Question
The amount of money in a savings account over time is given in the table below. Answer the questions below to explain what kind of function would better model the data, linear or exponential.
Number of Years Since Account Opened, x 11 22 33 44
Money in account ($), f(x) 179, point, 9, 2179.92 217, point, 6, 3217.63 259, point, 0, 8259.08 312, point, 1, 5312.15
Answer
Attempt 1 out of 2
function would better model the data because as xx increases, the yy values change
. The
of this function is approximately
Number of Years Since Account Opened, x 11 22 33 44
Money in account ($), f(x) 179, point, 9, 2179.92 217, point, 6, 3217.63 259, point, 0, 8259.08 312, point, 1, 5312.15
Answer
Attempt 1 out of 2
function would better model the data because as xx increases, the yy values change
. The
of this function is approximately
Answers
GPT-4o mini
To determine whether a linear or exponential function better models the data provided, we can analyze the pattern of the values.
### Step 1: Examine the values
- At \( x = 11 \): \( f(x) = 1799.79 \)
- At \( x = 22 \): \( f(x) = 2179.92 \)
- At \( x = 33 \): \( f(x) = 2590.82 \)
- At \( x = 44 \): \( f(x) = 3121.15 \)
### Step 2: Determine the rate of change
**Linear Functions**: In linear models, the rate of change (slope) between consecutive values is constant. Let's calculate the differences between the \( f(x) \) values:
1. From \( x = 11 \) to \( x = 22 \):
\[
2179.92 - 1799.79 = 380.13
\]
2. From \( x = 22 \) to \( x = 33 \):
\[
2590.82 - 2179.92 = 410.90
\]
3. From \( x = 33 \) to \( x = 44 \):
\[
3121.15 - 2590.82 = 530.33
\]
The differences of money in the account are changing: \( 380.13 \), \( 410.90 \), and \( 530.33 \). Since the differences are not constant, this suggests that a linear function does not adequately describe the data.
### Step 3: Check for exponential growth
**Exponential Functions**: In exponential models, the ratio of consecutive terms tends to be consistent. Let's compute the ratios:
1. From \( x = 11 \) to \( x = 22 \):
\[
\frac{2179.92}{1799.79} \approx 1.21
\]
2. From \( x = 22 \) to \( x = 33 \):
\[
\frac{2590.82}{2179.92} \approx 1.19
\]
3. From \( x = 33 \) to \( x = 44 \):
\[
\frac{3121.15}{2590.82} \approx 1.20
\]
The ratios are approximately consistent (around 1.2), indicating that as \( x \) increases, \( f(x) \) grows at a rate that is proportional to its current size—this supports an exponential model.
### Conclusion
From the analysis, we conclude that an **exponential function** better models the data because as \( x \) increases, the \( y \) values change in a multiplicative manner rather than a fixed additive manner. The rate of growth is consistent across intervals but varies in absolute terms, characteristic of exponential growth.
### Final Response
**Answer:** An **exponential function** would better model the data because as \( x \) increases, the \( y \) values change multiplicatively. The growth factor of this function is approximately **1.2**.
### Step 1: Examine the values
- At \( x = 11 \): \( f(x) = 1799.79 \)
- At \( x = 22 \): \( f(x) = 2179.92 \)
- At \( x = 33 \): \( f(x) = 2590.82 \)
- At \( x = 44 \): \( f(x) = 3121.15 \)
### Step 2: Determine the rate of change
**Linear Functions**: In linear models, the rate of change (slope) between consecutive values is constant. Let's calculate the differences between the \( f(x) \) values:
1. From \( x = 11 \) to \( x = 22 \):
\[
2179.92 - 1799.79 = 380.13
\]
2. From \( x = 22 \) to \( x = 33 \):
\[
2590.82 - 2179.92 = 410.90
\]
3. From \( x = 33 \) to \( x = 44 \):
\[
3121.15 - 2590.82 = 530.33
\]
The differences of money in the account are changing: \( 380.13 \), \( 410.90 \), and \( 530.33 \). Since the differences are not constant, this suggests that a linear function does not adequately describe the data.
### Step 3: Check for exponential growth
**Exponential Functions**: In exponential models, the ratio of consecutive terms tends to be consistent. Let's compute the ratios:
1. From \( x = 11 \) to \( x = 22 \):
\[
\frac{2179.92}{1799.79} \approx 1.21
\]
2. From \( x = 22 \) to \( x = 33 \):
\[
\frac{2590.82}{2179.92} \approx 1.19
\]
3. From \( x = 33 \) to \( x = 44 \):
\[
\frac{3121.15}{2590.82} \approx 1.20
\]
The ratios are approximately consistent (around 1.2), indicating that as \( x \) increases, \( f(x) \) grows at a rate that is proportional to its current size—this supports an exponential model.
### Conclusion
From the analysis, we conclude that an **exponential function** better models the data because as \( x \) increases, the \( y \) values change in a multiplicative manner rather than a fixed additive manner. The rate of growth is consistent across intervals but varies in absolute terms, characteristic of exponential growth.
### Final Response
**Answer:** An **exponential function** would better model the data because as \( x \) increases, the \( y \) values change multiplicatively. The growth factor of this function is approximately **1.2**.