Chapter 6.5, Problem 6QR

Solution

a)

To find: The graph of the function is symmetric about $x-$ axis.

The function $y=\cos 4x$ is not symmetric about the $x-$ axis.

Given information:

The given function is $y=\cos 4x$ .

Formula used:

Even-odd identities: $\cos \left(-u\right)=\cos u$ .

Calculation:

Replace $y$ with $-y$ and check if the same function can be obtained.

If the same function is obtained then function is symmetric about the $x-$ axis.

  $\begin{array}{c}-y=\cos 4x\\ y=-\cos 4x\end{array}$

Same function is not obtained.

Therefore, the function $y=\cos 4x$ is not symmetric about the $x-$ axis.

b)

To find: The graph of the function is symmetric about $y-$ axis.

The function $y=\cos 4x$ is symmetric about the $y$ -axis.

Given information:

The given function is $y=\cos 4x$ .

Formula used:

Even-odd identities: $\cos \left(-u\right)=\cos u$ .

Calculation:

Replace $x$ with $-x$ and check if the same function can be obtained.

If the same function is obtained then function is symmetric about the $y-$ axis.

  $\begin{array}{c}y=\cos \left[4\left(-x\right)\right]\\ y=\cos \left(-4x\right)\\ =\cos 4x\end{array}$

Same function is obtained.

Therefore, the function $y=\cos 4x$ is symmetric about the $y$ -axis.

c)

To find: The graph of the function is symmetric about the origin.

The function $y=\cos 4x$ is not symmetric about the origin.

Given information:

The given function is $y=\cos 4x$ .

Formula used:

Even-odd identities: $\cos \left(-u\right)=\cos u$ .

Calculation:

Replace $y$ with $-y$ and check if the same function can be obtained.

  $\begin{array}{c}-y=\cos 4x\\ y=-\cos 4x\end{array}$

Replace $x$ with $-x$ and check if the same function can be obtained.

  $\begin{array}{c}y=\cos \left[4\left(-x\right)\right]\\ y=\cos \left(-4x\right)\\ =\cos 4x\end{array}$

If the same function is obtained then function is symmetric about the origin. .

From part (a) and part (b), the function is not symmetric about $x$ -axis but symmetric about the $y-$ axis.

Therefore, the function $y=\cos 4x$ is not symmetric about the origin.