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| Section | Objectives |
|---|---|
| Exponential Functions | - Interpret concavity for exponential functions - Interpret rates of change for exponential functions - Interpret inputs and outputs for exponential functions |
| Logistic Functions | - Interpret rates of change for logistic functions - Interpret inputs and outputs for logistic functions - Interpret asymptotes for logistic functions |
| Graphical Depictions | - Interpret inputs and outputs for situations - Interpret asymptotes for situations - Interpret maximum and minimum for situations - Interpret concavity for situations - Interpret rates of change for situations |
| Linear Functions | - Interpret inputs and outputs of linear functions - Interpret rates of change for linear functions |
| Validity of Models | - Determine validity - Determine fit - Examine utility |
| Functions and Algebra of Functions | - Derive conclusions based on data - Derive conclusions based on graphs - Derive conclusions based on notation |
| Polynomial Functions | - Interpret rates of change for polynomial functions - Interpret concavity for polynomial functions - Interpret inputs and outputs for polynomial functions |
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質問 # 142
The population of fish in a lake is changing according to the function P(t)=31t+438, where t is 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：D
解説：
The function P(t)=31t+438 is linear and follows the form P(t)=mt+b. In this form, m is the rate of change and b is the initial value. Here, the coefficient of t is 31, so the fish population changes by 31 fish per month.
Because 31 is positive, the population is increasing rather than decreasing. The number 438 is the initial fish population at the start of the year, not the rate of change. Therefore, the correct interpretation is that the number of fish in the lake is increasing at a constant rate of 31 fish per month. That matches option A.
質問 # 143
A contractor charges a flat rate plus a fixed rate per hour to come and make repairs. For a job that takes 3 hours, the contractor charges $220. For a job that takes 5 hours, the contractor charges $340.
Which function correctly represents the total charge Cfor a job that takes thours?
正解：D
解説：
The contractor charges a flat rate plus a fixed rate per hour, so this situation is modeled by a linear function:
C(t)=mt+b
where:
m= " hourly rate "
and
b= " flat rate "
We are given two points:
(3#220)
and
(5#340)
These mean:
A 3-hour job costs $220.
A 5-hour job costs $340.
First, find the hourly rate using slope:
m=(340-220)/(5-3)
m=120/2
m=60
So the contractor charges:
$60 " per hour "
Now the function has the form:
C(t)=60t+b
Use the point (3#220)to find the flat rate:
220=60(3)+b
220=180+b
b=40
So the function is:
C(t)=60t+40
Check with t=5:
C(5)=60(5)+40
C(5)=300+40
C(5)=340
This matches the given information.
質問 # 144
A person makes down quilts to sell.
The graph shows the functions that model the cost and revenue.
How many down quilts need to sell to break even/start making a profit?
正解：B
解説：
This question asks us to interpret a graph showing cost and revenue.
In Applied Algebra, the break-even point is where:
" Revenue " = " Cost "
On a graph, this happens where the revenue line and the cost line intersect.
From the graph:
The solid blue line represents revenue.
The dashed blue line represents cost.
The break-even point occurs where these two lines cross.
Looking carefully at the graph, the two lines intersect at approximately:
x=6
The horizontal axis represents the number of units, meaning the number of down quilts sold.
So the person needs to sell about:
6 " quilts "
to break even.
After selling more than 6 quilts, the revenue line is above the cost line, meaning the person begins making a profit.
質問 # 145
Based on collected data, the number of employees at a firm can be modeled using the function f(x)=10× # 1.09 #