To determine if the given set of points represent a linear function and to find its slope, we'll follow these steps:
### Step 1: Identify the Points
The given points are:
- [tex]\( (3, -11) \)[/tex]
- [tex]\( (6, 1) \)[/tex]
- [tex]\( (9, 13) \)[/tex]
- [tex]\( (12, 25) \)[/tex]
### Step 2: Calculate the Differences between [tex]\(x\)[/tex]-values ([tex]\(\Delta x\)[/tex])
Compute the differences between consecutive [tex]\(x\)[/tex]-values:
- [tex]\(6 - 3 = 3\)[/tex]
- [tex]\(9 - 6 = 3\)[/tex]
- [tex]\(12 - 9 = 3\)[/tex]
These differences are: [tex]\(3, 3, 3\)[/tex].
### Step 3: Calculate the Differences between [tex]\(y\)[/tex]-values ([tex]\(\Delta y\)[/tex])
Compute the differences between consecutive [tex]\(y\)[/tex]-values:
- [tex]\(1 - (-11) = 12\)[/tex]
- [tex]\(13 - 1 = 12\)[/tex]
- [tex]\(25 - 13 = 12\)[/tex]
These differences are: [tex]\(12, 12, 12\)[/tex].
### Step 4: Calculate the Slopes ([tex]\( \frac{\Delta y}{\Delta x} \)[/tex])
Now, compute the slopes between consecutive points, which is done by dividing the differences in [tex]\(y\)[/tex]-values by the differences in [tex]\(x\)[/tex]-values:
- [tex]\( \frac{12}{3} = 4 \)[/tex]
- [tex]\( \frac{12}{3} = 4 \)[/tex]
- [tex]\( \frac{12}{3} = 4 \)[/tex]
The slopes are: [tex]\(4, 4, 4 \)[/tex].
### Step 5: Verify Consistency of the Slope
All slopes are equal ([tex]\(4\)[/tex]), confirming that the points lie on the same line.
### Conclusion
Since the slope between each pair of points is consistently [tex]\(4\)[/tex], the points can be represented by a linear function with a slope of [tex]\(4\)[/tex]. This indicates that the slope of the line is not [tex]\(\frac{1}{4}\)[/tex], but rather [tex]\(4\)[/tex].
Thus, the linear function representing the given points has a slope of [tex]\(4\)[/tex], not [tex]\(\frac{1}{4}\)[/tex].
