Chapter 4.6, Problem 31E

Solution

To determine : The value of $a,b,\text{ and }h$ in $f\left(x\right)=\sin \pi x-2\cos \pi x$ so that $f\left(x\right)\approx a\sin \left(b\left(x-h\right)\right)$ .

The values are $a=2.24,b=\pi$ and $h=0.35$ and the function is $f\left(x\right)\approx 2.24\sin \left(\pi \left(x-0.35\right)\right)$ .

Given information :

The function $f\left(x\right)=\sin \pi x-2\cos \pi x$

Explanation :

Consider the function: $f\left(x\right)=\sin \pi x-2\cos \pi x$ .

The graph of the function is:

  

Graph indicates that the function is a sinusoid function.

The maximum value of the function is approximately $2.24$ . Thus the amplitude of the function is $2.24$ .

The $x-$ intercept of the function near to zero is, approximately $0.35$ . Therefore, the phase shift of the function is approximately $0.35$ .

Consider the function of the form,

  $f\left(x\right)\approx a\sin \left(b\left(x-h\right)\right)$

Here, $a$ is the amplitude of the function.

Hence, $a=2.24,b=\pi$

Also,

Here, $h$ is the phase shift of the function.

Hence, $h=0.35$

Therefore, the function is $f\left(x\right)\approx 2.24\sin \left(\pi \left(x-0.35\right)\right)$ .