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| Section | Objectives |
|---|---|
| Topic 1: Linear Functions | - Interpret inputs and outputs of linear functions - Interpret rates of change for linear functions |
| Topic 2: Graphical Depictions | - Interpret asymptotes for situations - Interpret rates of change for situations - Interpret maximum and minimum for situations - Interpret inputs and outputs for situations - Interpret concavity for situations |
| Topic 3: Logistic Functions | - Interpret asymptotes for logistic functions - Interpret rates of change for logistic functions - Interpret inputs and outputs for logistic functions |
| Topic 4: Functions and Algebra of Functions | - Derive conclusions based on data - Derive conclusions based on notation - Derive conclusions based on graphs |
| Topic 5: Exponential Functions | - Interpret concavity for exponential functions - Interpret rates of change for exponential functions - Interpret inputs and outputs for exponential functions |
| Topic 6: Polynomial Functions | - Interpret rates of change for polynomial functions - Interpret inputs and outputs for polynomial functions - Interpret concavity for polynomial functions |
| Topic 7: Validity of Models | - Determine validity - Examine utility - Determine fit |
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質問 # 108
The graphed function n(t)represents the number of people, n, at a park thours after 8:00 a.m. The plotted points Aand Bhave coordinates (2#110)and (6#182).
Which statement gives the correct interpretation of the average rate of change of the number of people over the interval from point Ato point B?
正解：A
解説：
The average rate of change measures how much the output changes per one unit of input.
Here:
t= " hours after 8:00 a.m. "
and
n(t)= " number of people at the park "
The two given points are:
A=(2,110)
and
B=(6,182)
This means:
At t=2hours after 8:00 a.m., there were 110people at the park.
At t=6hours after 8:00 a.m., there were 182people at the park.
Now calculate the average rate of change:
" Average rate of change " =(182-110)/(6-2)
=72/4
=18
Because the value is positive, the number of people is increasing.
So the correct interpretation is:
" The number of people at the park increases at an average rate of 18 people per hour. "
質問 # 109
The exponential function
f(t)=2900(1.13)
t
represents the size of a bacteria population, where t is the time in hours.
How many bacteria are in the population when t=18?
正解：D
解説：
This is an exponential growth problem because the base 1.13 is greater than 1. The function starts with 2,900 bacteria and grows by a factor of 1.13 each hour, which represents a 13% hourly increase. To find the population after 18 hours, substitute t=18 into the function: f(18)=2900(1.13)
18
Evaluating the exponent and multiplying by 2900 gives approximately 26,170.38. Since bacteria are counted as whole organisms, this rounds to about 26,170 bacteria. The other answer choices do not match the exponential model at t=18. Therefore, the correct answer is C.
質問 # 110
The population of fish in a lake is changing according to the function
P(t)=462-35t
where tis the number of months since the beginning of the year and P(t)is the fish population at time t.
Which interpretation of the rate of change is correct?
正解：A
解説：
The function is:
P(t)=462-35t
This is a linear function in the form:
P(t)=mt+b
where:
m= " rate of change "
and
b= " initial value "
In this function:
m=-35
The negative sign means the fish population is decreasing.
The number 35tells us the amount of decrease per month.
So the fish population is decreasing by:
35 " fish per month "
The value 462is not the rate of change. It represents the starting fish population at the beginning of the year, when t=0:
P(0)=462
Therefore, the correct interpretation is:
" The number of fish in the lake is decreasing at a constant rate of 35 fish per month. " So the correct answer is:
# ( " D " )
質問 # 111
As a drone travels away from an operator, the function G(x)=24+47x models the distance, in yards, between the drone and the operator at x minutes. What is the value of G(1.9)?
正解：B
解説：
This is a linear function evaluation problem. The function G(x)=24+47x gives the drone's distance from the operator after x minutes. The constant 24 is the starting distance, and 47 is the rate of change in yards per minute. To evaluate G(1.9), substitute 1.9 for x. Then compute G(1.9)=24+47(1.9). Multiplying first gives
47×1.9=89.3. Adding 24 gives 113.3. Because the function measures distance in yards, the result means the drone is 113.3 yards from the operator after 1.9 minutes. The other options do not result from correctly substituting 1.9 into the function. Therefore, the correct answer is D.
質問 # 112
The graph shows the progress of a manufacturing team, modeling the number of widgets the team is able to produce each day since the team was formed.
How should the average rate of change from day 16 to day 18 be interpreted?
正解：A
解説：
The graph models the number of widgets the manufacturing team can produce each day.
The horizontal axis represents:
" Days "
The vertical axis represents:
" Widgets produced each day "
The two labeled points are:
(16#9.1)
and
(18#12.4)
This means:
On day 16, the team could produce about 9.1widgets per day.
On day 18, the team could produce about 12.4widgets per day.
The average rate of change is:
(12.4-9.1)/(18-16)
=3.3/2
=1.65
Rounded to the nearest tenth:
1.7
So from day 16 to day 18, the team's production increased by an average of about:
1.7 " widgets each day "
質問 # 113
......
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