Question 19·Easy·Two-Variable Data: Models and Scatterplots
Jon deposits $500 into a savings account that earns 2% interest per month, compounded monthly. He makes no additional deposits or withdrawals. What type of function best models the relationship between the account balance and time, in months?
For word problems asking what type of function models a situation, first decide if the quantity is going up or down over time (increasing vs decreasing). Next, focus on how it changes: if there is a constant difference (like “+5 each year”), it’s linear; if there is a constant percent or a repeated multiplication by the same factor (like “2% per month” or “multiplied by 1.02 each time”), it’s exponential. Match these two features—direction and type of change—to the answer choice labels quickly without overcomplicating the algebra.
Hints
Direction of change
Ask yourself: As months pass and interest is added, will the account balance generally go up or go down?
Type of change
Think about whether the account gains the same number of dollars every month, or whether it grows by the same percentage every month.
Link to function type
Recall: a constant difference over equal time intervals suggests one type of function, while a constant percent change (multiplying by the same factor) suggests another.
Combine the ideas
First decide if the balance is increasing or decreasing; then decide whether the pattern is better described by repeated addition or repeated multiplication.
Desmos Guide
Enter a possible model
In Desmos, type the expression B(t) = 500*(1.02)^t (you can use x instead of t if you prefer, like B(x) = 500*(1.02)^x).
Adjust the viewing window
Set the -axis to cover several months, for example to , and the -axis to start near and go a bit above the balance after 24 months. This will let you clearly see how the graph behaves over time.
Observe the graph’s shape and direction
Look at whether the graph goes up or down as increases, and whether it is a straight line or a curved line that gets steeper. Then compare what you see to the descriptions in the answer choices (increasing vs decreasing, linear vs exponential) to decide which label best matches the graph.
Step-by-step Explanation
Determine if the balance increases or decreases
The account earns 2% interest every month and Jon does not withdraw any money.
- Interest is extra money added to the account.
- Because money is being added every month and nothing is taken out, the balance will go up over time, not down.
So the relationship between time and balance is increasing, not decreasing.
Decide if the change is additive or multiplicative
A key idea:
- Linear models: add or subtract the same amount each time (constant difference).
- Exponential models: multiply by the same factor (or same percentage) each time (constant percent change).
Here, the account earns 2% interest per month. That means each month the new balance is the old balance multiplied by (100% + 2%), not the old balance plus a fixed dollar amount.
So this situation involves repeated multiplication by the same factor, which matches an exponential model.
Write a possible model (optional check)
You could model the balance after months by
- is the initial deposit.
- is the monthly growth factor (2% increase each month).
- The exponent shows that the factor is applied repeatedly.
This confirms the relationship is exponential.
Match the behavior to the answer choice
From the earlier steps:
- The balance is increasing over time (interest adds money).
- The change is exponential because the balance is multiplied by the same factor each month.
Therefore, the relationship is best modeled by an increasing exponential function, which corresponds to Choice D) Increasing exponential.