# How do you graph y=2cos6pix?

$y = 2 \cos 6 \pi x$
Considering the salient points,

At cos6pix=0;
y=0,
when $2 \cos 6 \pi x = 0$
when$\cos 6 \pi x = 0$
when$6 \pi x = \frac{- 7 \pi}{2} , \frac{- 5 \pi}{12} , \frac{- 3 \pi}{12} , \frac{- \pi}{12} , \frac{\pi}{2} , \frac{3 \pi}{2} , \frac{5 \pi}{2} , \frac{7 \pi}{2} , \ldots$
$x = \ldots - \frac{7}{12} , - \frac{5}{12} , - \frac{3}{12} , - \frac{1}{12} , \frac{1}{12} , \frac{3}{12} , \frac{5}{12} , \frac{7}{12} , \ldots$

At cos6pix=1;
$y = 2 \times 1 = 2$,
when$6 \pi x = \ldots , - 8 \pi , - 6 \pi , - 4 \pi , + 2 \pi , 0 , 2 \pi , 4 \pi , 6 \pi , 8 \pi , \ldots .$
$x = \ldots , - \frac{4}{3} , - \frac{3}{3} , - \frac{2}{3} , - \frac{1}{3} , 0 , \frac{1}{3} , \frac{2}{3} , \frac{3}{3} , \frac{4}{3} , \ldots .$
$x = \ldots , - \frac{16}{12} , - \frac{12}{12} , - \frac{8}{12} , 0 , \frac{4}{12} , \frac{8}{12} , \frac{12}{12} , \frac{16}{12} , \ldots$

At cos6pix=-1;
$y = 2 \times - 1 = - 2$
2hen $6 \pi x = \ldots , - 7 \pi , - 5 \pi , - 3 \pi , - \pi , \pi , 3 \pi , 5 \pi , 7 \pi , \ldots .$
$x = \ldots , - \frac{7}{6} , - \frac{5}{6} , - \frac{3}{6} , - \frac{1}{6} , \frac{1}{6} , \frac{3}{6} , \frac{5}{6} , \frac{7}{6} , \ldots .$
$x = \ldots , - \frac{14}{12} , - \frac{10}{12} , - \frac{6}{12} , - \frac{2}{12} , \frac{2}{12} , \frac{6}{12} , \frac{10}{12} , \frac{14}{12} , \ldots .$

We have a family of points;
$x = \ldots - \frac{7}{12} , - \frac{5}{12} , - \frac{3}{12} , - \frac{1}{12} , \frac{1}{12} , \frac{3}{12} , \frac{5}{12} , \frac{7}{12} , \ldots$ where $y = 0$
$x = \ldots , - \frac{16}{12} , - \frac{12}{12} , - \frac{8}{12} , - \frac{4}{12} , 0 , \frac{4}{12} , \frac{8}{12} , \frac{12}{12} , \frac{16}{12} , \ldots$ where $y = 1$
$x = \ldots , - \frac{14}{12} , - \frac{10}{12} , - \frac{6}{12} , - \frac{2}{12} , \frac{2}{12} , \frac{6}{12} , \frac{10}{12} , \frac{14}{12} , \ldots$ where $y = - 1$