Question
A customer purchases an electronic device for $650. Each year the value of the electronic device decreases by 30%.
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?
Question 2 options:
The y-intercept of the graph is 650.
The y-intercept of the graph is 455.
The graph has a horizontal asymptote at y = 30.
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?
Question 2 options:
The y-intercept of the graph is 650.
The y-intercept of the graph is 455.
The graph has a horizontal asymptote at y = 30.
Answers
GPT-4o mini
To analyze the depreciation of the electronic device, we can use the formula for exponential decay:
\[ y = P \cdot (1 - r)^x \]
where:
- \( y \) is the value of the device after \( x \) years,
- \( P \) is the initial value ($650),
- \( r \) is the rate of depreciation (30% or 0.30),
- \( x \) is the number of years.
In this case, substituting the values gives us:
\[ y = 650 \cdot (1 - 0.30)^x \]
\[ y = 650 \cdot (0.70)^x \]
### Analyzing the statements:
1. **The y-intercept of the graph is 650.**
- The y-intercept occurs when \( x = 0 \). Plugging in \( x = 0 \):
\[
y = 650 \cdot (0.70)^0 = 650 \cdot 1 = 650
\]
- This statement is **true**.
2. **The y-intercept of the graph is 455.**
- As shown above, the y-intercept is not 455, so this statement is **false**.
3. **The graph has a horizontal asymptote at y = 30.**
- Since the value of the device is decreasing but never reaches zero, the horizontal asymptote is at \( y = 0 \), not at \( y = 30 \). This statement is **false**.
### Conclusion:
The correct answer is: **The y-intercept of the graph is 650.**
\[ y = P \cdot (1 - r)^x \]
where:
- \( y \) is the value of the device after \( x \) years,
- \( P \) is the initial value ($650),
- \( r \) is the rate of depreciation (30% or 0.30),
- \( x \) is the number of years.
In this case, substituting the values gives us:
\[ y = 650 \cdot (1 - 0.30)^x \]
\[ y = 650 \cdot (0.70)^x \]
### Analyzing the statements:
1. **The y-intercept of the graph is 650.**
- The y-intercept occurs when \( x = 0 \). Plugging in \( x = 0 \):
\[
y = 650 \cdot (0.70)^0 = 650 \cdot 1 = 650
\]
- This statement is **true**.
2. **The y-intercept of the graph is 455.**
- As shown above, the y-intercept is not 455, so this statement is **false**.
3. **The graph has a horizontal asymptote at y = 30.**
- Since the value of the device is decreasing but never reaches zero, the horizontal asymptote is at \( y = 0 \), not at \( y = 30 \). This statement is **false**.
### Conclusion:
The correct answer is: **The y-intercept of the graph is 650.**