4. Graphing Trigonometric Functions

Given below is the graph of the function $y=\(\sin\]\left\)(bx\(\right\))$. Determine the correct value for b.

In Exercises 37–40, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, graph one period of the equation. Then find the following: a. the maximum displacement b. the frequency c. the time required for one cycle d. the phase shift of the motion. d = − 1/2 sin(πt/4 − π/2)

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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Determine the amplitude of each function. Then graph the function and y = sin x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = -3 sin x

Determine the value of $y=\(\sin\[\left\)(-\(\frac{\pi}{2}\]\right\))+50$ without using a calculator or the unit circle.

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

The graph of y = sec x in Figure 37 suggests that sec(-x) = sec x for all x in the domain of sec x.

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

y = -sin (x - 3π/4)

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 53–60, use a vertical shift to graph one period of the function. y = sin x + 2

Determine the value of $y=-2\(\cdot\]\sin\[\left\)(-\(\frac{3\pi}{2}\]\right\))+10$ without using a calculator or the unit circle.

In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = -4 cos 1/2 x