Question 5.4.5: A secant function with a transformation Sketch two cycles of...

Precalculus Functions and Graphs

A secant function with a transformation

Sketch two cycles of the function y = 2 sec(x – π/2), and determine the period and the range of the function.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

Since

y=2 \sec \left(x-\frac{\pi}{2}\right)=\frac{2}{\cos (x-\pi / 2)}

we first graph y = cos(x – π/2), as shown in Fig. 5.71. The function y = sec(x – π/2) goes through the maximum and minimum points on the graph of y = cos(x – π/2), but y = 2 sec(x – π/2) stretches y = sec(x – π/2) by a factor of 2. So the portions of the curve that open up do not go lower than 2, and the portions that open down do not go higher than -2, as shown in the figure. The period is 2π, and the range is $(-\infty,-2] \cup[2, \infty)$ .

5.71

Related Answered Questions

Question: 5.6.7

Finding the height of an object from a distance The angle of elevation of the top of a water tower from point A on the ground is 19.9°. From point B, 50.0 feet closer to the tower, the angle of elevation is 21.8°. What is the height of the tower? ...

Verified Answer:

Let y represent the height of the tower and x repr...

Question: 5.6.9

Graphing calculator required A 20-inch belt connects a pulley with a very small shaft on a motor, as shown in Fig. 5.91. The distance between the shaft and the large pulley is 4 inches. Assuming the shaft is a single point, find the radius of the large pulley to the nearest tenth of an inch. ...

Verified Answer:

Let x be the radius of the circle and w be the dis...

Question: 5.6.8

Photography from a spy plane In the late 1950s, the Soviets labored to develop a missile that could stop the U-2 spy plane. On May 1, 1960, Nikita S. Khrushchev announced to the world that the Soviets had shot down Francis Gary Powers while Powers was photographing the Soviet Union from a U-2 at ...

Verified Answer:

Figure 5.90 shows the line of sight to the horizon...

Question: 5.6.6

Finding the height of an object A guy wire of length 108 meters runs from the top of an antenna to the ground. If the angle of elevation of the top of the antenna, sighting along the guy wire, is 42.3°, then what is the height of the antenna? ...

Verified Answer:

Let y represent the height of the antenna, as show...

Question: 5.6.5

Significant digits Assume that the given numbers are measurements. Perform each computation, and give the answer with the appropriate number of significant digits. a. sin(123.4°) b. cos(3°)/sin(1.9°) c. 544 . sin(25°) ...

Verified Answer:

.a. Since 123.4° has four significant digits, we r...

Question: 5.5.8

Evaluating compositions of functions Find the exact value of each composition without using a table or a calculator. a. sin(tan^-1(0)) b. arcsin(cos(π/6)) c. tan(sec^-1(√2)) d. arcsin(sin(4π/3)) ...

Verified Answer:

a. Since

\tan (0)=0, \tan ^{-1}(0)=0[/late...

Question: 5.5.7

Evaluating the inverse functions with a calculator Find the approximate value of each expression rounded to four decimal places. a. sin^-1(0.88) b. arccot(2.4) c. csc^-1(4) d. cot^-1(0) ...

Verified Answer:

a. Use the inverse sine function in radian mode to...

Question: 5.5.5

Finding an angle given its cosine In each case find the angle α to the nearest tenth of a degree, given that 0° < α < 180°. a. cos α = 0.23 b. cos α = -0.82 ...

Verified Answer:

a. Since

\cos ^{-1}(0.23)

is the u...

Question: 5.5.3

Finding an angle given its sine Let α be an angle such that -90° < α < 90°. In each case, find α to the nearest tenth of a degree. a. sin α = 0.88 b. sin α = -0.27 ...

Verified Answer:

a. The value of

\sin ^{-1}(0.88)

i...

Question: 5.4.6

A cosecant function with a transformation Sketch two cycles of the graph of y = csc(2x – 2π/3) and determine the period and the range of the function. ...

Verified Answer:

Since

\begin{aligned} y &=\csc (2 x-2 ...