Step 1

The equation \(\displaystyle{r}=\frac{18}{{{6}-{6} \cos{\theta}}}\) is not in standard form as the constant tern in the denominator is not 1.

To obtain 1 as the constant term in the denominator divide the numerator and denominator by 4.

Thus,

\(\displaystyle{r}=\frac{18}{{{6}-{6} \cos{\theta}}}=\frac{3}{{{1}- \cos{\theta}}}\)

The equation \(\displaystyle{r}=\frac{3}{{{1}- \cos{\theta}}}\)

is in the standard form of \(\displaystyle{r}=\frac{{{e}{p}}}{{{1}-{e} \cos{\theta}}}.\)

On comparing the provided equation to the standard form it can be obtained that \(\displaystyle{e}={1}.\)

Since \(\displaystyle{e}={1},\) thus the given polar equation represents a parabola.

Step 2

Now, use Ti-83 to plot the graph of the function:

a) Press the [MODE] key then select the polar function and the radian mode.

b) Press the [Y=] key, then there will appear the equations for y.

c) Enter the equations in \(\displaystyle{r}_{{1}}\).

Here, \(\displaystyle{r}_{{1}}=\frac{18}{{{6}-{6} \cos{\theta}}}.\)

d) Press [WINDOW] and then edit the values as:

\(\displaystyle{X}\min=-{7},{X}\max={7},{X}\text{Scale}={1},{Y}\min=-{7}{\quad\text{and}\quad}{Y}\text{Scale}={1}\)

e) press the [GRAPH] key to plot the graph.

The graph:

The equation \(\displaystyle{r}=\frac{18}{{{6}-{6} \cos{\theta}}}\) is not in standard form as the constant tern in the denominator is not 1.

To obtain 1 as the constant term in the denominator divide the numerator and denominator by 4.

Thus,

\(\displaystyle{r}=\frac{18}{{{6}-{6} \cos{\theta}}}=\frac{3}{{{1}- \cos{\theta}}}\)

The equation \(\displaystyle{r}=\frac{3}{{{1}- \cos{\theta}}}\)

is in the standard form of \(\displaystyle{r}=\frac{{{e}{p}}}{{{1}-{e} \cos{\theta}}}.\)

On comparing the provided equation to the standard form it can be obtained that \(\displaystyle{e}={1}.\)

Since \(\displaystyle{e}={1},\) thus the given polar equation represents a parabola.

Step 2

Now, use Ti-83 to plot the graph of the function:

a) Press the [MODE] key then select the polar function and the radian mode.

b) Press the [Y=] key, then there will appear the equations for y.

c) Enter the equations in \(\displaystyle{r}_{{1}}\).

Here, \(\displaystyle{r}_{{1}}=\frac{18}{{{6}-{6} \cos{\theta}}}.\)

d) Press [WINDOW] and then edit the values as:

\(\displaystyle{X}\min=-{7},{X}\max={7},{X}\text{Scale}={1},{Y}\min=-{7}{\quad\text{and}\quad}{Y}\text{Scale}={1}\)

e) press the [GRAPH] key to plot the graph.

The graph: