Question 1

Consider the function $$y = \cos ( 6 x - \pi )$$ . Specify the period.

Accepted Answer

A)  $$\frac { \pi } { 6 }$$ 
B)  $$1$$ 
C)  $$2 \pi$$ 
D)  $$\frac { \pi } { 3 }$$ 
E)  $$\frac { 4 \pi } { 3 }$$ 
A)  $$\frac { \pi } { 6 }$$ 
B)  $$1$$ 
C)  $$2 \pi$$ 
D)  $$\frac { \pi } { 3 }$$ 
E)  $$\frac { 4 \pi } { 3 }$$ D

Question 2

Graph the function for one period, and show (or specify) intercepts and asymptotes. $$y = \cot \left( x + \frac { \pi } { 3 } \right)$$

Accepted Answer

A) x-intercept: none; y-intercept: 
$$\frac { 2 \sqrt { 3 } } { 3 }$$; asymptotes: $x = - \frac { \pi } { 3 }$ and $$x = \frac { 2 \pi } { 3 }$$
  
B)  x-intercept: none; y-intercept: 
$$- \frac { 2 \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$   
C)  x-intercept: $$\frac { \pi } { 6 }$$ ; y-intercept: $$\frac { \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$   
D)  x-intercept: $$\frac { \pi } { 6 }$$ ; y-intercept: $$\frac { \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$   
E)  x-intercept: $$\frac { \pi } { 6 }$$ ; y-intercept: $$- \frac { \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$   
A) x-intercept: none; y-intercept: 
$$\frac { 2 \sqrt { 3 } } { 3 }$$; asymptotes: $x = - \frac { \pi } { 3 }$ and $$x = \frac { 2 \pi } { 3 }$$
  
B)  x-intercept: none; y-intercept: 
$$- \frac { 2 \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$   
C)  x-intercept: $$\frac { \pi } { 6 }$$ ; y-intercept: $$\frac { \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$   
D)  x-intercept: $$\frac { \pi } { 6 }$$ ; y-intercept: $$\frac { \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$   
E)  x-intercept: $$\frac { \pi } { 6 }$$ ; y-intercept: $$- \frac { \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$   C

Question 3

Consider the function $$y = 8 \sin \frac { \pi x } { 8 }$$ on the interval $$[ 0,16 ]$$ . Determine the $$\chi$$ -intercepts by giving the $$\chi$$ -coordinate(s).

Accepted Answer

A)  $$0,8,24$$ 
B)  $$0,8,16$$ 
C)  $$8,16$$ 
D)  $$0,16$$ 
E)  $$0,8 \pi , 16 \pi$$ 
A)  $$0,8,24$$ 
B)  $$0,8,16$$ 
C)  $$8,16$$ 
D)  $$0,16$$ 
E)  $$0,8 \pi , 16 \pi$$ B

Question 4

State whether the function $$y = - \sin x$$ is increasing or decreasing on the interval. $$- \frac { 3 \pi } { 2 } < x < - \pi$$

Accepted Answer

A)  increasing 
B)  decreasing 
A)  increasing 
B)  decreasing

Question 5

Given: $$\cot \theta + \tan \theta + 1$$ . Determine the right side of the identity equation.

Accepted Answer

A)  $$\frac { \tan \theta } { 1 - \tan \theta } + \frac { \cot \theta } { 1 - \cot \theta }$$ 
B)  $$\frac { \cot \theta } { 1 - \tan \theta } + \frac { \tan \theta } { 1 - \cot \theta }$$ 
C)  $$\frac { \tan \theta } { 1 + \tan \theta } + \frac { \cot \theta } { 1 + \cos \theta }$$ 
D)  $$\frac { \cot \theta } { 1 + \tan \theta } + \frac { \tan \theta } { 1 + \cot \theta }$$ 
E)  $$\frac { \cos \theta } { 1 - \tan \theta } + \frac { \cos \theta } { 1 - \cot \theta }$$ 
A)  $$\frac { \tan \theta } { 1 - \tan \theta } + \frac { \cot \theta } { 1 - \cot \theta }$$ 
B)  $$\frac { \cot \theta } { 1 - \tan \theta } + \frac { \tan \theta } { 1 - \cot \theta }$$ 
C)  $$\frac { \tan \theta } { 1 + \tan \theta } + \frac { \cot \theta } { 1 + \cos \theta }$$ 
D)  $$\frac { \cot \theta } { 1 + \tan \theta } + \frac { \tan \theta } { 1 + \cot \theta }$$ 
E)  $$\frac { \cos \theta } { 1 - \tan \theta } + \frac { \cos \theta } { 1 - \cot \theta }$$

Question 6

Graph the function for one period, and show (or specify) the intercepts and the asymptotes. $$y = - \tan \left( x - \frac { \pi } { 6 } \right)$$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 7

Graph the function for one period, and show (or specify) intercepts and asymptotes. $$y = - \frac { 1 } { 4 } \csc ( 4 \pi x )$$

Accepted Answer

A)  x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$   
B)  x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$   
C)  x-intercept: none; y-intercept: none; asymptotes: $$x = 0$$ and $$x = \frac { 1 } { 4 }$$  
Chapter 8
Answer Section
MULTIPLE CHOICE 
D)  x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$   
E)  x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$   
A)  x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$   
B)  x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$   
C)  x-intercept: none; y-intercept: none; asymptotes: $$x = 0$$ and $$x = \frac { 1 } { 4 }$$  
Chapter 8
Answer Section
MULTIPLE CHOICE 
D)  x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$   
E)  x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$

Question 8

Use the Pythagorean identities to simplify the expression. $$\frac { \cos ^ { 2 } t + \sin ^ { 2 } t } { \cot ^ { 2 } t + 1 }$$

Accepted Answer

A)  $$\sin ^ { 2 } t$$ 
B)  $$\csc ^ { 2 } \theta$$ 
C)  $$\cos ^ { 2 } \theta$$ 
D)  1 
E)  $$\sec ^ { 2 } \theta$$ 
A)  $$\sin ^ { 2 } t$$ 
B)  $$\csc ^ { 2 } \theta$$ 
C)  $$\cos ^ { 2 } \theta$$ 
D)  1 
E)  $$\sec ^ { 2 } \theta$$

Question 9

Consider the function $$f ( x ) = \sin \left( x - \frac { \pi } { 4 } \right)$$ . Specify the phase shift.

Accepted Answer

A)  $$\frac { \pi } { 4 }$$ 
B)  $$\frac { \pi } { 8 }$$ 
C)  $$- \frac { \pi } { 4 }$$ 
D)  $$\frac { 3 \pi } { 4 }$$ 
E)  $$0$$ 
A)  $$\frac { \pi } { 4 }$$ 
B)  $$\frac { \pi } { 8 }$$ 
C)  $$- \frac { \pi } { 4 }$$ 
D)  $$\frac { 3 \pi } { 4 }$$ 
E)  $$0$$

Question 10

Consider the function $$y = 4 \sin x$$ . Specify the amplitude.

Accepted Answer

A)  1 
B)  4 
C)  5 
D)  8 
E)  2 
A)  1 
B)  4 
C)  5 
D)  8 
E)  2

Question 11

Consider the function $$y = \cos 3 x$$ . Specify the period.

Accepted Answer

A)  $$\frac { \pi } { 3 }$$ 
B)  $$\frac { 2 \pi } { 3 }$$ 
C)  $$\frac { 6 \pi } { 7 }$$ 
D)  $$2 \pi$$ 
E)  $$- \frac { 2 \pi } { 3 }$$ 
A)  $$\frac { \pi } { 3 }$$ 
B)  $$\frac { 2 \pi } { 3 }$$ 
C)  $$\frac { 6 \pi } { 7 }$$ 
D)  $$2 \pi$$ 
E)  $$- \frac { 2 \pi } { 3 }$$

Question 12

Graph the function for one period, and show (or specify) the intercepts and the asymptotes. $$y = \tan \left( x + \frac { \pi } { 2 } \right)$$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 13

Consider the function $$y = \sin \frac { x } { 4 } - 5$$ on the interval $$[ 0,8 \pi ]$$ . Specify the intervals in which the function is increasing.

Accepted Answer

A)  $$( 0,2 \pi )$$ 
B)  $$( 0,2 \pi ) , ( 6 \pi , 8 \pi )$$ 
C)  $$( 6 \pi , 8 \pi )$$ 
D)  $$( 2 \pi , 6 \pi ) , ( 8 \pi , 9 \pi )$$ 
E)  $$( 0,8 \pi )$$ 
A)  $$( 0,2 \pi )$$ 
B)  $$( 0,2 \pi ) , ( 6 \pi , 8 \pi )$$ 
C)  $$( 6 \pi , 8 \pi )$$ 
D)  $$( 2 \pi , 6 \pi ) , ( 8 \pi , 9 \pi )$$ 
E)  $$( 0,8 \pi )$$

Question 14

State whether the function $$y = \sin x$$ is increasing or decreasing on the interval. $$- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$

Accepted Answer

A)  decreasing 
B)  increasing 
A)  decreasing 
B)  increasing

Question 15

Refer to the graph of $$y = \cos x$$ in the figure. Specify the coordinates of the point I.

Accepted Answer

A)  $$( - 3 \pi , 1 )$$ 
B)  $$\left( - \frac { 3 \pi } { 2 } , 1 \right)$$ 
C)  $$\left( - \frac { 7 \pi } { 2 } , 1 \right)$$ 
D)  $$( - 5 \pi , 1 )$$ 
E)  $$( - 2 \pi , 1 )$$ 
A)  $$( - 3 \pi , 1 )$$ 
B)  $$\left( - \frac { 3 \pi } { 2 } , 1 \right)$$ 
C)  $$\left( - \frac { 7 \pi } { 2 } , 1 \right)$$ 
D)  $$( - 5 \pi , 1 )$$ 
E)  $$( - 2 \pi , 1 )$$

Question 16

Refer to the graph of $$y = - \sin x$$ in the figure. Specify the coordinates of the point I.

Accepted Answer

A)  $$( 15.71,0 )$$ 
B)  $$( 6.28,0 )$$ 
C)  $$( 12.56,0 )$$ 
D)  $$( 1.57,0 )$$ 
E)  $$( 4.71,0 )$$ 
A)  $$( 15.71,0 )$$ 
B)  $$( 6.28,0 )$$ 
C)  $$( 12.56,0 )$$ 
D)  $$( 1.57,0 )$$ 
E)  $$( 4.71,0 )$$

Question 17

Graph the function for one period, and show (or specify) intercepts and asymptotes. $$y = - \csc \left( \frac { x } { 4 } \right)$$

Accepted Answer

A)  x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$   
B)  x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ and $$x = 0$$   
C)  x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$   
D)  x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$   
E)  x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$   
A)  x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$   
B)  x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ and $$x = 0$$   
C)  x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$   
D)  x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$   
E)  x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$

Question 18

Specify the period and amplitude for the function.

Accepted Answer

A)  period: 8; amplitude: 8 
B)  period: 4; amplitude: 16 
C)  period: 2; amplitude: 16 
D)  period: 4; amplitude: 8 
E)  period: 2; amplitude: 8 
A)  period: 8; amplitude: 8 
B)  period: 4; amplitude: 16 
C)  period: 2; amplitude: 16 
D)  period: 4; amplitude: 8 
E)  period: 2; amplitude: 8

Question 19

Determine whether the equation for the graph has the form $$y = A \sin B x$$ or $$y = A \cos B x$$ ( with $$B > 0$$ ) and then find the values of $$A$$ and $$ { B }$$ .

Accepted Answer

A)  $$y = 5 \sin 6 x$$ 
B)  $$y = 5 \sin \frac { x } { 6 }$$ 
C)  $$y = - \frac { 1 } { 5 } \cos \frac { x } { 6 }$$ 
D)  $$y = 5 \cos \frac { x } { 6 }$$ 
E)  $$y = - 5 \cos 6 x$$ 
A)  $$y = 5 \sin 6 x$$ 
B)  $$y = 5 \sin \frac { x } { 6 }$$ 
C)  $$y = - \frac { 1 } { 5 } \cos \frac { x } { 6 }$$ 
D)  $$y = 5 \cos \frac { x } { 6 }$$ 
E)  $$y = - 5 \cos 6 x$$

Question 20

Consider the function $$y = 1 - \cos \frac { \pi x } { 4 }$$ on the interval $$[ 0,8 ]$$ . Determine the $$\chi$$ -intercepts by giving the $$\chi$$ -coordinate(s).

Accepted Answer

A)  $$0,4$$ 
B)  $$0,8$$ 
C)  $$- 4$$ 
D)  $$8$$ 
E)  there are no $$\chi$$ -intercepts 
A)  $$0,4$$ 
B)  $$0,8$$ 
C)  $$- 4$$ 
D)  $$8$$ 
E)  there are no $$\chi$$ -intercepts

Consider the function $y = \cos ( 6 x - \pi )$ . Specify the period.

Graph the function for one period, and show (or specify) intercepts and asymptotes. $y = \cot \left( x + \frac { \pi } { 3 } \right)$

Consider the function $y = 8 \sin \frac { \pi x } { 8 }$ on the interval $[ 0,16 ]$ . Determine the $\chi$ -intercepts by giving the $\chi$ -coordinate(s).

State whether the function $y = - \sin x$ is increasing or decreasing on the interval. $- \frac { 3 \pi } { 2 } < x < - \pi$

Given: $\cot \theta + \tan \theta + 1$ . Determine the right side of the identity equation.

Graph the function for one period, and show (or specify) the intercepts and the asymptotes. $y = - \tan \left( x - \frac { \pi } { 6 } \right)$

Graph the function for one period, and show (or specify) intercepts and asymptotes. $y = - \frac { 1 } { 4 } \csc ( 4 \pi x )$

Use the Pythagorean identities to simplify the expression. $\frac { \cos ^ { 2 } t + \sin ^ { 2 } t } { \cot ^ { 2 } t + 1 }$

Consider the function $f ( x ) = \sin \left( x - \frac { \pi } { 4 } \right)$ . Specify the phase shift.

Consider the function $y = 4 \sin x$ . Specify the amplitude.

Consider the function $y = \cos 3 x$ . Specify the period.

Graph the function for one period, and show (or specify) the intercepts and the asymptotes. $y = \tan \left( x + \frac { \pi } { 2 } \right)$

Consider the function $y = \sin \frac { x } { 4 } - 5$ on the interval $[ 0,8 \pi ]$ . Specify the intervals in which the function is increasing.

State whether the function $y = \sin x$ is increasing or decreasing on the interval. $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$

Refer to the graph of $y = \cos x$ in the figure. Specify the coordinates of the point I. $Refer to the graph of y = \cos x in the figure. Specify the coordinates of the point I.$

Refer to the graph of $y = - \sin x$ in the figure. Specify the coordinates of the point I. $Refer to the graph of y = - \sin x in the figure. Specify the coordinates of the point I.$

Graph the function for one period, and show (or specify) intercepts and asymptotes. $y = - \csc \left( \frac { x } { 4 } \right)$

Specify the period and amplitude for the function.

Consider the function $y = 1 - \cos \frac { \pi x } { 4 }$ on the interval $[ 0,8 ]$ . Determine the $\chi$ -intercepts by giving the $\chi$ -coordinate(s).

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

Exponential and Logarithmic Functions

An Introduction to Trigonometry Via Right Triangles

The Trigonometric Functions

Analytical Trigonometry

Additional Topics in Trigonometry

Systems of Equations

The Conic Sections

Roots of Polynomial Equations

Additional Topics in Algebra

Filters

Exam 8: Graphs of the Trigonometric Functions

Consider the function y=cos⁡(6x−π)y = \cos ( 6 x - \pi )y=cos(6x−π) . Specify the period.

Graph the function for one period, and show (or specify) intercepts and asymptotes. y=cot⁡(x+π3)y = \cot \left( x + \frac { \pi } { 3 } \right)y=cot(x+3π​)

Consider the function y=8sin⁡πx8y = 8 \sin \frac { \pi x } { 8 }y=8sin8πx​ on the interval [0,16][ 0,16 ][0,16] . Determine the χ\chiχ -intercepts by giving the χ\chiχ -coordinate(s).

State whether the function y=−sin⁡xy = - \sin xy=−sinx is increasing or decreasing on the interval. −3π2<x<−π- \frac { 3 \pi } { 2 } < x < - \pi−23π​<x<−π

Given: cot⁡θ+tan⁡θ+1\cot \theta + \tan \theta + 1cotθ+tanθ+1 . Determine the right side of the identity equation.

Graph the function for one period, and show (or specify) the intercepts and the asymptotes. y=−tan⁡(x−π6)y = - \tan \left( x - \frac { \pi } { 6 } \right)y=−tan(x−6π​)

Graph the function for one period, and show (or specify) intercepts and asymptotes. y=−14csc⁡(4πx)y = - \frac { 1 } { 4 } \csc ( 4 \pi x )y=−41​csc(4πx)

Use the Pythagorean identities to simplify the expression. cos⁡2t+sin⁡2tcot⁡2t+1\frac { \cos ^ { 2 } t + \sin ^ { 2 } t } { \cot ^ { 2 } t + 1 }cot2t+1cos2t+sin2t​

Consider the function f(x)=sin⁡(x−π4)f ( x ) = \sin \left( x - \frac { \pi } { 4 } \right)f(x)=sin(x−4π​) . Specify the phase shift.

Consider the function y=4sin⁡xy = 4 \sin xy=4sinx . Specify the amplitude.

Consider the function y=cos⁡3xy = \cos 3 xy=cos3x . Specify the period.

Graph the function for one period, and show (or specify) the intercepts and the asymptotes. y=tan⁡(x+π2)y = \tan \left( x + \frac { \pi } { 2 } \right)y=tan(x+2π​)

Consider the function y=sin⁡x4−5y = \sin \frac { x } { 4 } - 5y=sin4x​−5 on the interval [0,8π][ 0,8 \pi ][0,8π] . Specify the intervals in which the function is increasing.

State whether the function y=sin⁡xy = \sin xy=sinx is increasing or decreasing on the interval. −π2<x<π2- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }−2π​<x<2π​

Refer to the graph of y=cos⁡xy = \cos xy=cosx in the figure. Specify the coordinates of the point I.

Refer to the graph of y=−sin⁡xy = - \sin xy=−sinx in the figure. Specify the coordinates of the point I.

Graph the function for one period, and show (or specify) intercepts and asymptotes. y=−csc⁡(x4)y = - \csc \left( \frac { x } { 4 } \right)y=−csc(4x​)

Specify the period and amplitude for the function.

Determine whether the equation for the graph has the form y=Asin⁡Bxy = A \sin B xy=AsinBx or y=Acos⁡Bxy = A \cos B xy=AcosBx ( with B>0B > 0B>0 ) and then find the values of AAA and B { B }B .

Consider the function y=1−cos⁡πx4y = 1 - \cos \frac { \pi x } { 4 }y=1−cos4πx​ on the interval [0,8][ 0,8 ][0,8] . Determine the χ\chiχ -intercepts by giving the χ\chiχ -coordinate(s).

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

Exponential and Logarithmic Functions

An Introduction to Trigonometry Via Right Triangles

The Trigonometric Functions

Analytical Trigonometry

Additional Topics in Trigonometry

Systems of Equations

The Conic Sections

Roots of Polynomial Equations

Additional Topics in Algebra

Filters

Consider the function $y = \cos ( 6 x - \pi )$ . Specify the period.

Graph the function for one period, and show (or specify) intercepts and asymptotes. $y = \cot \left( x + \frac { \pi } { 3 } \right)$

Consider the function $y = 8 \sin \frac { \pi x } { 8 }$ on the interval $[ 0,16 ]$ . Determine the $\chi$ -intercepts by giving the $\chi$ -coordinate(s).

State whether the function $y = - \sin x$ is increasing or decreasing on the interval. $- \frac { 3 \pi } { 2 } < x < - \pi$

Given: $\cot \theta + \tan \theta + 1$ . Determine the right side of the identity equation.

Graph the function for one period, and show (or specify) the intercepts and the asymptotes. $y = - \tan \left( x - \frac { \pi } { 6 } \right)$

Graph the function for one period, and show (or specify) intercepts and asymptotes. $y = - \frac { 1 } { 4 } \csc ( 4 \pi x )$

Use the Pythagorean identities to simplify the expression. $\frac { \cos ^ { 2 } t + \sin ^ { 2 } t } { \cot ^ { 2 } t + 1 }$

Consider the function $f ( x ) = \sin \left( x - \frac { \pi } { 4 } \right)$ . Specify the phase shift.

Consider the function $y = 4 \sin x$ . Specify the amplitude.

Consider the function $y = \cos 3 x$ . Specify the period.

Graph the function for one period, and show (or specify) the intercepts and the asymptotes. $y = \tan \left( x + \frac { \pi } { 2 } \right)$

Consider the function $y = \sin \frac { x } { 4 } - 5$ on the interval $[ 0,8 \pi ]$ . Specify the intervals in which the function is increasing.

State whether the function $y = \sin x$ is increasing or decreasing on the interval. $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$

Refer to the graph of $y = \cos x$ in the figure. Specify the coordinates of the point I. $Refer to the graph of y = \cos x in the figure. Specify the coordinates of the point I.$

Refer to the graph of $y = - \sin x$ in the figure. Specify the coordinates of the point I. $Refer to the graph of y = - \sin x in the figure. Specify the coordinates of the point I.$

Graph the function for one period, and show (or specify) intercepts and asymptotes. $y = - \csc \left( \frac { x } { 4 } \right)$

Consider the function $y = 1 - \cos \frac { \pi x } { 4 }$ on the interval $[ 0,8 ]$ . Determine the $\chi$ -intercepts by giving the $\chi$ -coordinate(s).