Math SAT I Question--HELP!

Test Preparation
1

<p>I have looked at this question a few times, and still haven't figured out the solution...</p>

<p>" |x | f(x)|
________
|0 | a |
________
|1 | 24|</p>

<hr>

<pre><code> 2 | b |
</code></pre>

<p>The table above shows some values for the function f. If f is a linear function, what is the value of a + b?</p>

<p>(A) 24
(B) 36
(C) 48
(D) 72
(E) It cannot be determined by the given information.</p>

<hr>

<p>[sorry, my table-making skills are limited]</p>

<p>Any help would be appreciated. Thanks</p>

2

<p>what question?</p>

3

<p>Wait a second.. it's definitely C!</p>

<p>You have your points (0, a), (1, 24), (2, b)</p>

<p>Since this is a linear function, these points will lie all on the same line, therefore the slope between each point will be equal to one another.</p>

<p>So..</p>

<p>(24-a)/(1-0) = (24-b)/(1-2)</p>

<p>24-a = -24 + b</p>

<p>48 = a + b</p>

<p>I hope that's clear</p>

4

<p>yup, a=0 because f is linear. f(2) = 2f(1) = f*24 = 48. So a+b = 0 + 48 =48. (C)</p>

5

<p>mathwiz, how do you assume that the line passes through the origin? Can't a function still be linear but have a non-zero y-intercept? I thought a function is linear because it is dependent upon slope not y-intercept. Correct me if I'm wrong on this</p>

6

<p>Nope. A function is linear if f(cx) = c(f(x)) and f(a+b) = f(a) + f(b). The ones you are thinking of are straight lines.</p>

7

<p>I need to learn the definitions of some of these math terms. A linear function doesn't necessarily have to be straight? Geez! Why do they make everything so confusing lol..</p>

8

<p>No, a linear function has to be straight. But there are lots of "straight" functions that aren't linear, such as f=4x+4</p>

9

<p>Thanks! I understand it now!</p>

10

<p>O okay, thanks mathwiz. So basically:</p>

<p>f(1) = 24</p>

<p>f(1 + 0) = 24 + a (so 'a' is zero)</p>

<p>f(1 + 1) = 24 + b</p>

<p>so, f(2) = f(1) + f(1)</p>

<p>and f(1 + 1) = 24 + 24</p>

<p>f(2) = 48 or f(2) = b</p>

<p>and 'b' will be 48</p>

<p>Does a linear function always have to pass through the origin?</p>

11

<p>I personally like Krabble's explanation in post#3, it's clear and simple. A messier way is to do it via straight algebra:
Suppose f(x) = mx + c
Then f(0) = a = (m)(0) + c ; from which c=a
f(1) = (m)(1) + a = 24 ; from which m = 24-a
f(2) = (m)(2) + a = b
or (2)(24-a) + a = b
48 -2a + a = b
48 = a + b</p>

12

<p>why doesn't f=4x+4 a linear function?</p>

13

<p>See <a href="http://www.answers.com/topic/linear-equation%5B/url%5D"&gt;http://www.answers.com/topic/linear-equation&lt;/a&gt;&lt;/p>

<p>Strictly speaking, f(x) = mx + c is a linear equation or a linear function.</p>

<p>A function that also satisfies
f(x + y) = f(x) + f(y)
and f(ax) = af(x)
would be a 'linear operator', if my fading memory is correct. </p>

<p>For SAT I and SAT II questions, I'm pretty sure they mean f(x)=mx + c when they talk about linear functions.</p>

14

<p>Points A(0,a), B(1,24) and C(2,b) lie on a straight line.
Point B is exactly in the middle between points A and C (notice the order of x-coordinates of these points), so
24 is exactly in the middle between a and b on the number line, i.e.
24 is the mean of a and b
which means a + b = 2*24,
a + b = 48. That's (C).</p>

<p>PS.
Values of a and b CANNOT be determined from the information given.
That makes choice (E) tempting.
But: if SAT asks for the value of an expression with more than one variable, you almost always can't or needn't find the value of each variable separately.</p>

<p>In this question a and b could be any numbers as long as they add to 48.
Three points (0,0), (1,24), (2,48) on the straight line are as good as
(0,24), (1,24), (2,24), or
(0,48), (1,24), (2,0).</p>

15

<p>PPS.
See a school definition of a linear function
<a href="http://www.answers.com/main/ntquery?method=4&dsid=2222&dekey=Linear+function&gwp=8&curtab=2222_1%5B/url%5D"&gt;http://www.answers.com/main/ntquery?method=4&dsid=2222&dekey=Linear+function&gwp=8&curtab=2222_1&lt;/a>
vs. mathwiz's true math definition
<a href="http://www.answers.com/topic/linear%5B/url%5D"&gt;http://www.answers.com/topic/linear&lt;/a>
(scroll down to Mathematics).</p>

<p>And they say math is a precise science...</p>

16

<p>firstly, the SAT rule does NOT allow you to discuss any test questions, in any means, including but not limited to the Internet, forums, text messages. We should NOT discuss a specific question here.</p>

<p>Secondly, if you took 4 math sections, one of these is equating, and that equating section is the same with 1 Math section in Nov, 2005.</p>

17

<p>Hmmm.... This is a problem in the blue book, isn't it? I recognize it... and I'm pretty sure I would have remembered if it was in the Jan SATs.... In fact, I'm sure it's in the blue book, not the Jan SAT.</p>

<p>So, I really don't think we're violating any rules here.....</p>