4. Graphing Trigonometric Functions

Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts. (Midpoints and quarter points are identified by dots.)

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Graph each function over a one-period interval.

y = -1 + csc x

In Exercises 25–28, use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.

Graph each function over a one-period interval.

y = 3 sec [(1/4)x]

In Exercises 18–24, graph two full periods of the given tangent or cotangent function. y = −2 tan π/4 x

Graph each function over a one-period interval.

y = sec (x + π/4)

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = 1/3 tan (3x - π/3)

Graph each function over a one-period interval.

y = ½ cot (4x)

Decide whether each statement is true or false. If false, explain why.

The tangent and secant functions are undefined for the same values.

In Exercises 29–44, graph two periods of the given cosecant or secant function. y = sec x/3

Match each function with its graph in choices A - D.

y = sec (x - π/2)

Graph each function over a one-period interval.

y = csc((1/2)x - π/4)

Graph each function over a one-period interval.

y = (1/2) csc (2x + π/2)

Graph each function over a two-period interval.

y = cot (3x + π/4)

In Exercises 53–54, let f(x) = 2 sec x, g(x) = −2 tan x, and h(x) = 2x − π/2. Graph two periods of y = (f∘h)(x).