Bank Account Balance

Task

At the beginning of the week, Jessie had \$500 in her bank account. She deposited a check for \$50 on Monday and then paid \$250 in rent on Wednesday. On Friday, Jessie deposited \$200 in the account and then on Saturday she paid \$100 for groceries from her bank account. Jessie made the following graph for the balance in her bank account during this week:

1. Is the depiction of how the account balance varies over the week accurate? Explain.
2. How can Jessie graphically represent the bank account balance in a way that better shows how it changes?

IM Commentary

The purpose of this task is to study an example of a function which varies discretely over time. Step functions are often good examples for this type of function. In practice, instead of a step function, bar graphs are sometimes used. Alternatively, Jessie's method of ''smoothing'' over the jumps is also very common.

Many real world phenomena are discrete but the step functions that model them are not commonly used. There is a good opportunity for the instructor to discuss the advantages and disadvantages of the two graphs produced in this problem. Jessie's graph has the advantage of immediately communicating the general trend in the bank account balance. The step function graph is more accurate and communicates more information but it is perhaps not as easy to read since it is a less familiar format.

Solution

1. According to Jessie's graph, her bank account had \$500 at the beginning of Monday and steadily went up to about \$550 on Tuesday. There was no activity on Tuesday but on Wednesday the account went down to \$300. On Thursday the account remained at \$300 and then went up to about \$500 on Friday. On Saturday the account went back down to about \$400 where it stayed on Sunday. On each day where the balance changed, Jessie's graph shoes the balance changing continuously over that period.

   This is not realistic. When we make a withdrawal or deposit to a bank account, that amount is either deducted or added all at once rather than continuously over a period of time. Though not realistic, Jessie's graph does capture the overall up and down behavior of her bank balance.

2. According to the analysis in part (a), we could better capture the behavior of the bank balance by showing it changing (up or down) in jumps and then maintaining a stable value until the next transaction:

   

   This graph shows a deposit on Monday, a withdrawal on Wednesday, a deposit on Friday, and a withdrawal on Saturday. Notice also the closed and open points at the end of the intervals: for example, the closed point on Monday at about \$550 and the corresponding open point at \$500 indicate that this is the moment when the deposit was made so that the account balance has changed at this instant on Monday.