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The period is over the course of a year. There are about 52 weeks in a year.
Would I say that my period is 52? Or is it 360/52? I apologize in advance, it has been a while since sinusoidal functions.
I'm leaning more towards 52 being written in the equation, however.
Yes, the period is 52.
Remember that the period is given by $\displaystyle \frac{2\pi}{b}$.
So now solve $\displaystyle \frac{2\pi}{b} = 52$ for $\displaystyle b$.
Also, how much of a horizontal translation has there been? Remember that a regular sine function begins at the origin.
360/52 = 6.9
Horizontal translations are something I have always been pretty dreadful at, and something I need a lot of practice with. I will hazard a guess, however...
If the max starts at week 4, and the min at week 30, and a sine function always starts at the origin, then there has been a horizontal shift 30 to the right?
So x - 30 ?
No.
First, if you have trigonometric FUNCTIONS or GRAPHS, they are always measured in RADIANS.
So solve the equation $\displaystyle \frac{2\pi}{b} = 52$ for $\displaystyle b$.
You should find that $\displaystyle b = \frac{\pi}{26}$.
So now you have
$\displaystyle y = 5000\sin{\left(\frac{\pi x}{26} - c\right)} + 7000$.
You also have some points that you know lie on the graph, namely $\displaystyle (x, y) = (4, 12 000)$ and $\displaystyle (30, 2000)$.
So, let $\displaystyle x$ and $\displaystyle y$ equal some of these values.
$\displaystyle 12000 = 5000\sin{\left(\frac{4\pi}{26} - c\right)} + 7000$
$\displaystyle 12000 = 5000\sin{\left(\frac{2\pi}{13} - c\right)} + 7000$
$\displaystyle 5000 = 5000\sin{\left(\frac{2\pi}{13} - c\right)}$
$\displaystyle 1 = \sin{\left(\frac{2\pi}{13} - c\right)}$
$\displaystyle \frac{2\pi}{13} - c = \arcsin{1}$
$\displaystyle \frac{2\pi}{13} - c = \frac{\pi}{2}$
$\displaystyle c = \frac{2\pi}{13} - \frac{\pi}{2}$
$\displaystyle c = - \frac{9\pi}{26}$.
So this means there has been a horizontal translation of $\displaystyle \frac{9\pi}{26}$ units to the left.
And now, finally, your function is
$\displaystyle y = 5000\sin{\left(\frac{\pi x}{26} + \frac{9\pi}{26}\right)} + 7000$.
My apologies. We haven't been taught radians yet--that's grade 12 here in Canada--though I understand your equations. However we will not be asked to use radians on any sort of test situation, so I would like to make sure I am getting the same answer in the 'correct' format for my course.
So in non-radian-speak, we have...
pt. (4, 12000)
12000 = 5000sin [6.9(4) - d] + 7000
5000 = 5000sin (27.6 - d)
1 = sin (27.6 - d)
arcsin (1) = 27.6 - d
90 - 27.6 = d
d = 62.4
Therefore the equation is ...
y = 5000sin (6.9x - 62.4) + 7000 ?