please help!!I've been staring at this forever!!
1. Every year, Annie makes a graph showing the amount in her bank account over time. Sketch a graph showing the amount in her bank account over time from the description given. Draw a separate graph for each description. Explain each of your graphs and why you think each is correct.
(Note: You should have 5 graphs and an explanation for each graph when you are finished.)
(a) The bank account started at $0 and then started increasing. Halfway through the year, the amount started decreasing until it reached $0.
(b) The bank account started with $0 and then increased at a constant rate over time.
(c) The bank account started with money in it and then decreased at a constant rate over time.
(d) The bank account started with money in it, and the amount stayed the same for a period of time.
(e) The bank account started with money in it. Then it increased, then it decreased, then it increased again, and then it started decreasing again.
2. Annie says all of the graphs in Problem 1 are functions.
(a) Do you agree with Annie that all of the graphs are functions? Support your answer with reasoning by explaining what you think in your own words.
(b) Annie's brother claims, “A graph of the balance in a bank account over time will always be a function.” Do you agree or disagree with him? Explain your answer with support.
(c) Annie's sister says to her brother, “I think you got it backwards. I think you meant to say that any function could represent the amount of money in a bank account.” Do you think Annie's sister's statement is true or false? Explain your answer with support.
If you've only been staring at it for hours, you need to just start thinking about it. Some sample graphs might be
(a)
http://www.wolframalpha.com/input/?i=50-%7Cx-50%7C+for+0%3C%3Dx%3C%3D100
(b)
http://www.wolframalpha.com/input/?i=x+for+0%3C%3Dx%3C%3D100
(c)
http://www.wolframalpha.com/input/?i=100-x+for+0%3C%3Dx%3C%3D100
(d)
http://www.wolframalpha.com/input/?i=y%3D100+for+0%3C%3Dx%3C%3D100
(e)
http://www.wolframalpha.com/input/?i=%7C%7C100-%7Cx-50%7C%7C-30%7C+for+0%3C%3Dx%3C%3D170
#2
(a) Annie is right - they all pass the vertical line test
(b) Agree for the same reason. At no time can the balance be two different values.
(c) sister is wrong, since money must be discrete values, and many functions are continuous. But, if you allow for continuous functions, then she is right, if you allow negative values.
we'll not get into discontinuous domains...
(a) For the description of the bank account starting at $0 and then increasing, and halfway through the year decreasing back to $0, the correct graph would be a positive sloping line that reaches a peak at the halfway mark and then slopes down back to $0. This is because the account starts with no money and then steadily increases until it reaches a maximum value, at which point it starts decreasing until it reaches $0.
(b) For the description of the bank account starting with $0 and increasing at a constant rate over time, the correct graph would be a straight line with a positive slope. This is because the account starts with no money, but then increases at a constant rate over time.
(c) For the description of the bank account starting with money and decreasing at a constant rate over time, the correct graph would be a straight line with a negative slope. This is because the account starts with some money, but then steadily decreases at a constant rate over time.
(d) For the description of the bank account starting with money and the amount staying the same for a period of time, the correct graph would be a horizontal line. This is because the account starts with some money and then maintains that same amount over time.
(e) For the description of the bank account starting with money, then increasing, then decreasing, then increasing again, and finally decreasing again, the correct graph would have a pattern of ups and downs. It would start at some positive value, then increase to a peak, decrease to a lower value, increase to a higher peak, and finally decrease again. This could be shown as a series of peaks and valleys on a graph.
2. (a) No, I don't agree with Annie that all of the graphs are functions. A function is a mathematical relationship where each input (time in this case) corresponds to exactly one output (the amount in the bank account). However, in the case of the graph described in (e), there are multiple outputs for some inputs (time periods), which violates the definition of a function.
(b) I disagree with Annie's brother that a graph of the balance in a bank account over time will always be a function. As mentioned in my previous answer, the graph described in (e) demonstrates that not all bank account balances can be represented as functions, as there can be multiple balances for the same point in time.
(c) I disagree with Annie's sister's statement that any function could represent the amount of money in a bank account. While a function can definitely be used to represent the relationship between time and bank account balance, not all functions will be suitable. For example, a function that oscillates or has multiple outputs for the same input (time) would not accurately represent a realistic bank account balance.
(a) For the description given, the graph would look like an increasing line for the first half of the year, and then a decreasing line for the second half of the year until it reaches $0. This graph would represent the bank account starting at $0 and then increasing, but halfway through the year, the amount started decreasing until it reached $0.
(b) For this description, the graph would be a straight line starting at $0 and increasing at a constant rate over time. This graph represents the bank account starting with $0 and then increasing at a constant rate over time.
(c) In this case, the graph would be a straight line starting with a positive amount of money and decreasing at a constant rate over time. This graph represents the bank account starting with money in it and then decreasing at a constant rate over time.
(d) This description suggests that the graph would be a horizontal line at a certain amount, representing the bank account starting with money in it and the amount staying the same for a period of time.
(e) This description suggests a more complex pattern for the graph. It starts with an increasing trend, then decreases, then increases again, and finally starts decreasing again. This graph represents the bank account starting with money in it, then it increases, decreases, increases again, and finally starts decreasing again.
2. (a) I agree with Annie that all of the graphs in Problem 1 are functions. A function is a relation where each input has a unique output. In all the given descriptions, there is a clear correspondence between time and the amount in the bank account, meaning there is only one output (bank account balance) for each input (time).
(b) I agree with Annie's brother that a graph of the balance in a bank account over time will always be a function. As mentioned before, each point on the graph represents a unique balance at a specific time. There will not be two different balances for the same point in time.
(c) I disagree with Annie's sister's statement that any function could represent the amount of money in a bank account. While it is true that any function could represent a mathematical relationship, not all functions would accurately represent the amount of money in a bank account. The function needs to follow the constraints and conditions of real-world banking scenarios, such as not allowing negative balances or sudden drastic changes in the amount.
1. (a) Graph explanation:
For this graph, the amount in Annie's bank account starts at $0 and then starts increasing. Halfway through the year, the amount starts decreasing until it reaches $0. This can be represented by a line that slopes upward from $0 at the beginning, reaches a peak, and then slopes downward back to $0 at the end. This graph accurately represents the given description.
(b) Graph explanation:
In this scenario, the bank account starts at $0 and then increases at a constant rate over time. This can be represented by a straight line that starts at $0 and has a positive slope, indicating a constant rate of increase over time. This graph accurately represents the given description.
(c) Graph explanation:
In this case, the bank account starts with money in it and then decreases at a constant rate over time. This can be represented by a straight line that starts at a specific point above $0 and has a negative slope, indicating a constant rate of decrease over time. This graph accurately represents the given description.
(d) Graph explanation:
Here, the bank account starts with money in it, and the amount stays the same for a period of time. This can be represented by a horizontal line at a specific point, indicating no change in the amount over time. This graph accurately represents the given description.
(e) Graph explanation:
In this situation, the bank account starts with money in it, then increases, then decreases, then increases again, and finally starts decreasing again. This can be represented by a graph with a combination of increasing and decreasing lines or curves, showing the different changes in the amount over time. This graph accurately represents the given description.
2. (a) Yes, I agree with Annie that all of the graphs are functions. A function is a mathematical relationship where each input (x-value) is associated with a unique output (y-value). In each of the graphs, for any given time (x-value), there is only one corresponding amount in the bank account (y-value). Thus, the graphs satisfy the definition of a function.
(b) I agree with Annie's brother that a graph of the balance in a bank account over time will always be a function. This is because, according to the definition of a function, every x-value (time) will have only one corresponding y-value (balance in the bank account). In other words, at any specific point in time, there can only be one amount in the bank account.
(c) Annie's sister's statement is false. The sister claims that any function could represent the amount of money in a bank account. However, not all functions can represent the amount of money in a bank account over time. For example, a function that oscillates between positive and negative values or a function that has multiple outputs for the same input would not accurately represent the balance in a bank account, as the balance cannot be negative and there can only be one balance at any given point in time.