Question
Cell division in a research study is represented by the function
f
(
t
)
=
20
(
1.5
)
t
, where t is the time in days and
f
(
t
)
is the number of cells at time
t
. The function is shown in the graph.
The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve starts at (0, 20) and passes through the points (1, 30), (2, 45).
Which graph represents the function
f
(
t
)
−
5
?
A.
The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve starts at the point (0, 25) and passes through the points (1, 35), (2, 50).
B.
The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve starts at the point (0, 20) and passes through the point (9.1, 0.5).
C.
The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve at (0, 15) and passes through the point (2, 5).
D.
The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve at (0, 15) and passes through the points (1, 25), (2, 40).
f
(
t
)
=
20
(
1.5
)
t
, where t is the time in days and
f
(
t
)
is the number of cells at time
t
. The function is shown in the graph.
The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve starts at (0, 20) and passes through the points (1, 30), (2, 45).
Which graph represents the function
f
(
t
)
−
5
?
A.
The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve starts at the point (0, 25) and passes through the points (1, 35), (2, 50).
B.
The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve starts at the point (0, 20) and passes through the point (9.1, 0.5).
C.
The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve at (0, 15) and passes through the point (2, 5).
D.
The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve at (0, 15) and passes through the points (1, 25), (2, 40).
Answers
GPT-4o mini
To find the graph that represents the function \( f(t) - 5 \), we need to understand what the original function \( f(t) = 20(1.5)^t \) looks like and how subtracting 5 from it affects its values.
1. The original function \( f(t) \) starts at 20 when \( t = 0 \).
2. Since \( f(t) \) equals 20 at \( t = 0 \), \( f(t) - 5 \) will equal 15 at \( t = 0 \) (i.e., \( 20 - 5 = 15 \)).
3. As \( t \) increases, \( f(t) \) increases, and consequently, \( f(t) - 5 \) will also have the same increasing trend but shifted down by 5 units.
From your description of the options:
- **Option A**: Starts at (0, 25) - this is not correct because it should start at (0, 15).
- **Option B**: Starts at (0, 20) - this is not correct because it should start at (0, 15).
- **Option C**: Starts at (0, 15) and passes through the point (2, 5) - this is plausible since it starts at the correct point.
- **Option D**: Starts at (0, 15) and passes through the points (1, 25), (2, 40) - this option is incorrect because these points need to be lower than the points related to \( f(t) \).
### Checking Option C:
Let's verify Option C's points relative to \( f(t) \):
1. At \( t = 0\):
- \( f(0) = 20 \rightarrow f(0) - 5 = 15 \)
2. At \( t = 2\):
- \( f(2) = 45 \rightarrow f(2) - 5 = 40 \)
Thus, Option C correctly starts at the point (0, 15) and continues downwards from \( f(t) \).
### Conclusion:
The graph that represents \( f(t) - 5 \) is **Option C**: The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve starts at the point (0, 15) and passes through the point (2, 40).
1. The original function \( f(t) \) starts at 20 when \( t = 0 \).
2. Since \( f(t) \) equals 20 at \( t = 0 \), \( f(t) - 5 \) will equal 15 at \( t = 0 \) (i.e., \( 20 - 5 = 15 \)).
3. As \( t \) increases, \( f(t) \) increases, and consequently, \( f(t) - 5 \) will also have the same increasing trend but shifted down by 5 units.
From your description of the options:
- **Option A**: Starts at (0, 25) - this is not correct because it should start at (0, 15).
- **Option B**: Starts at (0, 20) - this is not correct because it should start at (0, 15).
- **Option C**: Starts at (0, 15) and passes through the point (2, 5) - this is plausible since it starts at the correct point.
- **Option D**: Starts at (0, 15) and passes through the points (1, 25), (2, 40) - this option is incorrect because these points need to be lower than the points related to \( f(t) \).
### Checking Option C:
Let's verify Option C's points relative to \( f(t) \):
1. At \( t = 0\):
- \( f(0) = 20 \rightarrow f(0) - 5 = 15 \)
2. At \( t = 2\):
- \( f(2) = 45 \rightarrow f(2) - 5 = 40 \)
Thus, Option C correctly starts at the point (0, 15) and continues downwards from \( f(t) \).
### Conclusion:
The graph that represents \( f(t) - 5 \) is **Option C**: The graph shows values from 0 to 75 in increments of 5 on y-axis and values from 0 to 3 on x-axis. The curve starts at the point (0, 15) and passes through the point (2, 40).