Simonne has been following the stock ratings of Carter's Pork Bellies. She has noticed that the pattern can be modeled by a cosine function. During her observations, she has found the stock to have a maximum of x = 6, minimum of x = −2, and period of π. Which of the following cosine functions could model Simonne's stock rating?

Question

sleepyjess · Answer

f(x) = 4 cos 2x − 4
f(x) = 6 cos pi over 2x + 2
f(x) = 6 cos pi over 2x − 4
f(x) = 4 cos 2x + 2

sleepyjess · Answer

@phi

phi · Answer

The first step could be, what is the range of the values from low to high ?

sleepyjess · Answer

6?

phi · Answer

x = 6, minimum of x = −2

so the distance from min to max is ?

sleepyjess · Answer

8

phi · Answer

a cosine has a min of -1 and a max of +1, and a range of 2
your cosine range is 8.  that means multiply the cosine by 4

4 cos(stuff)
will range from -4 to +4  or a total of 8

phi · Answer

however, we have to tweak that, so it ranges from -2 to +6
I would add +2 to it, like this:

4 cos(stuff) + 2
now it will range from -2 to +6

sleepyjess · Answer

ok thank you so much for your help! :-)

phi · Answer

It looks like we did not need to use the period to get to the correct answer.  But if we had to, we would do this:

inside a cosine, (the "stuff" ),  write it as
$$ \frac{2 \pi}{T} x $$
where T is the period

they tell use T is π
so the inside should be
$$ \frac{2 \pi}{\pi} x = 2x $$

the full answer is
$$ 4 \cos(2x)+2 $$