Question
the graph shows the number of game systems sold since 2015. Based on this information, which function best models the number of game systems sold in millions x years since 2015?
(0,30.5) (1,21.35) (2,14.95)
a y=21.35(0.7)^x
b y=30.5(21.35)^x
c y=30.5(1.3)^x
d y=30.5(0.7)^x
(0,30.5) (1,21.35) (2,14.95)
a y=21.35(0.7)^x
b y=30.5(21.35)^x
c y=30.5(1.3)^x
d y=30.5(0.7)^x
Answers
Answered by
GPT-4o mini
To determine which function best models the number of game systems sold based on the provided points \((0,30.5)\), \((1,21.35)\), and \((2,14.95)\), we need to analyze the values given in the choices provided:
1. **Point (0, 30.5)** indicates that in 2015 (when \(x = 0\)), the number of game systems sold was 30.5 million. This means the equation must start with \(30.5\) as the coefficient of the initial year.
2. **Point (1, 21.35)** indicates that in 2016 (when \(x = 1\)), the number of game systems sold was 21.35 million.
3. **Point (2, 14.95)** indicates that in 2017 (when \(x = 2\)), the number of game systems sold was 14.95 million.
Now let's evaluate the options based on the form of the points:
### Analyze Each Option
1. **Option (A)** \( y = 21.35(0.7)^x \)
- At \(x=0\), \(y = 21.35(0.7)^0 = 21.35\) (not correct for \(x=0\)).
2. **Option (B)** \( y = 30.5(21.35)^x \)
- At \(x=0\), \(y = 30.5(21.35)^0 = 30.5\) (correct).
- However, the function grows rapidly and does not align with \(x=1\) and \(x=2\) values.
3. **Option (C)** \( y = 30.5(1.3)^x \)
- At \(x=0\), \(y = 30.5(1.3)^0 = 30.5\) (correct).
- This will also grow rapidly, not matching \(x=1\) and \(x=2\) values.
4. **Option (D)** \( y = 30.5(0.7)^x \)
- At \(x=0\), \(y = 30.5(0.7)^0 = 30.5\) (correct).
- At \(x=1\), \(y = 30.5(0.7)^1 = 30.5(0.7) = 21.35\) (correct).
- At \(x=2\), \(y = 30.5(0.7)^2 = 30.5(0.49) = 14.95\) (correct).
### Conclusion
The function that best models the number of game systems sold in millions since 2015, considering all data points provided, is:
**Option (D)** \( y = 30.5(0.7)^x \).
1. **Point (0, 30.5)** indicates that in 2015 (when \(x = 0\)), the number of game systems sold was 30.5 million. This means the equation must start with \(30.5\) as the coefficient of the initial year.
2. **Point (1, 21.35)** indicates that in 2016 (when \(x = 1\)), the number of game systems sold was 21.35 million.
3. **Point (2, 14.95)** indicates that in 2017 (when \(x = 2\)), the number of game systems sold was 14.95 million.
Now let's evaluate the options based on the form of the points:
### Analyze Each Option
1. **Option (A)** \( y = 21.35(0.7)^x \)
- At \(x=0\), \(y = 21.35(0.7)^0 = 21.35\) (not correct for \(x=0\)).
2. **Option (B)** \( y = 30.5(21.35)^x \)
- At \(x=0\), \(y = 30.5(21.35)^0 = 30.5\) (correct).
- However, the function grows rapidly and does not align with \(x=1\) and \(x=2\) values.
3. **Option (C)** \( y = 30.5(1.3)^x \)
- At \(x=0\), \(y = 30.5(1.3)^0 = 30.5\) (correct).
- This will also grow rapidly, not matching \(x=1\) and \(x=2\) values.
4. **Option (D)** \( y = 30.5(0.7)^x \)
- At \(x=0\), \(y = 30.5(0.7)^0 = 30.5\) (correct).
- At \(x=1\), \(y = 30.5(0.7)^1 = 30.5(0.7) = 21.35\) (correct).
- At \(x=2\), \(y = 30.5(0.7)^2 = 30.5(0.49) = 14.95\) (correct).
### Conclusion
The function that best models the number of game systems sold in millions since 2015, considering all data points provided, is:
**Option (D)** \( y = 30.5(0.7)^x \).