posted Mar 1, 2010, 7:32 PM by Chuang Tina
[
updated Mar 18, 2010, 4:58 AM
]
Mar. 17, 2010 Read the following SAT test question, then click on a button to select your answer. ƒ(2n) = 2ƒ(n) for all integers n ƒ(4)= 4
If ƒ is a function defined for all positive integers n, and ƒ satisfies the two conditions above, which of the following could be the definition of ƒ?
 ƒ(n) = n – 2
 ƒ(n) = n
 ƒ(n) = 2n
 ƒ(n) = 4
 ƒ(n) = 2n – 4
<Hint> It is easy to check whether ƒ(4) = 4 for the functions given. First, eliminate those functions for which ƒ(4) 4. Then evaluate the functions for some values, such as n = 1 or n = 2, to see whether ƒ(2n) = 2ƒ(n) for those values.
<Correct Answer> 2 <Here's Why>
If ƒ(n) = n – 2, then ƒ(4) = 4 – 2 = 2 4, so the second condition fails. If ƒ(n) = 2n, then ƒ(4) = 8 4, so the second condition fails for this function also. The other three options satisfy ƒ(4) = 4, so it remains to check whether they satisfy the first condition.
If n = 1, and ƒ(n) = 4, then ƒ(2n) = ƒ(2) = 4 and 2ƒ(1) = 2(4) = 8, so it is not true that ƒ(2n) = 2ƒ(n) for all integers n. This means that the function ƒ(n) = 4 does not satisfy the first condition. If n = 1, and ƒ(n) = 2n – 4, then ƒ(2n) = ƒ(2) = 2(2)– 4 = 0 and 2ƒ(n) = 2ƒ(1) = 2(–2) = –4, so it is not true that ƒ(2n) = 2ƒ(n) for all integers n. This means that the function ƒ(n) = 2n – 4 does not satisfy the first condition.
However, if ƒ(n) = n, then ƒ(2n) = 2n = 2ƒ(n), for all integers n. Also, ƒ(4) = 4. Therefore, the function ƒ(n) = n is the only option that satisfies both conditions.
<Difficulty> Hard
<Question Type> Standard Multiple Choice
Mar. 12, 2010 Read the following SAT test question, then click on a button to select your answer.
If the graph of the function ƒ in the xyplane contains the points (0, –9), (1, –4), and (3, 0), which of the following CANNOT be true?
 The graph of ƒ has a maximum value.
 y 0 for all points (x, y) on the graph of ƒ.
 The graph of ƒ is symmetric with respect to a line.
 The graph of ƒ is a line.
 The graph of ƒ is a parabola.
<Hint>
If three points in the xyplane lie on a line, the slope of the segment connecting any pair of the points must be the same as the slope of the segment connecting any other pair of the points. <Correct Answer> D <Here's Why>
If the graph of the function ƒ, which contains the points (0, –9), (1, –4), and (3, 0), were a line, then the slope of the segment connecting the points (0, –9) and (1, –4) would be the same as the slope of the segment connecting the points (1, –4) and (3, 0). However, the slope of the segment connecting the points (0, –9) and (1, –4) is 5, and the slope of the segment connecting the points (1, –4) and (3, 0) is 2. Therefore, the graph of ƒ cannot be a line.
The statements in the other four options could be true. For example, if the equation of ƒ were y = –(x – 3)2, then the graph of ƒ would contain the points (0, –9), (1, –4), and (3, 0). The graph of ƒ would be a parabola symmetric with respect to the line with equation x = 3. The maximum value of ƒ would occur at the point (3, 0), and y 0 would be true for all points (x, y) on the graph of ƒ. <Difficulty> Hard <Question Type> Standard Multiple Choice
Mar. 9, 2010
What is the equation of the line parallel to the xaxis and four units above the xaxis? 1. x = –4 2. x = 4 3. y = –4 4. y = 0 5. y = 4
<Hint> Any line that is parallel to the xaxis is a horizontal line.
<Correct Answer> 5
A line that is parallel to the xaxis and four units above the xaxis is the vertical translation of the xaxis four units upward. Since the xaxis is a horizontal line and has equation y = 0, it follows that the line parallel to the xaxis and four units above the xaxis has equation y = 4.
<Difficulty> Easy
<Question Type> Standard Multiple Choice
Mar. 4, 2010
If 0.036 = 3.6 × 10^{t},
what is the value of t?
 –3
 –2
 –1
 2
 3
<Hint> If t is a nonzero integer, multiplying a number in decimal form by 10t moves the decimal point to the right or to the left, depending on whether t is positive or negative, respectively.
<Correct Answer> B
If t is a nonzero integer, multiplying a number in decimal form by 10t moves the decimal point to the right or to the left, depending on whether t is positive or negative, respectively. (If t = 0, multiplying by 10t does not change the value of the number.) To get 0.036 from 3.6, you must move the decimal point two places to the left, so t = –2.  Difficulty: Easy
 Question Type: Standard Multiple Choice
Feb. 26, 2010
A florist buys roses at $0.50
apiece and sells them for $1.00 apiece. If there are no other expenses, how
many roses must be sold in order to make a profit of $300?
 100
 150
 200
 300
 600
<Hint>
Each rose brings a profit of $1.00 minus $0.50, or $0.50.
How many $0.50’s are there in $300?
<Correct Answer> E
Each
rose brings a profit of $1.00 minus $0.50, or $0.50. The number of roses that
need to be sold to make a profit of $300 is $300 ÷ $0.50.
This is 600 roses.
 Difficulty: Easy
 Question Type: Standard Multiple Choice

