trig question. please help?

Question

"Write a function for the sinusoid with maximum at A and minimum at B."

A(9pi,1)B(3pi,-5)


If you understand this problem, please explain it to me and show all work. Every little bit helps. I'm confused on how to get the period especially. Also, if you know of a website that explaisn sinusoids, please tell me. Thanks for the help!

Answer 1

It's helpful to plot this sort of problem on graph paper...

First, you must find the amplitude. This is the y-coordinate difference between the middle of the graph and the maximum (or minimum). The difference between the y-coordinates of the maximum and minimum is 6 (1 - -5 = 6), so the amplitude is one half of that (3).

Now, we must find the axis of the graph (the middle of the graph). The y-coordinate of the axis is constant, and it is equal to the midpoint of the y-coordinates of the maximum and minimum. (1 + -5) / 2 = -4/2 = -2. Thus, the axis has y-coordinate -2, giving the equation y = -2.

After that, we must find the period. A period is the x-distance that the graph travels before it repeats. Remember, from a maximum to a minimum is one half of a period. The x-distance from the minimum to the maximum is 6pi, so the period is 12pi.

Now, we must find the phase shift. The phase shift is how much the "phase" of the graph is shifted. In order to find this, we need to decide on the type of the graph. Let's call it a cosine graph. A cosine graph's maximum is located at the start of the graph before it is shifted/stretched, while a sine's is located in the center. The maximum is x-coordinate 9pi, which means that the graph was shifted 9pi to the right for a phase shift.

Finally, we must construct an equation. An equation for a cosine takes the standard form f(x)=a*cos(b*(x+c))+d. a is the amplitude, b is a value related to period, c is how much the graph is shifted to the LEFT, and d is how far the graph is moved upwards. To find b, we use the following formula: 2pi/b =period. Rearrange this to find 2pi/period = b. Our period is 12pi, so the value of b is 1/6. The value of a is 3. The value for c is 9pi to the right, or -9pi (to the left). The value for d is -2. Thus, the equation as a whole appears as the following:

f(x) = 3 cos (1/6 (x - 9pi)) - 2

Hope this is helpful (it took a very long time to type)! The best link I found was http://www.math.uga.edu/~hofmann/mat1113-8-5.pdf

EDIT: Graphing the equation using a graphing aid, I verified that this is the correct equation. A helpful hint about phase shift: remember that when looking at equations in the form y+a = f(x+b), the origin will be mapped to (-b,-a). That way, you can understand phase shift. Also, remember that one period is the x-distance that it takes for the function (a PERIODIC function) to repeat itself.

Answer 2

lol...
I recommend going to www.artofproblemsolving.com, check out their wiki and type in sinusoids at the left hand side.

btw, the amplitude would be the height of the y-coordinates, which is 6, so the amplitude is 3. The period would be 6 pi, (actually, it could be 3 pi as well, there are a lot of functions that work), so the period would be 2/3 pi if you're looking for the 3 pi function. so, the function would probably be something like y = 3sin((2/3)pi theta) - 2

-2 from vertical shift

Answer 3

The equation for a sinusoid is:
y = A sin(2pix/r + c) -d, where
A = amplitude
r = period
c = phase shift
d = shifts the graph up or down
r= 2*(x distance between max and min) =12pi
A = (1-(-5)(/2 = 3
c = -pi
d=-2
So equation is: y = 3sin(x/6-pi)-2

Answer 4

Here's how I do this kinda problem.
1. I look at my good old friend, y=1sin(x)
How tall is it from it's max to min? 2
Your curve is going to be 6 in height. That tells us that the amplitude is gonna be 3 (half of 6). And it's gonna be scooted up (-5 + 1)/2 units, or -2 units.

(Cause sin's amplitude is 1, half of the 2 difference from max to min)
Now, how long does it take for sin to go from its max (which is at pi/2) to it's min? (its min is at 3pi/2) Lessee... 3pi/2 - pi/2 = 2pi/2 which is... er... um... pi.
So sin does it with x advancing pi.
Your function needs to go SIX TIMES SLOWER.
So far, we're looking at something like
3sin(x/6)-2
I graph that, and, hooray! It meets the criteria!

Answer 5

i chanced on trig to be extra straightforward. Geometry in severe college replaced into the toughest for me, particularly, I nevertheless think of severe college geometry is fairly extra durable than a large number of the college math instructions i'm taking to be straightforward...I disliked it very lots.