What is the amplitude, period, and phase shift of f(x) = −3 sin(4x − π) − 5?
amplitude = −3; period = 2π; phase shift: x equals pi over four
amplitude = 3; period = pi over two; phase shift: x equals pi over four
amplitude = −3; period = 2π; phase shift: x equals negative pi over four
amplitude = 3; period = pi over two; phase shift: x equals negative pi over four

Second Option

Amplitude = 3; period = pi over two; phase shift: x equals pi over four

Step-by-step explanation:

By definition the sinusoidal function has the following form:

Where

is the Amplitude of the function

is the period of the function

is the phase shift

In this case the function is:

Therefore

2nd choice.

Step-by-step explanation:

Let's compare the following:

to

.

They are almost in the same form.

The amplitude is |a|, so it isn't going to be negative.

The period is .

The phase shift is .

If c is positive it has been shifted right c units.

If c is negative it has been shifted left c units.

d is the vertical shift.

If d is negative, it has been moved down d units.

If d is positive, it has been moved up d units.

So we already know two things:

The amplitude is |a|=|-3|=3.

The vertical shift is d=-5 which means it was moved down 5 units from the parent function.

Now let's find the others.

I'm going to factor out 4 from .

Like this:

Now if you compare this to

then b=4 so the period is .

Also in place of c you see which means the phase shift is .

The second choice is what we are looking for.

g(f(x)) = x² - 3x + 4

Step-by-step explanation:

Substitute x = f(x) into g(x) , that is

g(x² - 3x + 2)

= x² - 3x + 2 + 2

= x² - 3x + 4