Question
A customer purchases an electronic device for $200. Each year the value of the device decreases by 5%. Which statement is true about the graph that represents the relationship between the value of the device in dollars, y, and the number of years, x, since the device was purchased?
The graph has a horizontal asymptote at x = - 1
The y-intercept of the graph is 200
The graph has a vertical asymptote at x = - 1
The y-intercept of the graph is 100
The graph has a horizontal asymptote at x = - 1
The y-intercept of the graph is 200
The graph has a vertical asymptote at x = - 1
The y-intercept of the graph is 100
Answers
GPT-4o mini
To determine the true statements about the value of the electronic device as it depreciates over time, let's analyze the situation:
1. **Depreciation Model**: The value of the device decreases by 5% each year. This can be modeled by the equation:
\[
y = 200 \times (1 - 0.05)^x = 200 \times (0.95)^x
\]
where \(y\) is the value of the device after \(x\) years.
2. **Y-intercept**: The y-intercept of a function is found by setting \(x = 0\). Substituting \(x = 0\) into the equation:
\[
y = 200 \times (0.95)^0 = 200 \times 1 = 200
\]
Therefore, the y-intercept of the graph is 200.
3. **Horizontal Asymptote**: As \(x\) approaches infinity, \((0.95)^x\) approaches 0. Thus, the value of \(y\) approaches 0, meaning there is a horizontal asymptote at \(y = 0\), not at \(x = -1\).
4. **Vertical Asymptote**: There are no vertical asymptotes in this scenario, as vertical asymptotes occur in rational functions where the denominator approaches zero. The current function is an exponential decay function and does not exhibit vertical asymptotic behavior.
Now let's evaluate the given statements:
- **The graph has a horizontal asymptote at x = -1**: False. The horizontal asymptote is at \(y = 0\).
- **The y-intercept of the graph is 200**: True. As shown, the y-intercept is indeed 200.
- **The graph has a vertical asymptote at x = -1**: False. There are no vertical asymptotes in this function.
- **The y-intercept of the graph is 100**: False. As shown, the y-intercept is 200.
Therefore, the true statement is:
**The y-intercept of the graph is 200.**
1. **Depreciation Model**: The value of the device decreases by 5% each year. This can be modeled by the equation:
\[
y = 200 \times (1 - 0.05)^x = 200 \times (0.95)^x
\]
where \(y\) is the value of the device after \(x\) years.
2. **Y-intercept**: The y-intercept of a function is found by setting \(x = 0\). Substituting \(x = 0\) into the equation:
\[
y = 200 \times (0.95)^0 = 200 \times 1 = 200
\]
Therefore, the y-intercept of the graph is 200.
3. **Horizontal Asymptote**: As \(x\) approaches infinity, \((0.95)^x\) approaches 0. Thus, the value of \(y\) approaches 0, meaning there is a horizontal asymptote at \(y = 0\), not at \(x = -1\).
4. **Vertical Asymptote**: There are no vertical asymptotes in this scenario, as vertical asymptotes occur in rational functions where the denominator approaches zero. The current function is an exponential decay function and does not exhibit vertical asymptotic behavior.
Now let's evaluate the given statements:
- **The graph has a horizontal asymptote at x = -1**: False. The horizontal asymptote is at \(y = 0\).
- **The y-intercept of the graph is 200**: True. As shown, the y-intercept is indeed 200.
- **The graph has a vertical asymptote at x = -1**: False. There are no vertical asymptotes in this function.
- **The y-intercept of the graph is 100**: False. As shown, the y-intercept is 200.
Therefore, the true statement is:
**The y-intercept of the graph is 200.**