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Functions f(x) and g(x) are shown below:

f(x) = 2cos(x)

g(x) = (3)sin(x+pi), a graph of sine function which starts at 0 comma 0 and decreases to the minimum of pi over 2 then increases to the maximum 3 pi over 2 then decreases to 2pi where the cycle repeats.

Using complete sentences, explain how to find the maximum value for each function and determine which function has the largest maximum y-value.

Please please please help!!!

Respuesta :

evgeniylevi

Try this option:

the functions y=sin(x) and y=cos(x) are the standart functions; it means, the period of each is 2Ï€, maximum value is '1', minimum value is '-1'.

Using the properties described above, to find the max. value for the function y=2cos(x): 2*1=2, where 2- the amplitude of the given function, 1- maximum of the standart function;

the max. value for the function y=3sin(x+Ï€): 3*1=3, where 3 - the amplitude of the given function, 1- maximum of the standart function.

For more details see the attached picture.

In other words, common equation for such functions is y=A*sin(ωx+Ф), where A- amplitude, ω - frequency and Ф - initial phase of the given function.

Max. and min. values depend on A, all the points of the function repeat every 2Ï€, including max. and min. values.

Ver imagen evgeniylevi

Answer with explanation:

  • The function f(x) is given by:

[tex]f(x)=2\cos x[/tex]

We know that the function f(x) is a cosine function and the maximum value is obtained by the function when this cosine function takes the maximum value.

We know that:

[tex]-1\leq \cos x\leq 1[/tex]

This means that the cosine function  takes the maximum values as: 1

and when [tex]\cos x=1[/tex] then

[tex]f(x)=2\times 1\\\\i.e.\\\\f(x)=2[/tex]

i.e. Maximum value of function f(x)=2

  • The function g(x) is given by:

[tex]g(x)=3\sin (x+\pi)[/tex]

Again the function g(x) will attain the maximum value when the function sine will takes the maximum value.

We know that the range of the sine function is: [-1,1]

This means that the maximum value of sine function is: 1

Hence, when [tex]\sin (x+\pi)=1[/tex] then,

[tex]g(x)=3\times 1\\\\i.e.\\\\g(x)=3[/tex]

i.e. Maximum value of function g(x)=3

          The function g(x) has the largest maximum value.

                            ( Since 3>2)

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