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: $\frac{2\pi}{B}$

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 $3\pi$

$4\pi=\frac{2\pi}{B}$

$B=\frac{2\pi}{4\pi}=\frac{1}{2}$

$y=\sin(\frac{1}{2}x)$

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

$y=-\sin(\frac{1}{2}x)$

Therefore, our required function is $y=-\sin(\frac{1}{2}x)$.