SOLUTION: how to find the number of cycles in 2pi, the horizontal length of each of the 4 quadrants of each cycle, the horizontal shift of each cycle f(x)=-tan(2x)-3

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Question 1135335: how to find the number of cycles in 2pi, the horizontal length of each of the 4 quadrants of each cycle, the horizontal shift of each cycle

f(x)=-tan(2x)-3

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
y = -tan(2x)-3

the normal period for the tan function is pi, or 180 degrees.

to convert between radians and degrees, here's the formula.

degrees = radians * 180 / pi.

raedians = degrees * pi / 180.

pi radians * 180 / pi is equal to 180 degrees.

180 degrees * pi / 180 is equal to pi radians.

pi/2 radians * 180 / pi is equal to 90 degrees.

90 degrees * pi / 180 is equal to pi/2 radians.

the relationship between frequency and period is:

frequency = pi / period.

period = pi / frequency

the general form of the equation is y = a * tan(b * (x - c)) + d

a is the amplitude
b is the frequency
c is the horizontal displacement
d is the vertical displacement.

here's the graph of the normal tan function of y = tan(x).

$$$$$

here's the graph of y = tan(2x).

the normal cycle for the tan function is pi because the tan function repeats every pi radians, unlike the sine and cosine functions, which repeater every 2pi radians.

here's a graph of y = tan(x).

as you can see, the pattern repeats every pi radians.

here's a graph of y = tan(2x).

as you can see, the pattern repeats every pi/2 radians.

here's a graph of y = -tan(2x).

the pattern repeats every pi/2 radians and the pattern is reversed from top to bottom because the frequency is minus 1 rather than plus one.

this flips the graph vertically.

a full cycle of the tan function is between the vertical dashed lines in all the graphs.

here's a good reference that explains it all about as good as any.

https://www.purplemath.com/modules/grphtrig.htm

here's the final graph.

it's the same as the last graph except there has been a vertical displacment down 3 units.

this means the center line of the graph is at y = -3 rather than at y = 0 as it normally is.

here's your final graph of the function y = -tan(2x) - 3.

the general form is y = a * tan(b * (x - c)) + d.

the amplitude is reversed because a is equal to -1 rather than 1.

there is no horizontal shift because c is equal to 0.

the period is pi/2 because b is equal to 2 and period = pi / frequency for the tan function.

there is a vertical shift of the center line from y = 0 to y = -3 because d is equal to -3.