Question 1181063


 a) Given the range of a sinusoidal function is {y|y ∈ R,-6 ≤ y ≤ 0}, determine the amplitude and the equation of the central axis.

{{{f(x)=Asin(Bx+C)+D}}}

Where
{{{A}}}=Amplitude
{{{2pi/B}}}=Period
{{{C/B}}}=Phase shift
{{{D}}}=Vertical shift

Range=[{{{D-A}}},{{{A+D}}}]
or
Range=[{{{A+D}}},{{{D-A}}}]

{y|y ∈ R,-6 ≤ y ≤ 0}

{{{D-A=-6}}}
{{{D=A-6}}}............eq.1

{{{A+D=0}}}
{{{A=-D}}} ....eq.2

=>{{{D=A-6}}}=>{{{D=-D-6}}}=>{{{2D=-6}}}=>{{{D=-3}}}
=>{{{A=-(-3)=3}}}
Period of sine function is {{{2pi}}}

{{{2pi/B=2pi}}}=>{{{B=1}}}

Phase shift
{{{C/B=0/1}}}=>{{{C=0}}}

{{{f(x)=3sin(x)-3}}}

the equation of the central axis: {{{y=-3}}}


b) Determine the 2 transformations applied on the function {y|y ∈ R,-6 ≤ y ≤ 0}, to transform the range to {y|y ∈ R, 4 ≤ y ≤ 6}.

{y|y ∈ R,4 ≤ y ≤ 6}

{{{D-A=4}}}
{{{D=A+4}}}............eq.1

{{{A+D=6}}}
{{{D=6-A}}} ....eq.2
=> from eq.1 and eq.2 we have

{{{A+4=6-A}}}
{{{A+A=6-4}}}
{{{2A=2}}}
{{{A=1}}}

then
{{{D=A+4}}}=>{{{D=1+4}}}=>{{{D=5}}}

{{{f(x)=sin(x)+5}}}