Chapter 4.5, Problem 69E

To determine

To find: $a$ and $d$ for the given function, such that the graph of $f$ matches the figure.

Answer to Problem 69E

  $\begin{array}{c}a=-3\\ b=2\\ c=0\end{array}$

Explanation of Solution

Given information:

Given function is

  $f\left(x\right)=a\sin \left(bx-c\right)$

Given graph

  

Concept used:

For the standard equation is $y=a\sin \left(bx-c\right)+d$ .

Amplitude is $\left|a\right|$ and period is $\frac{2\pi }{b}$ .

Vertical shift is $d$ unit and horizontal shift is $\frac{c}{b}$

The value $c$ represents the horizontal shift. If $c$ is positive then the graph will be shifted by $c$ units towards right compared to the graph of $\sin x$ and if it is negative then the shift will be towards left.

Calculation:

Let us consider the following graph

  

The graph of $g\left(x\right)=\sin x$ has a maximum value or the amplitude as 1 and the minimum value is $-1$ and has a period $2\pi$

Therefore, for the function $f\left(x\right)=a\sin \left(bx-c\right)$ to have a graph like the given graph,

Where the maximum value lies at 3 and the minimum value lies at $-3$ and has a period if $\pi$ .

Increase the amplitude to 3, for which multiply $\sin x$ by $-3$ and for the graph to have a period equal to $\pi$ we have to multiply $x$ by 2.

Hence, the function should be $f\left(x\right)=-3\sin \left(2x\right),$ ,

  $\begin{array}{c}a=-3\\ b=2\\ c=0\end{array}$