Chapter 4.5, Problem 68E

To determine

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

Answer to Problem 68E

  $a=1,d=-3$

Explanation of Solution

Given information:

Given function is

  $f\left(x\right)=a\cos x+d$

Given graph

  

Concept used:

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

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 of $d$ gives the vertical shift of the graph, which is if $d$ is a positive value then the graph will be shifted upwards by $d$ units compared to the graph of $\cos x$ and if its negative the shift will be downwards.

Calculation:

Let us consider the following graph

  

Now, notice form the given graph that the peak occurs at $y=-2$ and the valley is at $y=-4$ . So the amplitude will be:

  $\begin{array}{c}\text{amplitude}=\frac{1}{2}\left(\left|-4\right|+\left|-2\right|\right)\\ =1\end{array}$

Hence, $a=1$

Notice that the graph is not cantered at the origin, it is shifted downwards by 3 unit. So here $d=-3$ . Therefore the given graph represents the following function:

  $f\left(x\right)=1\cos x-3$