To determine what type of function can model Albert’s bank account balance over time, we need to analyze how the balance changes from week to week.
Here’s the given data:
Week 1: \[tex]$1,426
Week 2: \$[/tex]1,528
Week 3: \[tex]$1,630
Week 4: \$[/tex]1,732
To determine whether the function is linear or exponential, we need to check either for a common difference (indicative of a linear function) or a common ratio (indicative of an exponential function) between consecutive week balances.
First, let's check for a common difference:
For Week 1 to Week 2:
\[tex]$1,528 - \$[/tex]1,426 = \[tex]$102
For Week 2 to Week 3:
\$[/tex]1,630 - \[tex]$1,528 = \$[/tex]102
For Week 3 to Week 4:
\[tex]$1,732 - \$[/tex]1,630 = \[tex]$102
Since the difference between each consecutive week is constant (\$[/tex]102), the changes indicate a common difference, which is characteristic of a linear function.
Next, let's verify there is no common ratio, just for thoroughness.
For Week 1 to Week 2:
[tex]$ \frac{1,528}{1,426} \approx 1.0715 $[/tex]
For Week 2 to Week 3:
[tex]$ \frac{1,630}{1,528} \approx 1.0668 $[/tex]
For Week 3 to Week 4:
[tex]$ \frac{1,732}{1,630} \approx 1.0626 $[/tex]
The ratios are not the same, meaning there is no common ratio.
Therefore, given our findings:
- There is a constant difference (\$102) between the balances each week.
- There is no constant ratio between the balances.
Hence, the balance is modeled by a linear function because there is a common difference in the balance between the weeks. So, the correct statement is:
This is a linear function because there is a common difference in the balance between the weeks.
