Question 1180527

Function 1: 

{{{y=7sin(x+5)}}}


Function 2:

{{{y=7cos(5x-1)}}}

in general: {{{y=Acos(Bx-C)+D}}}

the amplitude is {{{A}}}
{{{B=2pi*f}}}  where {{{f}}} is the frequency
{{{f=1/period}}}
the period is {{{2pi/B}}}
the phase shift is {{{C}}}
the vertical shift is {{{D}}}



Part A: Find the amplitude, period, y-minimum and maximum, and phase shift for Function 1.

{{{y=7sin(x+5)}}}

the amplitude is {{{A=7}}}
{{{B =  1}}}
the period is 2pi/1=2pi

max{ {{{7sin(x + 5)}}} } = {{{7}}} at {{{x = 2pi*n + pi/2 - 5}}} for integer {{{n}}}

min{ {{{7sin(x + 5)}}} } = {{{-7}}} at {{{x = 2pi*n + (3pi)/2 - 5}}} for integer{{{ n}}}

the phase shift is {{{C=-5}}}, {{{5}}} units to the left



Part B: Find the amplitude, period, y-minimum and maximum, and phase shift for Function 2

{{{y=7cos(5x-1)}}}

{{{y=7cos(5(x-1/5))}}}

the amplitude is {{{A=7}}}
{{{B = 5}}} 
the period is {{{2pi/B=2pi/5}}}
the phase shift is {{{C=1/5}}}, {{{1/5}}} units to the right
the vertical shift is {{{D=0}}}

max{ {{{7cos(5x - 1)}}} } = {{{7}}} at {{{x = 1/5 +(2pi*n)/5}}} for integer {{{n}}}
min{ {{{7 cos(5 x - 1)}}} } ={{{ -7}}} at {{{x = pi/5 + 1/5 +(2pi*n)/5 }}} for integer {{{n}}}


Part C: write a sentence to compare and contrast the two functions
Both functions have the same amplitude, {{{y}}} coordinate of max and min, and no vertical shift. Functions have different phase shift. 
Function 1 has the phase shift  {{{C=-5}}}, {{{5}}} units to the left, and Function 2 has the phase shift {{{C=1/5}}}, {{{ 1/5}}} units to the right