Question 1135335
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).


<img src = "http://theo.x10hosting.com/2019/022301.jpg" alt="$$$" >


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


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


<img src = "http://theo.x10hosting.com/2019/022302.jpg" alt="$$$" >


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


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


<img src = "http://theo.x10hosting.com/2019/022303.jpg" alt="$$$" >


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.


<a href = "https://www.purplemath.com/modules/grphtrig.htm" target = "_blank">https://www.purplemath.com/modules/grphtrig.htm</a>


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.


<img src = "http://theo.x10hosting.com/2019/022304.jpg" alt="$$$" >


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.