Question

Math question

Answer 1

since:
\[ - 1 \leqslant \cos \left( {4x + \pi } \right) \leqslant 1\]
then we can write:
\[3 \leqslant - 3\cos \left( {4x + \pi } \right) + 6 \leqslant 9\]
so the requested amplitude is

\[A = \frac{{9 - 3}}{2} = 3\]

Answer 2

for period T, we can write this:
\[\begin{gathered}
\cos \left( {4x + \pi } \right) = \cos \left( {4x} \right)\cos \pi - \sin \left( {4x} \right)\sin \pi = \hfill \\
\hfill \\
= - \cos \left( {4x} \right) \hfill \\
\end{gathered} \]
so our function becomes:
\[f\left( x \right) = 3\cos \left( {4x} \right) + 6\]

Answer 3

now, if we make this change of variable:
\[y = 4x\]
then our function becomes:
\[f\left( y \right) = 3\cos \left( y \right) + 6\]
As you can know, the period of that function is
\[T = 2\pi \]
So what can you conclude?

Answer 4

oops..as you well know...

Answer 5

@vera_ewing

Answer 6

the amplitude of a periodic function, has to be always positive

Answer 7

since the cos(x) function is such that:
-1 </ cos(x) </ 1,
here </ stands for "less or equal to"
then we can write:

3</ f(x) </ 9
so the amplitude is:
A=(9-3)/2=...?

Answer 8

furthermore, if we make this change of variable:
4x=y, then we can rewrite your fuunction as follows:
f(x)=3*cos(4x)+6, and therefore:
f(y) = 3 cos(y)+6
the period of that function is T= 2*pi, so what can you conclude?

Answer 9

I'm sorry, I can not give the direct answer, since I have to respect the Code of Conduct

Answer 10

I think that option D is a wrong answer!

Answer 11

the period can not be 2*pi, since as I wrote before, if we use this substitution:
y=4x
then your function as a function of y has period = 2*pi, whereas as function of x has the period equal to:
(2*pi)/4=...?

Answer 12

1.57

Answer 13

better is pi/2

Answer 14

now using the same substitution, we have these correspondences:
x--->y/4
and therefore:
pi---> pi/4
so what can you conclude about the phase shift?

Answer 15

The phase shift is x=pi/4 ?

Answer 16

the phase shift is a relative quantity, so we can assume taht it is - pi/4

Answer 17

that*

Answer 18

Oh so the answer is A!

Answer 19

yes!

Answer 20

Michele thank you so much! :)

Answer 21

:)