No calculator, leave in exact form. Determine the amplitude, period and phase shift of the function.

Question

Determine the amplitude, period, and phase shift of the function. Then graph one period of the function. Show your steps (basic graph and all transformations).

𝑓(𝑥)=−tan(𝑥+𝜋/3)

a.) Period

b.) Phase Shift

c.) Graph

Answer 1

Recall the general form of a trig function (here, let's use cosine)

y = acos(bx+c)+d,

where a,b,c,d are real numbers. Then

1. |a| is the amplitude, if a<0, then the graph of the parent function cos(x) has been completely flipped
2. -c/b is the phase shift (if -c/b<0 shift the graph of cos(x) |c/b| units left, and if -c/b >0, shift the graph of cos(x) c/b units right)
3. T/b is the period, where T is the period of the parent function, cos(x)
4. d is the vertical shift

So to tackle the given problem, we need to know some facts about the parent function, tan(x), mainly, it has period T=π. Hence the phase shift is -π/3 and the period is π. a=-1 in this problem, so the graph of tan(x) has been completely flipped. So to graph the new equation, first graph tan(x), flip it (ie, negate every value, and then shift the graph π/3 units to the left