) + 2 = 3 + 2 = 5 \][/tex]
Point: [tex]\( (-5, 5) \)[/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[ f(10) = -\frac{3}{5}(10) + 2 = -6 + 2 = -4 \][/tex]
Point: [tex]\( (10, -4) \)[/tex]
Plot these additional points to ensure the line is correctly drawn.
### Summary:
For further clarity, here is the visual representation:
1. Y-intercept point: [tex]\( (0, 2) \)[/tex]
2. Another calculated point using the slope: [tex]\( (5, -1) \)[/tex]
3. Optional additional points: [tex]\( (-5, 5) \)[/tex] and [tex]\( (10, -4) \)[/tex]
Finally, draw a straight line through these points on graph paper or a graphing tool. This line is the graph of the function [tex]\( f(x) = -\frac{3}{5}x + 2 \)[/tex].
### Graph:
You can create this graph on graph paper or use a graphing tool by plotting the points [tex]\( (0, 2) \)[/tex], [tex]\( (5, -1) \)[/tex], and additional points as described, then drawing a straight line through them.
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