Question 1

The function graphed is of the form y = a sin bx or y = a cos bx, where b > 0. Determine the equation of the graph. 
A) $y = 4 \cos \left( \frac { 1 } { 2 } x \right)$
B) $y = - 4 \cos ( 2 x )$
C) $y = - 4 \cos \left( \frac { 1 } { 2 } x \right)$
D) $y = 4 \sin ( 2 x )$
-The function graphed is of the form y = a sin bx or y = a cos bx, where b > 0. Determine the equation of the graph. 
A) $y = 4 \cos \left( \frac { 1 } { 2 } x \right)$
B) $y = - 4 \cos ( 2 x )$
C) $y = - 4 \cos \left( \frac { 1 } { 2 } x \right)$
D) $y = 4 \sin ( 2 x )$

Accepted Answer

The answer of The function graphed is of the form...

Question 2

Graph the function over a one-period interval.
-$$y = \frac { 1 } { 2 } \sin ( x + \pi )$$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the function over a one-period interval.
-\[y...

Question 3

Graph the function.
-$y=-2-\tan \left(x+\frac{\pi}{4}\right)$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the function.
-$y=-2-\tan \left(x+\frac{\pi}{4}\right)$

A)

B)

C)

D)

...

Question 4

Solve the problem.
-The minimum length $\mathrm { L }$ of a highway sag curve can be computed by
$$\mathrm { L } = \frac { \left( \theta _ { 2 } - \theta _ { 1 } \right) \mathrm { S } ^ { 2 } } { 200 ( \mathrm {~h} + \mathrm { S } \tan \alpha ) }$$
where $\theta _ { 1 }$ is the downhill grade in degrees $\left( \theta _ { 1 } < 0 ^ { \circ } \right) , \theta _ { 2 }$ is the uphill grade in degrees $\left( \theta _ { 2 } > 0 ^ { \circ } \right)$, $\mathrm { S }$ is the safe stopping distance for a given speed limit, $h$ is the height of the headlights, and $\alpha$ is the alignment of the headlights in degrees. Compute $L$ for a 55-mph speed limit, where $h = 1.9 \mathrm { ft }$, $\alpha = 0.7 ^ { \circ } , \theta _ { 1 } = - 5 ^ { \circ } , \theta _ { 2 } = 4 ^ { \circ }$, and $S = 336 \mathrm { ft }$. Round your answer to the nearest foot.
A) $824 \mathrm { ft }$
B) $893 \mathrm { ft }$
C) $846 \mathrm { ft }$
D) $869 \mathrm { ft }$

Accepted Answer

The correct length $L$ is calculated by substituting the given values into the formula: $L = \frac{(4 - (-5)) \times 336^2}{200(1.9 + 336 \tan 0.7^\circ)}$, which rounds to $846 \mathrm { ft }$.

Question 5

Solve the problem.
-A weight attached to a spring is pulled down 5 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 3 } \mathrm { sec }$, what is the frequency of the system?
A) 3 cycles per sec
B) $\frac { 2 } { 3 \pi }$ cycles per sec
C) $\frac { 2 \pi } { 3 }$ cycles per sec
D) 6 cycles per sec

Accepted Answer

The frequency of a system is the reciprocal of the period. Given the period is $\frac{1}{3}$ sec, the frequency is $\frac{1}{\frac{1}{3}} = 3$ cycles per sec.

Question 6

Solve the problem.
-A generator produces an alternating current according to the equation I = 48 sin 109πt, where t is time in seconds and I is the current in amperes. What is the smallest time t such that I = 24? 
A) $\frac { 1 } { 327 } \mathrm { sec }$
B) $\frac { 1 } { 436 } \mathrm { sec }$
C) $\frac { 1 } { 218 } \mathrm { sec }$
D) $\frac { 1 } { 654 } \mathrm { sec }$

Accepted Answer

The answer of Solve the problem.
-A generator produces an alternating...

Question 7

The function graphed is of the form y = cos x + c, y = sin x + c, y = cos(x - d), or y = sin(x - d), where d is the least possible
positive value. Determine the equation of the graph.
-
A) $\sin \left( x - \frac { \pi } { 4 } \right)$
B) $\sin x - \frac { \pi } { 2 }$
C) $\cos \left( x - \frac { \pi } { 2 } \right)$
D) $\sin \left( x - \frac { \pi } { 2 } \right)$

Accepted Answer

The answer of The function graphed is of the form...

Question 8

Match the function with its graph.
-1) $y = \sin x$
2) $y = \cos x$
3) $y = - \sin x$
4) $y = - \cos x$

a)

b)

c)

d)

A) $1 \mathrm { C } , 2 \mathrm {~A} , 3 \mathrm {~B} , 4 \mathrm { D }$
B) $1 \mathrm {~B} , 2 \mathrm { D } , 3 \mathrm { C } , 4 \mathrm {~A}$
C) $1 \mathrm {~A} , 2 \mathrm { D } , 3 \mathrm { C } , 4 \mathrm {~B}$
D) $1 \mathrm {~A} , 2 \mathrm {~B} , 3 \mathrm { C } , 4 \mathrm { D }$

Accepted Answer

The answer of Match the function with its graph.
-1) \(y...

Question 9

The function graphed is of the form y = cos x + c, y = sin x + c, y = cos(x - d), or y = sin(x - d), where d is the least possible
positive value. Determine the equation of the graph.
-
A) $y = \sin ( x - 1 )$
B) $y = \cos ( x + 1 )$
C) $y = \cos x - 1$
D) $y = \sin x - 1$

Accepted Answer

The answer of The function graphed is of the form...

Question 10

Graph the function.
-$y=2-3 \sec \left(x-\frac{\pi}{5}\right)$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the function.
-$y=2-3 \sec \left(x-\frac{\pi}{5}\right)$

A)

B)

C)

D)

...

Question 11

Solve the problem.
-The formula for the up and down motion of a weight on a spring is given by $s ( t ) = \sin \sqrt { \frac { k } { m } } t$. If the spring constant is 5 , then what mass $m$ must be used in order to produce a period of 6 seconds?
A) $\frac { 5 \pi } { 2 }$
B) $\frac { 2 \pi } { 5 }$
C) $\frac { 4 } { \pi ^ { 2 } }$
D) $\frac { 45 } { \pi ^ { 2 } }$

Accepted Answer

The answer of Solve the problem.
-The formula for the up...

Question 12

Graph the function.
-$y=\frac{3}{5} \tan \left(\frac{3}{4} x+\frac{\pi}{3}\right)$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the function.
-$y=\frac{3}{5} \tan \left(\frac{3}{4} x+\frac{\pi}{3}\right)$

A)

B)

C)

D)

...

Question 13

Solve the problem.
-The voltage E in an electrical circuit is given by E = 1.3 cos 100πt, where t is time measured in seconds. Find the period. 
A) 50
B) $50 \pi$
C) $\frac { \pi } { 50 }$
D) $\frac { 1 } { 50 }$

Accepted Answer

The period of a cosine function of the form $cos(bt)$ is given by $\frac{2\pi}{b}$. Here, $b = 100\pi$, so the period is $\frac{2\pi}{100\pi} = \frac{1}{50}$.

Question 14

Find the specified quantity.
-Find the period of $y = \sin \left( \frac { 1 } { 4 } x - \frac { \pi } { 2 } \right)$.
A) $\frac { \pi } { 2 }$
B) $8 \pi$
C) $4 \pi$
D) $5 \pi$

Accepted Answer

The period of a sine function in the form $y = \sin(bx - c)$ is given by $T = \frac{2\pi}{|b|}$. Here, $b = \frac{1}{4}$, so the period is $T = \frac{2\pi}{1/4} = 8\pi$.

Question 15

Solve the problem.
-The chart represents the amount of fuel consumed by a machine used in manufacturing. The machine is turned on at the beginning of the day, takes a certain amount of time to reach its full
Power (the point at which it uses the most fuel per hour), runs for a certain number of hours, and is
Shut off at the end of the work day. The fuel usage per hour of the machine is represented by a
Periodic function. When does the machine first reach its full power? 
A) After 2 hours
B) After 14 hours
C) After 7 hours
D) After 4 hours

Accepted Answer

The answer of Solve the problem.
-The chart represents the amount...

Question 16

The function graphed is of the form y = cos x + c, y = sin x + c, y = cos(x - d), or y = sin(x - d), where d is the least possible
positive value. Determine the equation of the graph.
-
A) $y = \cos x - 1$
B) $y = \sin x - 1$
C) $y = \cos ( x - 1 )$
D) $y = \sin x + 1$

Accepted Answer

The answer of The function graphed is of the form...

Question 17

Solve the problem.
-The position of a weight attached to a spring is $s ( t ) = - 5 \cos 5 \pi t$ inches after $t$ seconds. When does the weight first reach its maximum height?
A) After $\frac { 5 } { 4 } \mathrm { sec }$
B) After $\frac { 1 } { 5 } \sec$
C) After $\frac { \pi } { 5 } \sec$
D) After -

Accepted Answer

The maximum height is reached when the cosine function equals 1, which happens at $t = 0$. Since the cosine function has a period of $\frac{2\pi}{5\pi} = \frac{2}{5}$ seconds, the first maximum after $t = 0$ occurs at $\frac{1}{5}$ of the period, which is $\frac{1}{5} \times \frac{2}{5} = \frac{2}{25} = \frac{1}{5}$ seconds.

Question 18

Graph the function over a one-period interval.
-$y=1+\sin (2 x-\pi)$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the function over a one-period interval.
-\(y=1+\sin...

Question 19

The function graphed is of the form y = cos x + c, y = sin x + c, y = cos(x - d), or y = sin(x - d), where d is the least possible
positive value. Determine the equation of the graph.
-
A) $\cos \left( x - \frac { \pi } { 3 } \right)$
B) $\cos \left( x - \frac { \pi } { 6 } \right)$
C) $\cos x - \frac { \pi } { 3 }$
D) $\sin \left( x - \frac { \pi } { 3 } \right)$

Accepted Answer

The answer of The function graphed is of the form...

Question 20

Graph the function.
-$y=\frac{2}{5} \sec \left(\frac{1}{2} x-\frac{\pi}{4}\right)$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the function.
-$y=\frac{2}{5} \sec \left(\frac{1}{2} x-\frac{\pi}{4}\right)$

A)

B)

C)

D)

...

Quiz 4: Graphs of the Circular Functions

Graph the function over a one-period interval. - $y = \frac { 1 } { 2 } \sin ( x + \pi )$ $Graph the function over a one-period interval. - y = \frac { 1 } { 2 } \sin ( x + \pi ) A) B) C) D)$

Graph the function. - $y=-2-\tan \left(x+\frac{\pi}{4}\right)$ $Graph the function. - y=-2-\tan \left(x+\frac{\pi}{4}\right) A) B) C) D)$

Solve the problem. -A weight attached to a spring is pulled down 5 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 3 } \mathrm { sec }$ , what is the frequency of the system?

Solve the problem. -A generator produces an alternating current according to the equation I = 48 sin 109πt, where t is time in seconds and I is the current in amperes. What is the smallest time t such that I = 24?

Graph the function. - $y=2-3 \sec \left(x-\frac{\pi}{5}\right)$ $Graph the function. - y=2-3 \sec \left(x-\frac{\pi}{5}\right) A) B) C) D)$

Solve the problem. -The formula for the up and down motion of a weight on a spring is given by $s ( t ) = \sin \sqrt { \frac { k } { m } } t$ . If the spring constant is 5 , then what mass $m$ must be used in order to produce a period of 6 seconds?

Graph the function. - $y=\frac{3}{5} \tan \left(\frac{3}{4} x+\frac{\pi}{3}\right)$ $Graph the function. - y=\frac{3}{5} \tan \left(\frac{3}{4} x+\frac{\pi}{3}\right) A) B) C) D)$

Solve the problem. -The voltage E in an electrical circuit is given by E = 1.3 cos 100πt, where t is time measured in seconds. Find the period.

Find the specified quantity. -Find the period of $y = \sin \left( \frac { 1 } { 4 } x - \frac { \pi } { 2 } \right)$ .

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 5 \cos 5 \pi t$ inches after $t$ seconds. When does the weight first reach its maximum height?

Graph the function over a one-period interval. - $y=1+\sin (2 x-\pi)$ $Graph the function over a one-period interval. - y=1+\sin (2 x-\pi) A) B) C) D)$

Graph the function. - $y=\frac{2}{5} \sec \left(\frac{1}{2} x-\frac{\pi}{4}\right)$ $Graph the function. - y=\frac{2}{5} \sec \left(\frac{1}{2} x-\frac{\pi}{4}\right) A) B) C) D)$

Quiz 4: Graphs of the Circular Functions

Graph the function over a one-period interval. - y=12sin⁡(x+π) y = \frac { 1 } { 2 } \sin ( x + \pi ) y=21​sin(x+π)

Graph the function. - y=−2−tan⁡(x+π4) y=-2-\tan \left(x+\frac{\pi}{4}\right) y=−2−tan(x+4π​)

Solve the problem. -A weight attached to a spring is pulled down 5 inches below the equilibrium position. Assuming that the period of the system is 13sec\frac { 1 } { 3 } \mathrm { sec }31​sec , what is the frequency of the system?

Solve the problem. -A generator produces an alternating current according to the equation I = 48 sin 109πt, where t is time in seconds and I is the current in amperes. What is the smallest time t such that I = 24?

The function graphed is of the form y = cos x + c, y = sin x + c, y = cos(x - d) , or y = sin(x - d) , where d is the least possible positive value. Determine the equation of the graph. -

Match the function with its graph. -1) y=sin⁡xy = \sin xy=sinx 2) y=cos⁡xy = \cos xy=cosx 3) y=−sin⁡xy = - \sin xy=−sinx 4) y=−cos⁡xy = - \cos xy=−cosx a) b) c) d)

The function graphed is of the form y = cos x + c, y = sin x + c, y = cos(x - d) , or y = sin(x - d) , where d is the least possible positive value. Determine the equation of the graph. -

Graph the function. - y=2−3sec⁡(x−π5) y=2-3 \sec \left(x-\frac{\pi}{5}\right) y=2−3sec(x−5π​)

Solve the problem. -The formula for the up and down motion of a weight on a spring is given by s(t) =sin⁡kmts ( t ) = \sin \sqrt { \frac { k } { m } } ts(t) =sinmk​​t . If the spring constant is 5 , then what mass mmm must be used in order to produce a period of 6 seconds?

Graph the function. - y=35tan⁡(34x+π3) y=\frac{3}{5} \tan \left(\frac{3}{4} x+\frac{\pi}{3}\right) y=53​tan(43​x+3π​)

Solve the problem. -The voltage E in an electrical circuit is given by E = 1.3 cos 100πt, where t is time measured in seconds. Find the period.

Find the specified quantity. -Find the period of y=sin⁡(14x−π2) y = \sin \left( \frac { 1 } { 4 } x - \frac { \pi } { 2 } \right) y=sin(41​x−2π​) .

The function graphed is of the form y = cos x + c, y = sin x + c, y = cos(x - d) , or y = sin(x - d) , where d is the least possible positive value. Determine the equation of the graph. -

Solve the problem. -The position of a weight attached to a spring is s(t) =−5cos⁡5πts ( t ) = - 5 \cos 5 \pi ts(t) =−5cos5πt inches after ttt seconds. When does the weight first reach its maximum height?

Graph the function over a one-period interval. - y=1+sin⁡(2x−π) y=1+\sin (2 x-\pi) y=1+sin(2x−π)

The function graphed is of the form y = cos x + c, y = sin x + c, y = cos(x - d) , or y = sin(x - d) , where d is the least possible positive value. Determine the equation of the graph. -

Graph the function. - y=25sec⁡(12x−π4) y=\frac{2}{5} \sec \left(\frac{1}{2} x-\frac{\pi}{4}\right) y=52​sec(21​x−4π​)

Graph the function over a one-period interval. - $y = \frac { 1 } { 2 } \sin ( x + \pi )$ $Graph the function over a one-period interval. - y = \frac { 1 } { 2 } \sin ( x + \pi ) A) B) C) D)$

Graph the function. - $y=-2-\tan \left(x+\frac{\pi}{4}\right)$ $Graph the function. - y=-2-\tan \left(x+\frac{\pi}{4}\right) A) B) C) D)$

Solve the problem. -A weight attached to a spring is pulled down 5 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 3 } \mathrm { sec }$ , what is the frequency of the system?

Graph the function. - $y=2-3 \sec \left(x-\frac{\pi}{5}\right)$ $Graph the function. - y=2-3 \sec \left(x-\frac{\pi}{5}\right) A) B) C) D)$

Solve the problem. -The formula for the up and down motion of a weight on a spring is given by $s ( t ) = \sin \sqrt { \frac { k } { m } } t$ . If the spring constant is 5 , then what mass $m$ must be used in order to produce a period of 6 seconds?

Graph the function. - $y=\frac{3}{5} \tan \left(\frac{3}{4} x+\frac{\pi}{3}\right)$ $Graph the function. - y=\frac{3}{5} \tan \left(\frac{3}{4} x+\frac{\pi}{3}\right) A) B) C) D)$

Find the specified quantity. -Find the period of $y = \sin \left( \frac { 1 } { 4 } x - \frac { \pi } { 2 } \right)$ .

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 5 \cos 5 \pi t$ inches after $t$ seconds. When does the weight first reach its maximum height?

Graph the function over a one-period interval. - $y=1+\sin (2 x-\pi)$ $Graph the function over a one-period interval. - y=1+\sin (2 x-\pi) A) B) C) D)$

Graph the function. - $y=\frac{2}{5} \sec \left(\frac{1}{2} x-\frac{\pi}{4}\right)$ $Graph the function. - y=\frac{2}{5} \sec \left(\frac{1}{2} x-\frac{\pi}{4}\right) A) B) C) D)$