Answer :
To graph the equation [tex]\( f(x) = -\frac{5}{3} x - 2 \)[/tex], we need to follow these steps:
### 1. Identify the Slope and Y-Intercept
The given equation takes the form of a linear function [tex]\( f(x) = mx + b \)[/tex], where:
- [tex]\( m = -\frac{5}{3} \)[/tex] is the slope.
- [tex]\( b = -2 \)[/tex] is the y-intercept.
### 2. Plot the Y-Intercept
The y-intercept is the point where the graph crosses the y-axis (i.e., where [tex]\( x = 0 \)[/tex]).
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\frac{5}{3} \cdot 0 - 2 = -2 \][/tex]
- Plot the point [tex]\( (0, -2) \)[/tex] on the graph.
### 3. Use the Slope to Find Another Point
The slope [tex]\( m = -\frac{5}{3} \)[/tex] tells us that for every 3 units you move to the right (positive [tex]\( x \)[/tex]-direction), you move 5 units down (negative [tex]\( y \)[/tex]-direction).
Starting from the y-intercept [tex]\( (0, -2) \)[/tex]:
- Move 3 units to the right: [tex]\( 0 + 3 = 3 \)[/tex].
- Move 5 units down: [tex]\( -2 - 5 = -7 \)[/tex].
Thus, another point on the graph is [tex]\( (3, -7) \)[/tex]. Plot this point as well.
### 4. Draw the Line
With the two points [tex]\( (0, -2) \)[/tex] and [tex]\( (3, -7) \)[/tex] plotted, you can draw a straight line through these points. This line represents the function [tex]\( f(x) = -\frac{5}{3} x - 2 \)[/tex].
### 5. Extend the Line
To give a clear picture of the graph, extend the line in both directions, and you can use additional points to ensure accuracy if needed. For instance, using negative [tex]\( x \)[/tex] values:
- At [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = -\frac{5}{3} \cdot (-3) - 2 = 5 - 2 = 3 \][/tex]
- Plot the point [tex]\( (-3, 3) \)[/tex].
### Visual Description of the Graph
From the intercepts and slope calculations, the visual representation of the line will show it descending from left to right, indicating a negative slope. You should see the line crossing the y-axis at [tex]\( (0, -2) \)[/tex] and passing through other calculated points like [tex]\( (3, -7) \)[/tex].
### Summary
1. Plot the y-intercept point: [tex]\( (0, -2) \)[/tex].
2. Use the slope to find additional points, such as [tex]\( (3, -7) \)[/tex] and [tex]\( (-3, 3) \)[/tex].
3. Draw a straight line through these points.
4. Extend the line in both directions on the graph.
By following these steps, you’ll have the graph of the equation [tex]\( f(x) = -\frac{5}{3} x - 2 \)[/tex].
### 1. Identify the Slope and Y-Intercept
The given equation takes the form of a linear function [tex]\( f(x) = mx + b \)[/tex], where:
- [tex]\( m = -\frac{5}{3} \)[/tex] is the slope.
- [tex]\( b = -2 \)[/tex] is the y-intercept.
### 2. Plot the Y-Intercept
The y-intercept is the point where the graph crosses the y-axis (i.e., where [tex]\( x = 0 \)[/tex]).
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\frac{5}{3} \cdot 0 - 2 = -2 \][/tex]
- Plot the point [tex]\( (0, -2) \)[/tex] on the graph.
### 3. Use the Slope to Find Another Point
The slope [tex]\( m = -\frac{5}{3} \)[/tex] tells us that for every 3 units you move to the right (positive [tex]\( x \)[/tex]-direction), you move 5 units down (negative [tex]\( y \)[/tex]-direction).
Starting from the y-intercept [tex]\( (0, -2) \)[/tex]:
- Move 3 units to the right: [tex]\( 0 + 3 = 3 \)[/tex].
- Move 5 units down: [tex]\( -2 - 5 = -7 \)[/tex].
Thus, another point on the graph is [tex]\( (3, -7) \)[/tex]. Plot this point as well.
### 4. Draw the Line
With the two points [tex]\( (0, -2) \)[/tex] and [tex]\( (3, -7) \)[/tex] plotted, you can draw a straight line through these points. This line represents the function [tex]\( f(x) = -\frac{5}{3} x - 2 \)[/tex].
### 5. Extend the Line
To give a clear picture of the graph, extend the line in both directions, and you can use additional points to ensure accuracy if needed. For instance, using negative [tex]\( x \)[/tex] values:
- At [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = -\frac{5}{3} \cdot (-3) - 2 = 5 - 2 = 3 \][/tex]
- Plot the point [tex]\( (-3, 3) \)[/tex].
### Visual Description of the Graph
From the intercepts and slope calculations, the visual representation of the line will show it descending from left to right, indicating a negative slope. You should see the line crossing the y-axis at [tex]\( (0, -2) \)[/tex] and passing through other calculated points like [tex]\( (3, -7) \)[/tex].
### Summary
1. Plot the y-intercept point: [tex]\( (0, -2) \)[/tex].
2. Use the slope to find additional points, such as [tex]\( (3, -7) \)[/tex] and [tex]\( (-3, 3) \)[/tex].
3. Draw a straight line through these points.
4. Extend the line in both directions on the graph.
By following these steps, you’ll have the graph of the equation [tex]\( f(x) = -\frac{5}{3} x - 2 \)[/tex].
