vera_ewing one year ago Math question

1. vera_ewing

2. Michele_Laino

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 $A = \frac{{9 - 3}}{2} = 3$

3. Michele_Laino

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: $f\left( x \right) = 3\cos \left( {4x} \right) + 6$

4. Michele_Laino

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?

5. Michele_Laino

oops..as you well know...

6. Michele_Laino

@vera_ewing

7. Michele_Laino

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

8. Michele_Laino

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=...?

9. Michele_Laino

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?

10. Michele_Laino

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

11. Michele_Laino

I think that option D is a wrong answer!

12. Michele_Laino

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=...?

13. vera_ewing

1.57

14. Michele_Laino

better is pi/2

15. Michele_Laino

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?

16. vera_ewing

The phase shift is x=pi/4 ?

17. Michele_Laino

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

18. Michele_Laino

that*

19. vera_ewing

Oh so the answer is A!

20. Michele_Laino

yes!

21. vera_ewing

Michele thank you so much! :)

22. Michele_Laino

:)