Blue Book Questions

<p>2 days until the SAT, so I’m going to at least finish up the Math practice tests. Reading and Writing I’ve already got 700 and 730, but Math I should be worried about if I want to bring it up 100 points :p</p>

<p>Here are 2 more:
Page 790, #19 and #20</p>

<p>I’m going to try #19 one more time but I don’t think I’ll figure it out by the time you answer it >.<</p>

<p>I know x and y are inversely proportional, so (n^2 k^2)^-2/3 = xy ^-2/3? </p>

<p>But if that was true then the answer would be -2/(cuberoot n^2 k^2) when the real answer is 1/nk lol</p>

<p>I probably won’t be able to answer for a while. I don’t have the BB with me at the moment. Would you mind posting the question?</p>

<p>Ok. </p>

<h1>19</h1>

<p>if k, n, x and y are positive numbers satisfying x^(-4/3) = k^(-2) and y^(4/3) = n^(2), what is (xy)^(-2/3) in terms of n and k?</p>

<p>Answer: 1/nk</p>

<h1>20</h1>

<p>The figures above show the graphs of the functions f and g. The function f is defined by f(x) = x^3 - 4x. The function g is defined by g(x) = f(x+h) +k, where h and k are constants. What is the value of hk?</p>

<p>Graph of f(x): -1,3 and 1,-3 are the maximum and minimum values (the peak and bottom of the hills). Hills is the professional math term ofc :p</p>

<p>Graph of g(x): 2,1 and 4,-5 are the maximum and minimum values respectively.</p>

<p>Answer: 6</p>

<p>Since x^(-4/3)=k^(-2), you can raise both sides to 1/2 in order to bring x to be raised to x^(-2/3) The equation is now x^(-2/3) = k^(-1).</p>

<p>y^(4/3) =n^2; raise both sides by -1/2.
y^(-2/3) = n^(-1)</p>

<p>Since (xy)^(-2/3) = x^(-2/3) * y^(-2/3), you can substitute the derived equations into each variable.</p>

<p>k^(-1) * n^(-1) = 1/nk</p>

<p>I wrote this on a phone… So there might be errors… I’m not sure this was clear… I’ll repost a better response when I get home if you didn’t get this.</p>

<p>The other one is going to be long, so I’ll post an explanation when I am home…</p>

<p>I understand 19 now. :)</p>

<p>If n = 3p, for what value of p is n = p?</p>

<p>A) 0
B) 1/3
C) 1
D) 3
E) Never Equal</p>

<p>This is really easy but I don’t understand it still :(</p>

<p>I put 1/3 because that makes sense but apparently it’s 0</p>

<p>Also #16: Let [x] be defined as [x] = x^2 - x for all values of x. If [a] = [a-2], what is the value of a? I plugged in a-2 for x^2-x and got </p>

<p>a = (a-3)(a-2) Then I guessed that a = 3/2 even though I had no basis for guessing that other that that the 2 values of a in that are 3 and 2. I got it right but I didn’t do it right either.</p>

<p>

</p>

<p>This is much simpler geometrically. From (-1,3) to (2,1), you move right 3 times (-3) and down 2 times (-2).
(-3)(-2) = 6</p>

<p>I can explain the algebraic approach, but it will probably be unnecessary. for this problem, you just had to understand that the shift over the x-axis is the opposite (when you move right 3 times, it’s actually -3 not +3).</p>

<p>How do I know h and k are -3 and -2… shouldn’t they be x and y like they always are? Also, why did you take the maximum points on two different graphs to find h and k when h and k are only in the second function?</p>

<p>

</p>

<p>You can either substitute 3p for n, or you can use some logic.</p>

<p>n = 3p;
n = p;
3p = p; Subtract p from both sides
2p = 0;
p = 0</p>

<p>When you substitute 1/3 for p, it doesn’t work.
n = 3(1/3) = 1
n = 1/3
1/3 ≠ 1</p>

<p>

</p>

<p>[a] = [a-2]
a²-a = (a-2)² - (a-2); expand
a²-a = a²-4a+4-a+2;
a²-a = a²-5a+6; move everything to one side
0 = -4a+6; solve for a
-6 = -4a;
a = 3/2</p>

<p>

</p>

<p>Do you remember the shift of a graph?
For example, f(x) = (x+h)² is a horizontal shift while f(x) = x²+k is a vertical shift.
In the problem, f(x) is given. g(x) is a shift of f(x). You pick one point that the graph has in common (I guess the maximum). Maximum for graph of f(x) is (-1,3) and maximum for graph of g(x) is (2,1). It shifted 3 units to the right and 2 units down. (-3 is the 3 units to the right and -2 is 2 units down) The h in function g(x) is the horizontal shift (-3). The k in function g(x) is the vertical shift (-2).</p>

<p>Here is the long algebraic solution:</p>

<p>f(x) = x³ - 4x; Let’s put this equation so that it matches with f(x+h).
f(x+h) = (x+h)³ - 4(x+h)</p>

<p>g(x) = f(x+h) + k; Plug in
g(x) = [(x+h)³ - 4(x+h)] + k; We are given two points to work with for g(x): (2,1) and (4,-5)</p>

<p>Plug in for (2,1)
1 = [(2+h)³ - 4(2+h)] + k;
1 = (8+12h+6h²+h³) - (8+4h) + k;
1 = 8+12h+6h²+h³-8-4h+k; isolate k
1-8-12h-6h²-h³+8+4h = k;
1-8h-6h²-h³ = k</p>

<p>Plug in for (4,-5)
-5 = [(4+h)³ - 4(4+h)] + k;
-5 = (64+48h+12h²+h³) - (16+4h) + k;
-5 = 64+48h+12h²+h³-16-4h+k; isolate k
-5-64-48h-12h²-h³+16+4h = k;
-53-44h-12h²-h³ = k</p>

<p>1-8h-6h²-h³ = -53-44h-12h²-h³; Solve for h
54+36h+6h² = 0;
6(h²+6h+9) = 0;
(6)(h+3)(h+3) = 0
h = -3</p>

<p>Now… plug in h = -3 in 1-8h-6h²-h³ = k
1-8(-3)-6(-3)²-(-3)³ = k
1+24-54+27 = k
k = -2</p>

<p>hk = (-3)(-2) = 6</p>

<p>I assume the CB wants people to do the first solution seeing as they only give us 20-25 minutes ;)</p>

<p>Yes, especially since the graphs are provided.</p>

<p>This one I know I could have gotten, but I kept drawing a blank as to how to figure it out. </p>

<p>The price of ground coffee beans is d dollars for 8 ounces and each ounce makes c cups of brewed coffee. In terms of c and d, what is the dollar cost of the ground coffee beans required to make 1 cup of brewed coffee.</p>

<p>Answer: d/8c</p>

<p>

</p>

<p>For these problems, it’s best if you choose numbers to plug in if you’re not 100% confident in your algebraic reasoning skills.</p>

<p>Let d = 1 and c = 1: This should let you cancel at least 2-3 of the choices. Then, choose another set of numbers.</p>

<p>Let d = 2 and c = 2: This should bring you to one final answer.</p>

<p>Or, if you know about dimensional analysis, it might help.
The question asks, “In terms of c and d, what is the dollar cost of the ground coffee beans required to make 1 cup of brewed coffee?”</p>

<p>You know your answer has to end up in dollar/cup</p>

<p>Given:
d dollars = 8 ounces
1 ounce = c cups</p>

<p>d dollars | 1 ounce |
-------------------- = d/8c
8 ounces | c cups |</p>

<ol>
<li><p>The survey (showed that) most shoppers who drive prefer the mall (more than) downtown stores (simply because) finding parking is (less difficult) at the mall.</p></li>
<li><p>Professor Chen repeated (her point that) the hero, if (given) the chance (to relive) the moment, would choose to (do it).</p></li>
</ol>

<ol>
<li>Prefer…to is the correct structure.</li>
<li>Do it should be do so.</li>
</ol>

<p>Blue Book Page 842, #9</p>

<p>How could E possibly not be right? I was 100% sure on that one and when I saw that I was wrong, I got pretty mad.</p>

<p>Edit: I understand why D is another good choice and the answer, but why is E NOT viable?</p>