Answer :
To graph the function [tex]\( y = \frac{2x}{3} - 2 \)[/tex], we need to understand its behavior and the significant points that define it. Let's go through the steps detailedly.
### Step 1: Identify the Function Type
This function is a linear equation of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- [tex]\( m = \frac{2}{3} \)[/tex]
- [tex]\( b = -2 \)[/tex]
### Step 2: Determine the Y-Intercept
The y-intercept [tex]\( b \)[/tex] is the point where the graph crosses the y-axis (i.e., where [tex]\( x = 0 \)[/tex]).
- Set [tex]\( x = 0 \)[/tex] in the equation:
[tex]\[ y = \frac{2(0)}{3} - 2 = -2 \][/tex]
- Thus, the y-intercept is [tex]\( (0, -2) \)[/tex].
### Step 3: Determine the Slope
The slope [tex]\( m \)[/tex] is [tex]\( \frac{2}{3} \)[/tex], which means that for every increase of [tex]\( 3 \)[/tex] units in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by [tex]\( 2 \)[/tex] units.
### Step 4: Find Another Point Using the Slope
To plot the line accurately, find another point on the line using the slope.
- Start from the y-intercept [tex]\( (0, -2) \)[/tex].
- Move 3 units to the right (increase [tex]\( x \)[/tex] by 3) and then 2 units up (increase [tex]\( y \)[/tex] by 2):
[tex]\[ \text{New point: } (3, -2 + 2) = (3, 0) \][/tex]
### Step 5: Sketch the Line
Use the two points found:
- [tex]\( (0, -2) \)[/tex]
- [tex]\( (3, 0) \)[/tex]
Connect these points with a straight line. This line will be the graph of the function [tex]\( y = \frac{2x}{3} - 2 \)[/tex].
### Step 6: General Features of the Graph
The graph should:
- Pass through the y-intercept at [tex]\( (0, -2) \)[/tex].
- Pass through the point [tex]\( (3, 0) \)[/tex].
- Show a linear relationship where the slope is [tex]\( \frac{2}{3} \)[/tex].
### Detailed Coordinates List
From the numerical results, let's confirm a few values:
- For [tex]\( x = -10 \)[/tex]:
[tex]\[ y = \frac{2(-10)}{3} - 2 = -\frac{20}{3} - 2 \approx -8.67 \][/tex]
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = \frac{2(-5)}{3} - 2 = -\frac{10}{3} - 2 \approx -5.33 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -2 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = \frac{2(5)}{3} - 2 = \frac{10}{3} - 2 \approx 0.67 \][/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[ y = \frac{2(10)}{3} - 2 = \frac{20}{3} - 2 \approx 4.67 \][/tex]
Using these values, we confirm the straight line behavior.
### Graph Visualization
The graph should look like a diagonal line cutting through the points mentioned:
- [tex]\( (0, -2) \)[/tex]
- [tex]\( (3, 0) \)[/tex]
- Extending through other coordinates in similar pattern.
This graph represents the linear equation [tex]\( y = \frac{2x}{3} - 2 \)[/tex].
#### Expected Graph:
```
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|-|-------------------
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```
In summary, the graph of [tex]\( y = \frac{2x}{3} - 2 \)[/tex] will be a straight line with a slope of [tex]\( \frac{2}{3} \)[/tex], intersecting the y-axis at [tex]\( -2 \)[/tex], and passing through points like [tex]\( (-10, -8.67) \)[/tex], [tex]\( (0, -2) \)[/tex], [tex]\( (5, 0.67) \)[/tex], and [tex]\( (10, 4.67) \)[/tex].
### Step 1: Identify the Function Type
This function is a linear equation of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- [tex]\( m = \frac{2}{3} \)[/tex]
- [tex]\( b = -2 \)[/tex]
### Step 2: Determine the Y-Intercept
The y-intercept [tex]\( b \)[/tex] is the point where the graph crosses the y-axis (i.e., where [tex]\( x = 0 \)[/tex]).
- Set [tex]\( x = 0 \)[/tex] in the equation:
[tex]\[ y = \frac{2(0)}{3} - 2 = -2 \][/tex]
- Thus, the y-intercept is [tex]\( (0, -2) \)[/tex].
### Step 3: Determine the Slope
The slope [tex]\( m \)[/tex] is [tex]\( \frac{2}{3} \)[/tex], which means that for every increase of [tex]\( 3 \)[/tex] units in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by [tex]\( 2 \)[/tex] units.
### Step 4: Find Another Point Using the Slope
To plot the line accurately, find another point on the line using the slope.
- Start from the y-intercept [tex]\( (0, -2) \)[/tex].
- Move 3 units to the right (increase [tex]\( x \)[/tex] by 3) and then 2 units up (increase [tex]\( y \)[/tex] by 2):
[tex]\[ \text{New point: } (3, -2 + 2) = (3, 0) \][/tex]
### Step 5: Sketch the Line
Use the two points found:
- [tex]\( (0, -2) \)[/tex]
- [tex]\( (3, 0) \)[/tex]
Connect these points with a straight line. This line will be the graph of the function [tex]\( y = \frac{2x}{3} - 2 \)[/tex].
### Step 6: General Features of the Graph
The graph should:
- Pass through the y-intercept at [tex]\( (0, -2) \)[/tex].
- Pass through the point [tex]\( (3, 0) \)[/tex].
- Show a linear relationship where the slope is [tex]\( \frac{2}{3} \)[/tex].
### Detailed Coordinates List
From the numerical results, let's confirm a few values:
- For [tex]\( x = -10 \)[/tex]:
[tex]\[ y = \frac{2(-10)}{3} - 2 = -\frac{20}{3} - 2 \approx -8.67 \][/tex]
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = \frac{2(-5)}{3} - 2 = -\frac{10}{3} - 2 \approx -5.33 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -2 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = \frac{2(5)}{3} - 2 = \frac{10}{3} - 2 \approx 0.67 \][/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[ y = \frac{2(10)}{3} - 2 = \frac{20}{3} - 2 \approx 4.67 \][/tex]
Using these values, we confirm the straight line behavior.
### Graph Visualization
The graph should look like a diagonal line cutting through the points mentioned:
- [tex]\( (0, -2) \)[/tex]
- [tex]\( (3, 0) \)[/tex]
- Extending through other coordinates in similar pattern.
This graph represents the linear equation [tex]\( y = \frac{2x}{3} - 2 \)[/tex].
#### Expected Graph:
```
|
|
| /
| /
| /
| /
| /
| /
| /
| /
| /
|-|-------------------
|
|
|
```
In summary, the graph of [tex]\( y = \frac{2x}{3} - 2 \)[/tex] will be a straight line with a slope of [tex]\( \frac{2}{3} \)[/tex], intersecting the y-axis at [tex]\( -2 \)[/tex], and passing through points like [tex]\( (-10, -8.67) \)[/tex], [tex]\( (0, -2) \)[/tex], [tex]\( (5, 0.67) \)[/tex], and [tex]\( (10, 4.67) \)[/tex].
