http://i.imgur.com/QQS7VCY.png
the standard form of a trigonometric function is:
y = a * (sin or cos or tan or sec or csc or cot)(b * (x - c)) + d
a is the amplitude
b is the frequency
c is the horizontal displacement.
d is the vertical displacement.
period is equal to 360 / frequency in degrees and 2*pi / frequency in radians.
frequency is equal to 360 / period in degrees and 2*pi / period in radians.
your first picture looks like a cotangent function.
one complete cycle goes from -4 * pi to 4 * pi.
that's a period of 8 * pi.
the frequency is therefore equal to (2 * pi) divided by (8 * pi) which is equal to 1/4.
the function of y = a * cot(b * (x - c)) + d becomes:
y = cot(1/4 * x)
a is equal to 1 and is therefore not shown.
b is equal to 1/4
c is equal to 0 and is therefore not shown.
d is equal to 0 and is therefore not shown.
the graph of y = cot(1/4 * x) looks like this:
your second picture looks like an upside sine function or a sine function that has been horizontally displaced by 180 degrees (equivalent to pi radians).
either assumption should yield the same graph.
assuming it's an upside down sine function, then we'll check the other data to determine what value to place there.
the basic function would be y = a * sin(b * (x - c)) + d
a is equal to -8.
this is because the normal amplitude is 1 and this one is 8, so a has to be equal to 8.
since it's upside down, then a also has to be negative, so a will be equal to -8.
the center of the graph is on the x-axis, so d is equal to 0 and therefore not shown.
one complete cycle goes from 0 to 2*pi/3, so the period is equal to 2pi/3.
the frequency is equal to (2 * pi) divided by (2 * pi / 3) which is equal to 3.
the equation looks like it should be:
y = -8 * sin(3*x)
there is no horizontal displacement (c = 0 and therefore not shown) and there is no vertical displacement (d = 0and therefore not shown).
the graph of y = -8 * sin(3*x) looks like this:
assuming it's a sine function that is horizontally displaced by 180 degrees (equivalent to pi radians) rather than an upside sine wave , then you would tackle it as follows:
the amplitude is still 8.
it is not negative, however, since we are not looking at an upside down sine wave anymore.
the frequency is still 3.
the horizontal displacement, however, is equal to pi.
this is because pi radians is equivalent to 180 degrees.
the general formula of y = a * sin(b * (x-c)) + d becomes:
y = 8 * sin(3 * (x - pi)).
the graph of y = 8 * sin(3 * (x - pi)) looks like this:
as you can see, the graph of y = -8 * sin(3x) looks exactly the same as the graph of y = 8 * sin(3 * (x-pi)).
they're identical.
you can further confirm that this is true by taking your calculator and solving for any random value of x using both formulas.
for example, assume x is equal to 15.
remember, we're dealing in radians and not degrees. your calculator should be set to radians.
y = -8 * sin(3 * 15) = -6.807228196
y = 8 * sin(3 * (15 - pi)) = -6.807228196
they're the same.
this and the graph provides pretty conclusive proof that the equations are identical.
you can graph these yourself using the online calculator found at:
https://www.desmos.com/calculator