Answer :
To graph the function [tex]\( f(x) = \frac{3}{4} x - 2 \)[/tex], let’s go through the steps required to understand and draw the graph:
### Step-by-Step Solution:
#### 1. Identify the Type of Function:
This is a linear function of the form [tex]\( f(x) = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope
- [tex]\( b \)[/tex] is the y-intercept
Here, [tex]\( m = \frac{3}{4} \)[/tex] and [tex]\( b = -2 \)[/tex].
#### 2. Determine the Slope and Y-Intercept:
- Y-intercept (b): The function intercepts the y-axis at [tex]\( (0, b) \)[/tex]. For this function, the y-intercept is at [tex]\( (0, -2) \)[/tex].
- Slope (m): The slope [tex]\( \frac{3}{4} \)[/tex] means that for every 4 units you move to the right on the x-axis, you move 3 units up on the y-axis (since the slope is positive).
#### 3. Find the X-Intercept:
The x-intercept is where [tex]\( f(x) = 0 \)[/tex]. Set the equation equal to zero and solve for [tex]\( x \)[/tex].
[tex]\[ 0 = \frac{3}{4} x - 2 \][/tex]
[tex]\[ \frac{3}{4} x = 2 \][/tex]
[tex]\[ x = \frac{2 \times 4}{3} \][/tex]
[tex]\[ x = \frac{8}{3} \approx 2.67 \][/tex]
So, the function intercepts the x-axis at [tex]\( \left( \frac{8}{3}, 0 \right) \)[/tex].
#### 4. Plot the Intercepts:
- Y-intercept at [tex]\( (0, -2) \)[/tex]
- X-intercept at [tex]\( \left( \frac{8}{3}, 0 \right) \)[/tex]
#### 5. Draw the Line Using the Slope:
From the y-intercept [tex]\( (0, -2) \)[/tex]:
- Move 4 units to the right (positive x direction).
- Move 3 units up (positive y direction).
So, from [tex]\( (0, -2) \)[/tex], if you move 4 units to the right, you end up at [tex]\( (4, -2) \)[/tex]. Moving 3 units up from there, you end up at [tex]\( (4, 1) \)[/tex]. This is another point on the line.
#### 6. Draw the Line:
Using the points [tex]\( (0, -2) \)[/tex] and [tex]\( (4, 1) \)[/tex], draw a straight line through these points. Extend the line in both directions to cover a suitable range of values for [tex]\( x \)[/tex].
### Graph Summary:
1. Y-intercept: [tex]\( (0, -2) \)[/tex]
2. X-intercept: [tex]\( \left( \frac{8}{3}, 0 \right) \approx (2.67, 0) \)[/tex]
3. Another point using the slope: [tex]\( (4, 1) \)[/tex]
4. Slope: [tex]\( \frac{3}{4} \)[/tex] (rise over run: up 3, right 4)
#### Visualization:
```plaintext
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----------------+-------------------------------
-2 2.67 x
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```
This line inclines upwards from left to right, crossing the y-axis at [tex]\( (0, -2) \)[/tex] and the x-axis at approximately [tex]\( (2.67, 0) \)[/tex].
By placing these points and following the slope, you should be able to accurately graph the function [tex]\( f(x) = \frac{3}{4} x - 2 \)[/tex].
### Step-by-Step Solution:
#### 1. Identify the Type of Function:
This is a linear function of the form [tex]\( f(x) = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope
- [tex]\( b \)[/tex] is the y-intercept
Here, [tex]\( m = \frac{3}{4} \)[/tex] and [tex]\( b = -2 \)[/tex].
#### 2. Determine the Slope and Y-Intercept:
- Y-intercept (b): The function intercepts the y-axis at [tex]\( (0, b) \)[/tex]. For this function, the y-intercept is at [tex]\( (0, -2) \)[/tex].
- Slope (m): The slope [tex]\( \frac{3}{4} \)[/tex] means that for every 4 units you move to the right on the x-axis, you move 3 units up on the y-axis (since the slope is positive).
#### 3. Find the X-Intercept:
The x-intercept is where [tex]\( f(x) = 0 \)[/tex]. Set the equation equal to zero and solve for [tex]\( x \)[/tex].
[tex]\[ 0 = \frac{3}{4} x - 2 \][/tex]
[tex]\[ \frac{3}{4} x = 2 \][/tex]
[tex]\[ x = \frac{2 \times 4}{3} \][/tex]
[tex]\[ x = \frac{8}{3} \approx 2.67 \][/tex]
So, the function intercepts the x-axis at [tex]\( \left( \frac{8}{3}, 0 \right) \)[/tex].
#### 4. Plot the Intercepts:
- Y-intercept at [tex]\( (0, -2) \)[/tex]
- X-intercept at [tex]\( \left( \frac{8}{3}, 0 \right) \)[/tex]
#### 5. Draw the Line Using the Slope:
From the y-intercept [tex]\( (0, -2) \)[/tex]:
- Move 4 units to the right (positive x direction).
- Move 3 units up (positive y direction).
So, from [tex]\( (0, -2) \)[/tex], if you move 4 units to the right, you end up at [tex]\( (4, -2) \)[/tex]. Moving 3 units up from there, you end up at [tex]\( (4, 1) \)[/tex]. This is another point on the line.
#### 6. Draw the Line:
Using the points [tex]\( (0, -2) \)[/tex] and [tex]\( (4, 1) \)[/tex], draw a straight line through these points. Extend the line in both directions to cover a suitable range of values for [tex]\( x \)[/tex].
### Graph Summary:
1. Y-intercept: [tex]\( (0, -2) \)[/tex]
2. X-intercept: [tex]\( \left( \frac{8}{3}, 0 \right) \approx (2.67, 0) \)[/tex]
3. Another point using the slope: [tex]\( (4, 1) \)[/tex]
4. Slope: [tex]\( \frac{3}{4} \)[/tex] (rise over run: up 3, right 4)
#### Visualization:
```plaintext
|
| .
| .
|
|
|
| .
----------------+-------------------------------
-2 2.67 x
|
|
|
|
|
|
|
```
This line inclines upwards from left to right, crossing the y-axis at [tex]\( (0, -2) \)[/tex] and the x-axis at approximately [tex]\( (2.67, 0) \)[/tex].
By placing these points and following the slope, you should be able to accurately graph the function [tex]\( f(x) = \frac{3}{4} x - 2 \)[/tex].
