Chapter 4, Problem 68RE

Solution

To calculate: The values of $a,b\text{ \& }h.$

  $\begin{array}{c}a=3.61\\ b=\pi \\ h=0.49\end{array}$

Given information: $f(x)=3\text{cos }2x-2\text{sin }2x$

Graph:

  

Calculation: The period of $f(x)$ is the same as the period of $\sin x$ and $\sin x$ , namely $2\pi$ .

Comparing the standard sinusoid $\sin bx$ and $\cos bx$ with the given sinusoid $\text{sin }2x$ and $\text{cos }2x$ we get $b=\pi$

The maximum value the sinusoid has reached, rounded to the nearest hundredth, is $3.61$ which the amplitude is of $f(x)$ is about $a=3.61$ .

The $x$ -intercept closest to $x=0$ , rounded to the nearest hundredth is $0.49$ so the phase shift of the sine function is about $h=0.49$

All the required values for the standard equation

  $f(x)\approx a\text{ sin}(b(x-h))$

Now by substituting $a=3.61$ , $b=\pi$ , $h=0.49$ ,

  $f\left(x\right)\approx 3.61\text{ sin }\left(x=0.49\right)$

   $\begin{array}{c}a=3.61\\ b=\pi \\ h=0.49\end{array}$