Cumulative Assessment
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Exercise 4 Page 404
a To determine the type of function represented by the data in each table, you have two methods to apply. You will either find the differences between the y-values or you will take the ratio of the y-values.
b Use your results in Part A.
c Begin by writing a function for each data.
a Cheking Account: Linear
Savings Account: Exponential
b Checking Account: Linear growth, see solution.
Savings Account: Exponential growth, see solution.
c See solution.
Practice makes perfect
a To determine the type of function represented by the data in each table, we have two methods to apply. We will either find the differences between the y-values or we will take the ratio of the y-values.
- If the first difference is the same value, the model will be linear.
- If the second difference is the same value, the model will be quadratic.
- If the ratio of the y-values is the same, then the data is modeled by an exponential function.
| Checking Account | ||||
|---|---|---|---|---|
| t | 1 | 2 | 3 | 4 |
| A | 5000 | 5110 | 5220 | 5330 |
Since the x-values increase by the same amount of each time, we can begin by examining the first difference between the y-values. 5000 +110 ⟶ 5110 +110 ⟶ 5220 +110 ⟶ 5330 The first difference is the same value, so the model is linear. Next, we will examine the table that represents the savings account.
| Savings Account | ||||
|---|---|---|---|---|
| t | 1 | 2 | 3 | 4 |
| A | 5000 | 5100 | 5202 | 5306.04 |
Again, the x-values increase by the same amount over regular intervals. However, it seems like neither the first difference nor the second difference will be the same value because after 4 years, the amount in the account is $5306.04. Therefore, let's first check the ratio. 5000 *1.02 ⟶ 5100 *1.02 ⟶ 5202 *1.02 ⟶ 5306.04 As we can see, the ratio between the y-values is the same value, so the data can be modeled by an exponential function.
b The amount in the checking account grows by the same amount in each time, so it is a linear growth. On the other hand, the amount in the savings account grows by a fixed percent in each time. Therefore, it is an exponential growth.
c Let's first write a function for each set of data. The amount in the checking account grows by the same amount of $110 and the initial amount is $5000. From here, we can write a function in slope-intercept form.
For the savings account, the initial amount is also $5000 and it grows by a factor of 1.02. With this information we can write an exponential function for the data. Savings Account y=5000( 1.02)^t Now that we have the functions, we can compare the amount in each account after 10 years and after 15 years.
| t | 110t+5000 | Amount in the Checking Account | 5000(1.02)^t | |
|---|---|---|---|---|
| 10 | 110( 10)+5000 | $6100 | 5000(1.02)^(10) | $6095 |
| 15 | 110( 15)+5000 | $6650 | 5000(1.02)^(15) | $6729 |
Looking at the table, we can say that the amount in the checking account will be greater after 10 years. However, after 15 years the amount in the savings account will be greater.
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