Answer :
To graph the equation [tex]\( y + 2 = -\frac{3}{4}(x + 4) \)[/tex], we first need to convert it to the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Let's transform the equation step-by-step:
1. Start with the given equation:
[tex]\[ y + 2 = -\frac{3}{4}(x + 4) \][/tex]
2. Distribute the [tex]\(-\frac{3}{4}\)[/tex]:
[tex]\[ y + 2 = -\frac{3}{4}x - 3 \][/tex]
3. Isolate [tex]\( y \)[/tex] by subtracting 2 from both sides:
[tex]\[ y = -\frac{3}{4}x - 3 - 2 \][/tex]
[tex]\[ y = -\frac{3}{4}x - 5 \][/tex]
Now, the equation is in the form [tex]\( y = -\frac{3}{4}x - 5 \)[/tex]. This tells us that the slope ([tex]\( m \)[/tex]) is [tex]\(-\frac{3}{4}\)[/tex] and the y-intercept ([tex]\( b \)[/tex]) is [tex]\(-5\)[/tex].
Next, we can create a table of values for [tex]\( x \)[/tex] and calculate the corresponding [tex]\( y \)[/tex] values.
### Table of Values:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -6 & -0.5 \\ -5 & -1.25 \\ -4 & -2.0 \\ -3 & -2.75 \\ -2 & -3.5 \\ \hline \end{array} \][/tex]
Now, let's plot these points on a graph:
1. For [tex]\( x = -6 \)[/tex]:
[tex]\[ y = -\frac{3}{4}(-6) - 5 = 4.5 - 5 = -0.5 \][/tex]
2. For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = -\frac{3}{4}(-5) - 5 = 3.75 - 5 = -1.25 \][/tex]
3. For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = -\frac{3}{4}(-4) - 5 = 3 - 5 = -2 \][/tex]
4. For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = -\frac{3}{4}(-3) - 5 = 2.25 - 5 = -2.75 \][/tex]
5. For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = -\frac{3}{4}(-2) - 5 = 1.5 - 5 = -3.5 \][/tex]
### Plotting the Points:
- Point [tex]\((-6, -0.5)\)[/tex]
- Point [tex]\((-5, -1.25)\)[/tex]
- Point [tex]\((-4, -2.0)\)[/tex]
- Point [tex]\((-3, -2.75)\)[/tex]
- Point [tex]\((-2, -3.5)\)[/tex]
### Graph:
```plaintext
y
|
| (-6, -0.5)
|
| (-5, -1.25)
|
| (-4, -2)
| (-3, -2.75)
| *(-2, -3.5)
|
|
|------------------------------------ x
```
To finish, draw a straight line through these points, as they lie on the line given by the equation [tex]\( y = -\frac{3}{4}(x + 4) - 2 \)[/tex].
Let's transform the equation step-by-step:
1. Start with the given equation:
[tex]\[ y + 2 = -\frac{3}{4}(x + 4) \][/tex]
2. Distribute the [tex]\(-\frac{3}{4}\)[/tex]:
[tex]\[ y + 2 = -\frac{3}{4}x - 3 \][/tex]
3. Isolate [tex]\( y \)[/tex] by subtracting 2 from both sides:
[tex]\[ y = -\frac{3}{4}x - 3 - 2 \][/tex]
[tex]\[ y = -\frac{3}{4}x - 5 \][/tex]
Now, the equation is in the form [tex]\( y = -\frac{3}{4}x - 5 \)[/tex]. This tells us that the slope ([tex]\( m \)[/tex]) is [tex]\(-\frac{3}{4}\)[/tex] and the y-intercept ([tex]\( b \)[/tex]) is [tex]\(-5\)[/tex].
Next, we can create a table of values for [tex]\( x \)[/tex] and calculate the corresponding [tex]\( y \)[/tex] values.
### Table of Values:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -6 & -0.5 \\ -5 & -1.25 \\ -4 & -2.0 \\ -3 & -2.75 \\ -2 & -3.5 \\ \hline \end{array} \][/tex]
Now, let's plot these points on a graph:
1. For [tex]\( x = -6 \)[/tex]:
[tex]\[ y = -\frac{3}{4}(-6) - 5 = 4.5 - 5 = -0.5 \][/tex]
2. For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = -\frac{3}{4}(-5) - 5 = 3.75 - 5 = -1.25 \][/tex]
3. For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = -\frac{3}{4}(-4) - 5 = 3 - 5 = -2 \][/tex]
4. For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = -\frac{3}{4}(-3) - 5 = 2.25 - 5 = -2.75 \][/tex]
5. For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = -\frac{3}{4}(-2) - 5 = 1.5 - 5 = -3.5 \][/tex]
### Plotting the Points:
- Point [tex]\((-6, -0.5)\)[/tex]
- Point [tex]\((-5, -1.25)\)[/tex]
- Point [tex]\((-4, -2.0)\)[/tex]
- Point [tex]\((-3, -2.75)\)[/tex]
- Point [tex]\((-2, -3.5)\)[/tex]
### Graph:
```plaintext
y
|
| (-6, -0.5)
|
| (-5, -1.25)
|
| (-4, -2)
| (-3, -2.75)
| *(-2, -3.5)
|
|
|------------------------------------ x
```
To finish, draw a straight line through these points, as they lie on the line given by the equation [tex]\( y = -\frac{3}{4}(x + 4) - 2 \)[/tex].
