We have been given graph of a sinusoidal function. We are asked to write function formula for our given graph.
We know that [tex]\sin(0)=0[/tex] and [tex]\cos(0)=1[/tex]. We can see that our given function is starting at origin, so our graph is sine function.
We know that general form of sine is [tex]y=A\sin[B(x-c)]+D[/tex], where,
A = Amplitude,
Period: [tex]\frac{2\pi}{B}[/tex]
C = Horizontal shift,
D = Vertical shift.
We have been given that function has no horizontal shift, so the value of C is 0.
We can also see that midline of our function is x-axis, so there is no vertical shift as well that is [tex]D=0[/tex].
We can see that function goes up to 1 from midline and goes down to [tex]-1[/tex] from midline, so amplitude of function is 1 that is [tex]A=1[/tex].
We can see that period of our given function is [tex]4\pi[/tex] because it completes one cycle from [tex]-\pi[/tex] to [tex]3\pi[/tex]
[tex]4\pi=\frac{2\pi}{B}[/tex]
[tex]B=\frac{2\pi}{4\pi}=\frac{1}{2}[/tex]
[tex]y=\sin(\frac{1}{2}x)[/tex]
We can see that our function is reflected about x-axis, so our function will be [tex]y=-f(x)[/tex].
[tex]y=-\sin(\frac{1}{2}x)[/tex]
Therefore, our required function is [tex]y=-\sin(\frac{1}{2}x)[/tex].
