Question
The table below shows the number of customers who downloaded movies in the last 4 years from a popular internet media site.
Year Customers (millions)
2011 0.8
2012 1.44
2013 2.592
2014 4.6656
If the year 2011 is
x
=
0
, which of the following represent the function described in this table? Choose all that are correct.
A.
The graph shows horizontal axis, x, titled 'year', from 0 to 4 and vertical axis M of x, titled 'customers in millions', from 0 to 6 has a curve starting from (0, 0.8) and sloping down.
B.
M
(
x
)
=
0.8
(
1.8
)
x
C.
M
(
x
)
=
0.6
x
+
0.8
D.
The graph shows horizontal axis, x, titled 'year', from 0 to 4 and vertical axis M of x, titled 'customers in millions', from 0 to 6 has an upward sloping line passing through the points (0, 0.8), and (2, 2).
E.
The graph shows horizontal axis, x, titled 'year', from 0 to 4 and vertical axis M of x, titled 'customers in millions', from 0 to 6 has a curve starting from (0, 0.8) and sloping up passing through the points (1, 1.44), (2, 2.592), and (3, 4.6656).
Year Customers (millions)
2011 0.8
2012 1.44
2013 2.592
2014 4.6656
If the year 2011 is
x
=
0
, which of the following represent the function described in this table? Choose all that are correct.
A.
The graph shows horizontal axis, x, titled 'year', from 0 to 4 and vertical axis M of x, titled 'customers in millions', from 0 to 6 has a curve starting from (0, 0.8) and sloping down.
B.
M
(
x
)
=
0.8
(
1.8
)
x
C.
M
(
x
)
=
0.6
x
+
0.8
D.
The graph shows horizontal axis, x, titled 'year', from 0 to 4 and vertical axis M of x, titled 'customers in millions', from 0 to 6 has an upward sloping line passing through the points (0, 0.8), and (2, 2).
E.
The graph shows horizontal axis, x, titled 'year', from 0 to 4 and vertical axis M of x, titled 'customers in millions', from 0 to 6 has a curve starting from (0, 0.8) and sloping up passing through the points (1, 1.44), (2, 2.592), and (3, 4.6656).
Answers
GPT-4o mini
To determine which options correctly represent the data provided in the table, let's analyze the given years and corresponding customer downloads:
- \(M(0) = 0.8\) (for the year 2011)
- \(M(1) = 1.44\) (for the year 2012)
- \(M(2) = 2.592\) (for the year 2013)
- \(M(3) = 4.6656\) (for the year 2014)
From the values, we can observe that the number of customers downloading movies is increasing significantly each year, suggesting an exponential growth pattern.
Let's evaluate each option:
**A.** The description of the graph is incorrect because it's stated to be sloping down, while the data shows that the downloads are increasing.
**B.** The function \(M(x) = 0.8(1.8)^x\) is a candidate for an exponential growth function. However, we notice that it does not accurately represent the specific values from the table. Testing this against the actual data:
- For \(x = 1\): \(0.8(1.8)^1 = 1.44\) (correct)
- For \(x = 2\): \(0.8(1.8)^2 = 2.592\) (correct)
- For \(x = 3\): \(0.8(1.8)^3 = 4.6656\) (correct)
This fits the trend, making it valid.
**C.** The equation \(M(x) = 0.6x + 0.8\) is a linear function. Given the exponential nature of the data points shown (growing rapidly), this is not a correct representation.
**D.** Similarly, this description indicates a linear function, which is again incorrect as the data suggests an exponential growth pattern.
**E.** This option suggests that the graph starts at (0, 0.8) and curves upward, passing through the points (1, 1.44), (2, 2.592), and (3, 4.6656). This correctly describes the nature of the growth observed in the data, making it true.
Thus, the correct options from the list given the analysis above are:
- **B** (the exponential function that fits the data correctly)
- **E** (correct description of the graphical representation and points of the data)
Final answers: **B and E**.
- \(M(0) = 0.8\) (for the year 2011)
- \(M(1) = 1.44\) (for the year 2012)
- \(M(2) = 2.592\) (for the year 2013)
- \(M(3) = 4.6656\) (for the year 2014)
From the values, we can observe that the number of customers downloading movies is increasing significantly each year, suggesting an exponential growth pattern.
Let's evaluate each option:
**A.** The description of the graph is incorrect because it's stated to be sloping down, while the data shows that the downloads are increasing.
**B.** The function \(M(x) = 0.8(1.8)^x\) is a candidate for an exponential growth function. However, we notice that it does not accurately represent the specific values from the table. Testing this against the actual data:
- For \(x = 1\): \(0.8(1.8)^1 = 1.44\) (correct)
- For \(x = 2\): \(0.8(1.8)^2 = 2.592\) (correct)
- For \(x = 3\): \(0.8(1.8)^3 = 4.6656\) (correct)
This fits the trend, making it valid.
**C.** The equation \(M(x) = 0.6x + 0.8\) is a linear function. Given the exponential nature of the data points shown (growing rapidly), this is not a correct representation.
**D.** Similarly, this description indicates a linear function, which is again incorrect as the data suggests an exponential growth pattern.
**E.** This option suggests that the graph starts at (0, 0.8) and curves upward, passing through the points (1, 1.44), (2, 2.592), and (3, 4.6656). This correctly describes the nature of the growth observed in the data, making it true.
Thus, the correct options from the list given the analysis above are:
- **B** (the exponential function that fits the data correctly)
- **E** (correct description of the graphical representation and points of the data)
Final answers: **B and E**.