4. Graphing Trigonometric Functions

Graph each function over a one-period interval.

y = -4 sin(2x - π)

Graph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.

y = ⅔ sin x

Fill in the blank(s) to correctly complete each sentence.

The graph of y = 6 + 3 sin x is obtained by shifting the graph of y = 3 sin x ________ unit(s) __________ (up/down).

Fill in the blank(s) to correctly complete each sentence.

The graph of y = 3 + 5 cos (x + π/5) is obtained by shifting the graph of y = cos x horizontally ________ unit(s) to the __________, (right/left) stretching it vertically by a factor of ________, and then shifting it vertically ________ unit(s) __________. (up/down)

Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.

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In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 2 cos (2πx + 8π)

Determine the amplitude and period of each function. Then graph one period of the function. y = -3 sin 2πx

Graph each function over a two-period interval.

y = 1 - 2 cos ((1/2)x)

Graph each function over a two-period interval.

y = sin [2(x + π/4) ] + 1/2

In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = cos 2x

Graph the function $y=-3\(\cdot\[\cos\]\left\)(x\(\right\))$.

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = sin(x − π)

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

y = 3 cos [π/2 (x - ½)]

In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the graph of h by adding or subtracting the corresponding y-coordinates on the graphs of f and g. f(x) = cos x, g(x) = sin 2x, h(x) = (f − g)(x)

Graph each function over a one-period interval. See Example 3.

y = (3/2) sin [2(x + π/4)]