determine the amplitude, period, and phase shift of the following functions: y=3cos(2x-pi), y=3sin(2x-pi),y=-5sin(2x=pi over 2)
if a sine of cosine curve is of the form
y = a sin k(θ + c) + p
you should memorize what what each of the a, k, c, and p represent
None of your given functions have any verical shifts
for all of them |a| is the amplitude
the period is 2π/coefficient of x
the only difficulty I see is the phase shift.
Notice that in the general form we have
sin k(θ + c) , that is, the k must be divided out
So for y=3cos(2x-pi)
we change it to
y=3 cos 2(x-π/2) , so the sphase shift is π/2 to the right.
the other equations present the same problem
reiny r u a tutor ?
Yes, I am one of the volunteer tutors on this site, we do not any definite time when you will find us.
I answer mostly Math questions
To determine the amplitude, period, and phase shift of the given functions, we can use the general form of the trigonometric function:
y = A sin(B(x - C)) + D
For y = 3cos(2x - π):
Amplitude (A):
- The amplitude is the absolute value of the coefficient in front of the trigonometric function.
- In this case, the coefficient is 3.
- Therefore, the amplitude is |3| = 3.
Period (P):
- The period of a trigonometric function is given by P = 2π/B, where B is the coefficient of x.
- In this case, B = 2.
- Hence, the period is P = 2π/2 = π.
Phase Shift (C):
- The phase shift determines any horizontal shift to the left or right.
- To find the phase shift, we set (B(x - C)) equal to zero and solve for x.
- In this case, (B(x - C)) = 2x - π = 0.
- Solving for x, we get 2x = π, and dividing both sides by 2, we have x = π/2.
- Therefore, the phase shift is C = π/2.
In summary, for y = 3cos(2x - π):
- Amplitude (A) = 3
- Period (P) = π
- Phase Shift (C) = π/2
For y = 3sin(2x - π):
Amplitude (A):
- The amplitude is the absolute value of the coefficient in front of the trigonometric function.
- In this case, the coefficient is 3.
- Therefore, the amplitude is |3| = 3.
Period (P):
- The period of a trigonometric function is given by P = 2π/B, where B is the coefficient of x.
- In this case, B = 2.
- Hence, the period is P = 2π/2 = π.
Phase Shift (C):
- The phase shift determines any horizontal shift to the left or right.
- To find the phase shift, we set (B(x - C)) equal to zero and solve for x.
- In this case, (B(x - C)) = 2x - π = 0.
- Solving for x, we get 2x = π, and dividing both sides by 2, we have x = π/2.
- Therefore, the phase shift is C = π/2.
In summary, for y = 3sin(2x - π):
- Amplitude (A) = 3
- Period (P) = π
- Phase Shift (C) = π/2
For y = -5sin(2x + π/2):
Amplitude (A):
- The amplitude is the absolute value of the coefficient in front of the trigonometric function.
- In this case, the coefficient is -5.
- Therefore, the amplitude is |-5| = 5.
Period (P):
- The period of a trigonometric function is given by P = 2π/B, where B is the coefficient of x.
- In this case, B = 2.
- Hence, the period is P = 2π/2 = π.
Phase Shift (C):
- The phase shift determines any horizontal shift to the left or right.
- To find the phase shift, we set (B(x - C)) equal to zero and solve for x.
- In this case, (B(x - C)) = 2x + π/2 = 0.
- Solving for x, we get 2x = -π/2, and dividing both sides by 2, we have x = -π/4.
- Therefore, the phase shift is C = -π/4.
In summary, for y = -5sin(2x + π/2):
- Amplitude (A) = 5
- Period (P) = π
- Phase Shift (C) = -π/4