4. Graphing Trigonometric Functions

Below is a graph of the function $y=\tan\left(bx\right)$. Determine the value of b.

Below is a graph of the function $y=\cot\left(bx+\frac{\pi}{2}\right)$. Determine the value of b.

Graph each function over a one-period interval.

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

Concept Check Match each function with its graph in choices A–F.

7.

8.

9.

10.

11. y = cot (x + π )

4

12.

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D. <IMAGE> E. <IMAGE> F. <IMAGE>

Graph each function over a one-period interval.

y = - (1/2) csc (x + π/2)

Graph each function over a one-period interval. See Examples 1–3.

y = tan 4x

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

y = (1/2)csc (2x - π/4)

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

y = 2 sec(πx - 2π)

Graph each function over a one-period interval.

y = csc (x - π/4)

Graph each function over a one-period interval. See Examples 1–3.

y = 2 tan x

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

y = 1/3 tan (3x - π/3)

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

y = cot (x/2 + 3π/4)

Graph each function over a one-period interval.

y = sec (x + π/4)

Graph each function over a one-period interval.

y = 2 tan (¼ x)

Graph each function over a one-period interval.

y = csc((1/2)x - π/4)

Graph each function over a one-period interval.

y = csc((1/2)x - π/4)

Graph each function over a one-period interval.

y = cot (3x)

Graph each function over a one-period interval.

y = (1/2) csc (2x + π/2)

Graph each function over a one-period interval.

y = (1/2) csc (2x + π/2)

Graph each function over a one-period interval.

y = -2 tan (¼ x)

Identify the circular function that satisfies each description.

​period is π; function is decreasing on the interval (0, π)

Match each function in Column I with the appropriate description in Column II.

I

y = -4 sin(3x - 2)

II

A. amplitude = 2, period = π/2, phase shift = ¾

B. amplitude = 3, period = π, phase shift = 2

C. amplitude = 4, period = 2π/3, phase shift = ⅔

D. amplitude = 2, period = 2π/3, phase shift = 4⁄3

Graph each function over a one-period interval.

y = 2 + 3 sec (2x - π)

Graph each function over a one-period interval.

y = 2 + 3 sec (2x - π)

Graph each function over a one-period interval.

y = ½ cot (4x)

Graph each function over a one-period interval.

y = ½ sec x

Graph each function over a one-period interval.

y = 1 - (1/2) csc (x - 3π/4)

Graph each function over a two-period interval.

y = tan(2x - π)

Determine an equation for each graph.

Graph each function over a two-period interval.

y = cot (3x + π/4)

Graph each function over a one-period interval.

y = -1 + csc x

Determine an equation for each graph.

Graph each function over a two-period interval.

y = 1 + tan x

Determine an equation for each graph.

Graph each function over a two-period interval.

y = 1 - cot x

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

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

Graph each function over a two-period interval.

y = -1 + 2 tan x

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.

Graph each function over a two-period interval.

y= -1 + (1/2) cot (2x - 3π)

Graph each function over a two-period interval.

y = 1 - 2 cot [2(x + π/2)]

Consider the function g(x) = -2 csc (4x + π). What is the domain of g? What is its range?

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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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.)

<IMAGE> ​​

A rotating beacon is located at point A, 4 m from a wall. The distance a is given by

a = 4 |sec 2πt|,

where t is time in seconds since the beacon started rotating. Find the value of a for each time t. Round to the nearest tenth if applicable.

<IMAGE>

t = 1.24

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.)

<IMAGE>

Consider the following function from Example 5. Work these exercises in order.

y = -2 - cot (x - π/4)

Based on the answer in Exercise 58 and the fact that the cotangent function has period π, give the general form of the equations of the asymptotes of the graph of y = -2 - cot (x - π/4).

Let n represent any integer.

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

y = tan 3x

Consider the following function from Example 5. Work these exercises in order.

y = -2 - cot (x - π/4)

Use the fact that the period of this function is π to find the next positive x-intercept. Round to the nearest hundredth.

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)

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

y = sec (x - π/2)

Match each function with its graph in choices A–F.

y = tan (x - π )

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D. <IMAGE> E. <IMAGE> F. <IMAGE>