Answer :
Let's analyze the data point by point using the function [tex]\( y = 5 \cdot 4^x \)[/tex] given by Julia.
1. Data Point: (0, 5)
- [tex]\( x = 0 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^0 = 5 \cdot 1 = 5 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
2. Data Point: (1, 20)
- [tex]\( x = 1 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^1 = 5 \cdot 4 = 20 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
3. Data Point: (2, 80)
- [tex]\( x = 2 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^2 = 5 \cdot 16 = 80 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
4. Data Point: (4, 320)
- [tex]\( x = 4 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^4 = 5 \cdot 256 = 1280 \)[/tex]
- The calculated [tex]\( y \)[/tex] does not match the given [tex]\( y \)[/tex] (given [tex]\( y = 320 \)[/tex]).
5. Data Point: (8, 640)
- [tex]\( x = 8 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^8 = 5 \cdot 65536 = 327680 \)[/tex]
- The calculated [tex]\( y \)[/tex] does not match the given [tex]\( y \)[/tex] (given [tex]\( y = 640 \)[/tex]).
### Conclusion
Based on the calculations, we observe that the given function [tex]\( y = 5 \cdot 4^x \)[/tex] correctly models the data points [tex]\( (0, 5) \)[/tex], [tex]\( (1, 20) \)[/tex], and [tex]\( (2, 80) \)[/tex], but it does not match the points [tex]\( (4, 320) \)[/tex] and [tex]\( (8, 640) \)[/tex].
Now we will determine which of the given statements is true:
1. Julia is correct because the distance starts at 5 feet and increases by a factor of 4.
- This statement is misleading because, although the distance starts at 5 feet and initially increases by a factor of 4, it does not increase consistently by a factor of 4 for all points.
2. Julia is correct because the function is true for [tex]\((0, 5)\)[/tex] and [tex]\((1, 20)\)[/tex].
- This statement is partially true but incomplete since the function is also true for [tex]\((2, 80)\)[/tex]. However, it fails for other points, so it is not completely true.
3. Julia is not correct because the function is not true for the point [tex]\((2, 80)\)[/tex].
- This statement is false because the function does correctly predict the point [tex]\((2, 80)\)[/tex].
4. Julia is not correct because the distance does not increase by a constant factor each minute.
- This statement is true. Although the function predicts an initial consistent multiplication factor, it does not hold for all data points provided in the table, indicating that Julia's model [tex]\( y = 5 \cdot 4^x \)[/tex] is not consistently accurate across all data points.
Thus, the correct statement is:
Julia is not correct because the distance does not increase by a constant factor each minute.
1. Data Point: (0, 5)
- [tex]\( x = 0 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^0 = 5 \cdot 1 = 5 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
2. Data Point: (1, 20)
- [tex]\( x = 1 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^1 = 5 \cdot 4 = 20 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
3. Data Point: (2, 80)
- [tex]\( x = 2 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^2 = 5 \cdot 16 = 80 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
4. Data Point: (4, 320)
- [tex]\( x = 4 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^4 = 5 \cdot 256 = 1280 \)[/tex]
- The calculated [tex]\( y \)[/tex] does not match the given [tex]\( y \)[/tex] (given [tex]\( y = 320 \)[/tex]).
5. Data Point: (8, 640)
- [tex]\( x = 8 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^8 = 5 \cdot 65536 = 327680 \)[/tex]
- The calculated [tex]\( y \)[/tex] does not match the given [tex]\( y \)[/tex] (given [tex]\( y = 640 \)[/tex]).
### Conclusion
Based on the calculations, we observe that the given function [tex]\( y = 5 \cdot 4^x \)[/tex] correctly models the data points [tex]\( (0, 5) \)[/tex], [tex]\( (1, 20) \)[/tex], and [tex]\( (2, 80) \)[/tex], but it does not match the points [tex]\( (4, 320) \)[/tex] and [tex]\( (8, 640) \)[/tex].
Now we will determine which of the given statements is true:
1. Julia is correct because the distance starts at 5 feet and increases by a factor of 4.
- This statement is misleading because, although the distance starts at 5 feet and initially increases by a factor of 4, it does not increase consistently by a factor of 4 for all points.
2. Julia is correct because the function is true for [tex]\((0, 5)\)[/tex] and [tex]\((1, 20)\)[/tex].
- This statement is partially true but incomplete since the function is also true for [tex]\((2, 80)\)[/tex]. However, it fails for other points, so it is not completely true.
3. Julia is not correct because the function is not true for the point [tex]\((2, 80)\)[/tex].
- This statement is false because the function does correctly predict the point [tex]\((2, 80)\)[/tex].
4. Julia is not correct because the distance does not increase by a constant factor each minute.
- This statement is true. Although the function predicts an initial consistent multiplication factor, it does not hold for all data points provided in the table, indicating that Julia's model [tex]\( y = 5 \cdot 4^x \)[/tex] is not consistently accurate across all data points.
Thus, the correct statement is:
Julia is not correct because the distance does not increase by a constant factor each minute.
