Question

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin.

r = 4 cos 3θ

Answer 1

Answer:

It is symmetric about the x-axis because cos(3θ) = cos(-3θ).

It is not symmetric about the y-axis because cos(3θ) is not equal to cos(3(pi-θ)).

It is not symmetric about the origin because cos(3θ) is not equal to -cos(3θ).

Answer 2

Answer:

The graph is symmetric about x- axis.

Step-by-step explanation:

We are given that an equation

$r=4 cos 3\theta$

We have to find the graph is symmetric  about x- axis , y-axis or origin.

We taking r along y-axis and [tex\theta [/tex] along x- axis

When the graph is symmetric about x= axis then (x,y)=(-x,y)

$\theta$ is replaced by $-\theta$ and r remain same then we get

$r=4cos (-3\theta)$

We know that cos (-x)=cos x

Therefore, $r=4cos 3\theta$

Hence, the graph is symmetric about x- axis.

When the graph is symmetric about y- axis then (x,y)=(x,-y)

Now, r is replaced by -r then we get

$-r=4cos 3\theta$

$r=-4 cos {3\theta}$

$r=4 cos {\pi-3\theta}$

Therefore, the graph is not symmetric about y-axis.

When the graph is symmetric about y- axis then (x,y)=(x,-y)

Now, r is replaced by -r then we get

$-r=4cos 3\theta$

$r=-4 cos3\theta$

$r=4 cos (\pi-3\theta)$

$(\theta,r)\neq (\theta,-r)$

Therefore, the graph is not symmetric about y-axis.

When the graph is symmetric about origin then (-x,-y)=(x,y)

Replaced r by -r and $\theta$ by-$\theta$

Then we get $-r=4cos3(-\theta)$

$-r= 4 cos 3\theta$

Because cos(-x)=cos x

$r=-4 cos 3\theta$

$(-\theta,-r)\neq(\theta,r)$

Hence, the graph is not symmetric about origin.