Question

Write a function with no horizontal shift for the sinusoid shown.

Answer 1

We have been given graph of a sinusoidal function. We are asked to write function formula for our given graph.

We know that
`\sin(0)=0` and
`\cos(0)=1`. 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
`y=A\sin[B(x-c)]+D`, where,

A = Amplitude,

Period:
`(2\pi)/(B)`

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
`D=0`.

We can see that function goes up to 1 from midline and goes down to
`-1` from midline, so amplitude of function is 1 that is
`A=1`.

We can see that period of our given function is
`4\pi` because it completes one cycle from
`-\pi` to
`3\pi`

`4\pi=(2\pi)/(B)`

`B=(2\pi)/(4\pi)=(1)/(2)`

`y=\sin((1)/(2)x)`

We can see that our function is reflected about x-axis, so our function will be
`y=-f(x)`.

`y=-\sin((1)/(2)x)`

Therefore, our required function is
`y=-\sin((1)/(2)x)`.