Functions As Models
The old SAT constantly got hammered for being a test that
didn’t have much to do with the real world. The test-writers decided
that they didn’t want to hear it anymore with the new SAT, and so
they created Functions as Models questions, which have nothing to
do with Giselle or Tyson Beckford but present information about
a real-life scenario and then ask you to pick a function in equation
or graph form that best describes the scenario.
A Function as Models question with graphs looks something
like this:
|
|
|
If
temperature is f(x) and
time is x, which of the following
best describes a bucket of cold water left outside on a hot day?
|
| (A) |
|
| (B) |
|
| (C) |
|
| (D) |
|
| (E) |
|
|
So, what’ll happen to a bucket of cold water left outside
on a hot day? It’ll heat up, of course, so the answer’s B ...
except that this is the real world and the question contains a trick. B shows
that bucket heating up forever, when in fact, the
water in the bucket can’t actually get any hotter than the day itself.
So at some point, that rise in temperature has to hit a plateau: A is
the answer.
A Functions as Models problem that deals with
written-out functions looks like this:
|
|
|
A
bookstore is selling a particular book for $15 per copy. At this
price it has been selling 20 copies of the book each day. The store
owner estimates that for every dollar reduction in the selling price
of the book, daily sales will increase by 20 copies. What is the
daily sales, S, as a function of price, p? |
| (A) |
S = –20p + 320 |
| (B) |
S = 15p + 20 |
| (C) |
S =  p |
| (D) |
S = –20p – 15 |
| (E) |
S = p + 5 |
|
The key to solving this sort of problem is to first define
what kind of mathematical function this “real-world” scenario is
describing. Take another look at the question: At $15, the book
sells 20 copies, and for each dollar the price goes down, the book sells
another 20 copies. In other words, for each dollar decrease in price,
sales increase by a fixed amount. Sound like anything to you? How
about a linear function, like S = mp + b,
where m is the slope and b is
the y-intercept. The “fixed increase”
for every change of p is the slope.
Since the slope increases 20 units
for each dollar decrease in price, that slope must
be negative: –20. You could now eliminate all the answers
but A and D. To find the definitive right
answer, though, you have to find b.
You can do that by plugging in numbers from the question. You know that
when the book costs $15 dollars (p),
it sells 20 copies (S).
So, 20 = –20(15) + b; 20 = –300
+ b; b = 320.
There you go. A is the answer.
If you’re a little algebra-phobic, all this might seem
very hard to you. But there’s another way to go about it. Use the
information in the question to build a graph. From the information
in the question, you know that at the price of $15, the store sells
20 copies, and that for each dollar less, the store sells 20 more
copies. So, in other words, you know the points on this graph: (15,20),
(14,40), (13,60)...
This graph doesn’t give you the answer, but it does make
it clear that you’re dealing with a linear graph of slope –20,
and that if you just keep on counting back to where p =
0, you’ll get the y-intercept,
or 320.
