Answer :
To determine which graph represents the function [tex]\( y = \frac{2}{3}x - 2 \)[/tex], let's analyze the function step-by-step and identify key characteristics.
### Step-by-Step Solution:
1. Slope-Intercept Form:
The given function [tex]\( y = \frac{2}{3}x - 2 \)[/tex] is already in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Identifying the Slope and Y-Intercept:
- The slope [tex]\( m \)[/tex] of the line is [tex]\( \frac{2}{3} \)[/tex]. This means for every 3 units the graph moves horizontally to the right, it moves up 2 units vertically.
- The y-intercept [tex]\( b \)[/tex] is [tex]\(-2\)[/tex]. This is the point where the line crosses the y-axis.
3. Key Points on the Line:
We can use the function to identify some specific points that lie on the line for clarity:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{2}{3}(0) - 2 = -2 \][/tex]
So, the point is [tex]\( (0, -2) \)[/tex].
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ y = \frac{2}{3}(3) - 2 = 2 - 2 = 0 \][/tex]
So, the point is [tex]\( (3, 0) \)[/tex].
- When [tex]\( x = -3 \)[/tex]:
[tex]\[ y = \frac{2}{3}(-3) - 2 = -2 - 2 = -4 \][/tex]
So, the point is [tex]\( (-3, -4) \)[/tex].
4. Graph Characteristics:
Using the points identified:
- The line passes through [tex]\( (0, -2) \)[/tex], which is the y-intercept.
- The line passes through [tex]\( (3, 0) \)[/tex], indicating that it crosses the x-axis at this point.
- The line also passes through [tex]\( (-3, -4) \)[/tex].
Additionally, the slope of [tex]\( \frac{2}{3} \)[/tex] tells us that for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by [tex]\( \frac{2}{3} \)[/tex]. This upward inclination from left to right confirms the positive slope.
### Conclusion:
The graph of the function [tex]\( y = \frac{2}{3}x - 2 \)[/tex] is a straight line passing through the points [tex]\( (0, -2) \)[/tex], [tex]\( (3, 0) \)[/tex], and [tex]\( (-3, -4) \)[/tex], with an upward slope of [tex]\( \frac{2}{3} \)[/tex]. Make sure the graph you select exhibits these points and the correct slope behavior.
### Step-by-Step Solution:
1. Slope-Intercept Form:
The given function [tex]\( y = \frac{2}{3}x - 2 \)[/tex] is already in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Identifying the Slope and Y-Intercept:
- The slope [tex]\( m \)[/tex] of the line is [tex]\( \frac{2}{3} \)[/tex]. This means for every 3 units the graph moves horizontally to the right, it moves up 2 units vertically.
- The y-intercept [tex]\( b \)[/tex] is [tex]\(-2\)[/tex]. This is the point where the line crosses the y-axis.
3. Key Points on the Line:
We can use the function to identify some specific points that lie on the line for clarity:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{2}{3}(0) - 2 = -2 \][/tex]
So, the point is [tex]\( (0, -2) \)[/tex].
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ y = \frac{2}{3}(3) - 2 = 2 - 2 = 0 \][/tex]
So, the point is [tex]\( (3, 0) \)[/tex].
- When [tex]\( x = -3 \)[/tex]:
[tex]\[ y = \frac{2}{3}(-3) - 2 = -2 - 2 = -4 \][/tex]
So, the point is [tex]\( (-3, -4) \)[/tex].
4. Graph Characteristics:
Using the points identified:
- The line passes through [tex]\( (0, -2) \)[/tex], which is the y-intercept.
- The line passes through [tex]\( (3, 0) \)[/tex], indicating that it crosses the x-axis at this point.
- The line also passes through [tex]\( (-3, -4) \)[/tex].
Additionally, the slope of [tex]\( \frac{2}{3} \)[/tex] tells us that for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by [tex]\( \frac{2}{3} \)[/tex]. This upward inclination from left to right confirms the positive slope.
### Conclusion:
The graph of the function [tex]\( y = \frac{2}{3}x - 2 \)[/tex] is a straight line passing through the points [tex]\( (0, -2) \)[/tex], [tex]\( (3, 0) \)[/tex], and [tex]\( (-3, -4) \)[/tex], with an upward slope of [tex]\( \frac{2}{3} \)[/tex]. Make sure the graph you select exhibits these points and the correct slope behavior.
