Answer :
To graph the equation [tex]\( f(x) = -\frac{5}{3}x - 2 \)[/tex], we will follow these steps:
1. Determine the y-intercept:
The y-intercept is the point where the graph intersects the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = -\frac{5}{3}(0) - 2 = -2 \][/tex]
So the y-intercept is [tex]\( (0, -2) \)[/tex].
2. Determine the slope:
The slope is given by the coefficient of [tex]\( x \)[/tex] in the equation of the line. In this case, the slope [tex]\( m = -\frac{5}{3} \)[/tex]. This tells us that for every increase of 3 units in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 5 units.
3. Find another point on the line:
To find another point, you can use the slope to move from the y-intercept. Starting at [tex]\( (0, -2) \)[/tex]:
[tex]\[ \text{If } x = 3 \text{, then} \][/tex]
[tex]\[ f(3) = -\frac{5}{3}(3) - 2 = -5 - 2 = -7. \][/tex]
This gives us the point [tex]\( (3, -7) \)[/tex].
4. Draw the graph:
- Plot the y-intercept [tex]\( (0, -2) \)[/tex] on the graph.
- From [tex]\( (0, -2) \)[/tex], move 3 units to the right along the x-axis, and 5 units down along the y-axis to plot the point [tex]\( (3, -7) \)[/tex].
- Draw a straight line through these two points, extending it in both directions beyond them.
5. Confirm the linearity:
- Since we are graphing a linear equation, the line should be straight and extend infinitely in both directions. Adding more points helps to assure that the line is correct. As a further step, you might want to verify by adding other points such as [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = -\frac{5}{3}(-3) - 2 = \frac{15}{3} - 2 = 5 - 2 = 3. \][/tex]
This gives us the point [tex]\( (-3, 3) \)[/tex], which should also lie on the line.
Now that we have plotted these points and drawn the line through them, we can label the axes and any significant points:
- [tex]\( (0, -2) \)[/tex] is our y-intercept.
- [tex]\( (3, -7) \)[/tex] and [tex]\( (-3, 3) \)[/tex] can be used to check the linearity.
Here is a step-by-step graphical representation:
1. Plot the point [tex]\( (0, -2) \)[/tex] on the y-axis.
2. Use the slope [tex]\( -\frac{5}{3} \)[/tex]:
- From [tex]\( (0, -2) \)[/tex], move 3 units right to [tex]\( x = 3 \)[/tex] and down 5 units to [tex]\( y = -7 \)[/tex]. Plot [tex]\( (3, -7) \)[/tex].
3. Draw a line through the points [tex]\( (0, -2) \)[/tex] and [tex]\( (3, -7) \)[/tex].
By following these steps, you will have the graph of the equation [tex]\( f(x) = -\frac{5}{3} x - 2 \)[/tex].
The visual representation confirms the linear relationship and slope of the function.
1. Determine the y-intercept:
The y-intercept is the point where the graph intersects the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = -\frac{5}{3}(0) - 2 = -2 \][/tex]
So the y-intercept is [tex]\( (0, -2) \)[/tex].
2. Determine the slope:
The slope is given by the coefficient of [tex]\( x \)[/tex] in the equation of the line. In this case, the slope [tex]\( m = -\frac{5}{3} \)[/tex]. This tells us that for every increase of 3 units in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 5 units.
3. Find another point on the line:
To find another point, you can use the slope to move from the y-intercept. Starting at [tex]\( (0, -2) \)[/tex]:
[tex]\[ \text{If } x = 3 \text{, then} \][/tex]
[tex]\[ f(3) = -\frac{5}{3}(3) - 2 = -5 - 2 = -7. \][/tex]
This gives us the point [tex]\( (3, -7) \)[/tex].
4. Draw the graph:
- Plot the y-intercept [tex]\( (0, -2) \)[/tex] on the graph.
- From [tex]\( (0, -2) \)[/tex], move 3 units to the right along the x-axis, and 5 units down along the y-axis to plot the point [tex]\( (3, -7) \)[/tex].
- Draw a straight line through these two points, extending it in both directions beyond them.
5. Confirm the linearity:
- Since we are graphing a linear equation, the line should be straight and extend infinitely in both directions. Adding more points helps to assure that the line is correct. As a further step, you might want to verify by adding other points such as [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = -\frac{5}{3}(-3) - 2 = \frac{15}{3} - 2 = 5 - 2 = 3. \][/tex]
This gives us the point [tex]\( (-3, 3) \)[/tex], which should also lie on the line.
Now that we have plotted these points and drawn the line through them, we can label the axes and any significant points:
- [tex]\( (0, -2) \)[/tex] is our y-intercept.
- [tex]\( (3, -7) \)[/tex] and [tex]\( (-3, 3) \)[/tex] can be used to check the linearity.
Here is a step-by-step graphical representation:
1. Plot the point [tex]\( (0, -2) \)[/tex] on the y-axis.
2. Use the slope [tex]\( -\frac{5}{3} \)[/tex]:
- From [tex]\( (0, -2) \)[/tex], move 3 units right to [tex]\( x = 3 \)[/tex] and down 5 units to [tex]\( y = -7 \)[/tex]. Plot [tex]\( (3, -7) \)[/tex].
3. Draw a line through the points [tex]\( (0, -2) \)[/tex] and [tex]\( (3, -7) \)[/tex].
By following these steps, you will have the graph of the equation [tex]\( f(x) = -\frac{5}{3} x - 2 \)[/tex].
The visual representation confirms the linear relationship and slope of the function.
