Pre Calculus 12
posted by Shreya .
Graphing trigonometric functions
A sinusoidal function has a maximum at (2,10) and its next minimum (5,2) find an equation that represents this situation
A sinuosoidal function has a zero at (5,0) and its next minimum is (7,4). find an equation that represents this situation.
I really don't get how to come up with an equation, this chapter was really the hardest for me. I don't get how to find amplitude, vertical displacement, phase shift
Can someone please explain step by step

Pre Calculus 12 
Reiny
Notice the max is 10 and the min is 2, for a range of 12
which makes the amplitude = 6
the axis would be 4 units high
so far we have something like
y = 6 sin (.......) + 4
the distance from the max to the min along the horizontal is 3 (from 2 to 5 is 3 units)
but the period would be twice that
so the period is 6
remember 2π/k = period
2π/k = 6
6k = 2π
k = 2π/6 = π/3
Ok, now also have the period
and our equation must look something like this:
y = 6 sin (π/3)(x + d) + 4
almost done....
we just have to sub in one of our points to find d , the phase shift
let's use (2,10)
10 = 6sin π/3(2  d) + 4
1 = sin π/3(2d)
I know that sin π/2 = 1
so π/3(2d) = π/2
divide both sides by π
(1/3)(2d) = 1/2
times 6
2(2d) = 3
2d = 3/2
d = 3/2  2 = 1/2
d = 1/2
y = 6 sin (π/3)(x + 1/2) + 4
check for the other point:
if x = 5, we should get y = 2
y = 6 sin (π/3)(5+1/2) + 4
= 6 sin ((π/3)(11/2)) + 4
= 6 sin (11π/2) + 4
= 6(1) + 4 = 2 , YEAHHHHH
I chose to use a sine curve for no specific reason
We could have used a cosine as well.
The period and amplitude and vertical shift would be the same
the change would be in the value of d
Try the second one, it is slightly different.
Make a sketch to see what you are doing. 
Pre Calculus 12 
Shreya
so you added 10 and 2 to get 12, and amplitude is the middle of max and min?
how did you get the axis 4 units?
a period is always twice of something?
the phase shift is sort of confusing me, because isnt that just between the x values? 2, and 5? is there some other way to figure it out? 
Pre Calculus 12 
Reiny
"how did you get the axis 4 units? "
where is the middle value between 2 and 10
sketch on a number line to see.
the amplitude, or the a value, is 1/2 the difference between the max and the min
difference = 10  (2) = 12
half of that is 6
I illustrated with detailed calculations, how I got the phase shift d
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