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| Section | Objectives |
|---|---|
| Foundations of Algebra | - Equations and Inequalities
|
| Polynomial and Rational Expressions | - Rational Expressions
|
| Linear Functions and Systems | - Linear Equations
|
| Functions and Graphs | - Graphing Functions
|
| Exponential and Logarithmic Concepts | - Exponential Functions
|
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問題 #104
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?
答案：A
解題說明：
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.
問題 #105
The function P(t)represents the weekly profit, in thousands of dollars, for a rental store since opening. The graph of P(t)is shown.
When did the store have its maximum weekly profit?
答案：A
解題說明：
The graph shows:
P(t)= " weekly profit in thousands of dollars "
where:
t= " years since opening "
The graph is a downward-opening curve, so it has a maximum value at its highest point. In Applied Algebra, the highest point of this kind of polynomial graph is called the vertex.
To answer the question, we need the t-value of the vertex because the question asks when the store had its maximum weekly profit.
From the graph, the highest point occurs between:
t=5
and
t=6
More specifically, it is approximately:
t=5.5
So the store had its maximum weekly profit approximately:
5.5 " years after opening "
問題 #106
The scatterplot shows data on the number of shirts sold at a gift shop per week over time. The graphed regression function has an r
2
value of 0.09.
Which range of x-values is appropriate for extrapolation?
答案：C
解題說明：
The r
2
value tells how closely the regression model fits the scatterplot data. In this problem, r
2
=0.09, which is extremely low. This means the regression model explains only a very small portion of the variation in the shirt-sales data. When r
2
is this weak, the model is not reliable for predictions outside the data range. Extrapolation is already less certain than interpolation because it predicts beyond the observed data. With such a poor fit, extending the model beyond the data would not be appropriate. Since 0.09 < 0.3, the best justification is that extrapolation is not appropriate because r
2
< 0.3. Therefore, the correct answer is C.
問題 #107
The logistic function f(x), whose graph is shown, models the number of people who have created a website account, where xrepresents the number of days since the website started and f(x)represents the number of people who have created an account.
What is one range of values for which the graph is concave down?
答案：C
解題說明：
For a logistic graph, concavity changes at the inflection point.
Before the inflection point, the graph is usually concave up. This means the function is increasing faster and faster.
After the inflection point, the graph is concave down. This means the function is still increasing, but it is increasing slower and slower.
From the graph, the curve changes concavity around:
x=8
After x=8, the graph begins to level off as it approaches its upper limit.
So one interval where the graph is concave down is:
(8#16)
Therefore, the correct answer is:
# ( " A " )
問題 #108
The graph shows the number of people watching a live video. The number of minutes since the video began is on the horizontal axis, and the number of viewers is on the vertical axis. When were there 432 viewers?
Exhibit:
答案：A
解題說明：
This problem requires reading a value from a graph. The horizontal axis gives time in minutes, and the vertical axis gives the number of viewers. To solve it, find 432 on the vertical axis, locate where the graph reaches that value, and then move down to the horizontal axis. From the exhibit, the graph reaches 432 viewers at minute 1. That means the ordered pair is approximately (1, 432), so the input is 1 minute and the output is 432 viewers. The other listed minutes correspond to different viewer counts on the graph. In graphical depictions, matching the correct input-output pair is the key step. Therefore, the correct answer is B.
問題 #109
......
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