Answer :
To graph the function [tex]\( g(x) = \left(\frac{2}{3}\right) x - 2 \)[/tex], we need to determine several key points on the graph and plot them.
### Step-by-Step Solution:
1. Understanding the Function:
The given function is [tex]\( g(x) = \frac{2}{3} x - 2 \)[/tex], which is a linear function in the form [tex]\( g(x) = mx + b \)[/tex], where [tex]\( m = \frac{2}{3} \)[/tex] is the slope and [tex]\( b = -2 \)[/tex] is the y-intercept.
2. Determine Key Points:
To plot the graph accurately, we should calculate the values of [tex]\( g(x) \)[/tex] for a range of [tex]\( x \)[/tex]-values. Here is a table of values for [tex]\( x \)[/tex] from -10 to 10:
| [tex]\( x \)[/tex] | [tex]\( g(x) \)[/tex] |
|------|--------------------------------|
| -10 | [tex]\( -8.67 \)[/tex] |
| -9 | [tex]\( -8.0 \)[/tex] |
| -8 | [tex]\( -7.33 \)[/tex] |
| -7 | [tex]\( -6.67 \)[/tex] |
| -6 | [tex]\( -6.0 \)[/tex] |
| -5 | [tex]\( -5.33 \)[/tex] |
| -4 | [tex]\( -4.67 \)[/tex] |
| -3 | [tex]\( -4.0 \)[/tex] |
| -2 | [tex]\( -3.33 \)[/tex] |
| -1 | [tex]\( -2.67 \)[/tex] |
| 0 | [tex]\( -2.0 \)[/tex] |
| 1 | [tex]\( -1.33 \)[/tex] |
| 2 | [tex]\( -0.67 \)[/tex] |
| 3 | [tex]\( 0.0 \)[/tex] |
| 4 | [tex]\( 0.67 \)[/tex] |
| 5 | [tex]\( 1.33 \)[/tex] |
| 6 | [tex]\( 2.0 \)[/tex] |
| 7 | [tex]\( 2.67 \)[/tex] |
| 8 | [tex]\( 3.33 \)[/tex] |
| 9 | [tex]\( 4.0 \)[/tex] |
| 10 | [tex]\( 4.67 \)[/tex] |
3. Plotting the Points:
Using the table above, plot the points [tex]\((-10, -8.67)\)[/tex], [tex]\((-9, -8.0)\)[/tex], [tex]\((-8, -7.33)\)[/tex], [tex]\((-7, -6.67)\)[/tex], and so on up to [tex]\((10, 4.67)\)[/tex] on the Cartesian plane.
4. Drawing the Line:
By connecting these points with a straight line, you will clearly see the graphical representation of the linear function [tex]\( g(x) = \frac{2}{3} x - 2 \)[/tex]. Since the slope [tex]\( m = \frac{2}{3} \)[/tex] is positive, the line inclines upwards to the right. The y-intercept at [tex]\((0, -2)\)[/tex] means the line crosses the y-axis at -2.
### Graph Characteristics:
- Y-intercept: The line crosses the y-axis at [tex]\( y = -2 \)[/tex].
- Slope: The line rises by [tex]\( \frac{2}{3} \)[/tex] units for every 1 unit it moves to the right.
- Domain and Range: Being a linear function, the graph extends infinitely in both directions along the x-axis and y-axis, hence the domain and range are both all real numbers.
This graph should give a visual representation of the linear function [tex]\( g(x) = \frac{2}{3} x - 2 \)[/tex].
### Step-by-Step Solution:
1. Understanding the Function:
The given function is [tex]\( g(x) = \frac{2}{3} x - 2 \)[/tex], which is a linear function in the form [tex]\( g(x) = mx + b \)[/tex], where [tex]\( m = \frac{2}{3} \)[/tex] is the slope and [tex]\( b = -2 \)[/tex] is the y-intercept.
2. Determine Key Points:
To plot the graph accurately, we should calculate the values of [tex]\( g(x) \)[/tex] for a range of [tex]\( x \)[/tex]-values. Here is a table of values for [tex]\( x \)[/tex] from -10 to 10:
| [tex]\( x \)[/tex] | [tex]\( g(x) \)[/tex] |
|------|--------------------------------|
| -10 | [tex]\( -8.67 \)[/tex] |
| -9 | [tex]\( -8.0 \)[/tex] |
| -8 | [tex]\( -7.33 \)[/tex] |
| -7 | [tex]\( -6.67 \)[/tex] |
| -6 | [tex]\( -6.0 \)[/tex] |
| -5 | [tex]\( -5.33 \)[/tex] |
| -4 | [tex]\( -4.67 \)[/tex] |
| -3 | [tex]\( -4.0 \)[/tex] |
| -2 | [tex]\( -3.33 \)[/tex] |
| -1 | [tex]\( -2.67 \)[/tex] |
| 0 | [tex]\( -2.0 \)[/tex] |
| 1 | [tex]\( -1.33 \)[/tex] |
| 2 | [tex]\( -0.67 \)[/tex] |
| 3 | [tex]\( 0.0 \)[/tex] |
| 4 | [tex]\( 0.67 \)[/tex] |
| 5 | [tex]\( 1.33 \)[/tex] |
| 6 | [tex]\( 2.0 \)[/tex] |
| 7 | [tex]\( 2.67 \)[/tex] |
| 8 | [tex]\( 3.33 \)[/tex] |
| 9 | [tex]\( 4.0 \)[/tex] |
| 10 | [tex]\( 4.67 \)[/tex] |
3. Plotting the Points:
Using the table above, plot the points [tex]\((-10, -8.67)\)[/tex], [tex]\((-9, -8.0)\)[/tex], [tex]\((-8, -7.33)\)[/tex], [tex]\((-7, -6.67)\)[/tex], and so on up to [tex]\((10, 4.67)\)[/tex] on the Cartesian plane.
4. Drawing the Line:
By connecting these points with a straight line, you will clearly see the graphical representation of the linear function [tex]\( g(x) = \frac{2}{3} x - 2 \)[/tex]. Since the slope [tex]\( m = \frac{2}{3} \)[/tex] is positive, the line inclines upwards to the right. The y-intercept at [tex]\((0, -2)\)[/tex] means the line crosses the y-axis at -2.
### Graph Characteristics:
- Y-intercept: The line crosses the y-axis at [tex]\( y = -2 \)[/tex].
- Slope: The line rises by [tex]\( \frac{2}{3} \)[/tex] units for every 1 unit it moves to the right.
- Domain and Range: Being a linear function, the graph extends infinitely in both directions along the x-axis and y-axis, hence the domain and range are both all real numbers.
This graph should give a visual representation of the linear function [tex]\( g(x) = \frac{2}{3} x - 2 \)[/tex].
