To determine what type of function could model Abert's bank account balance over time, we need to look at the pattern in the given balances for each week.
The table provided is as follows:
| Week | Balance ($) |
|------|-------------|
| 1 | 428 |
| 2 | 620 |
| 3 | 812 |
| 4 | 1,004 |
We'll first calculate the differences in balance between consecutive weeks to see if there's a common difference:
1. Difference between Week 2 and Week 1:
[tex]\[
620 - 428 = 192
\][/tex]
2. Difference between Week 3 and Week 2:
[tex]\[
812 - 620 = 192
\][/tex]
3. Difference between Week 4 and Week 3:
[tex]\[
1,004 - 812 = 192
\][/tex]
We observe that the difference between the balances of consecutive weeks is the same (192). This suggests that the increase in balance is consistent over time.
A consistent difference in values suggests a linear function. A linear function has the form:
[tex]\[ f(x) = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope (the rate of change, which is 192 in this case), and [tex]\( b \)[/tex] is the y-intercept.
Since we have a common difference of 192 between each week, the balance increases by a constant amount each week.
Hence, the correct option is:
- This is a linear function because there is a common difference in the balance between the weeks.
