Answer:
2nd choice.
Step-by-step explanation:
Let's compare the following:
[tex]f(x)=a\sin(b(x-c))+d[/tex] to
[tex]f(x)=-3\sin(4x-\pi))-5[/tex].
They are almost in the same form.
The amplitude is |a|, so it isn't going to be negative.
The period is [tex]\frac{2\pi}{|b|}[/tex].
The phase shift is [tex]c[/tex].
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 [tex]4x-\pi[/tex].
Like this:
[tex]4(x-\frac{\pi}{4})[/tex]
Now if you compare this to [tex]b(x-c)[/tex]
then b=4 so the period is [tex]\frac{2\pi}{4}=\frac{\pi}{2}[/tex].
Also in place of c you see [tex]\frac{\pi}{4}[/tex] which means the phase shift is [tex]\frac{\pi}{4}[/tex].
The second choice is what we are looking for.
Answer: 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:
[tex]f(x) = asin(bx - c) +k[/tex]
Where
[tex]| a |[/tex] is the Amplitude of the function
[tex]\frac{2\pi}{b}[/tex] is the period of the function
[tex]-\frac{c}{b}[/tex] is the phase shift
In this case the function is:
[tex]f(x) = -3 sin(4x - \pi) - 5[/tex]
Therefore
[tex]Amplitude=|a|=3[/tex]
[tex]Period =\frac{2\pi}{b} = \frac{2\pi}{4}=\frac{\pi}{2}[/tex]
[tex]phase\ shift = -\frac{(-\pi)}{4}=\frac{\pi}{4}[/tex]
