how to graph trig functions

This is a part of guide to graphing functions.

sine, cosine and tangent

we will graph a function in the form $f(x)=A\cdot \sin (n(x-b))+c$,
and $f(x)=A\cdot \cos (n(x-b))+c$, and $f(x)=A\cdot \tan (n(x-b))+c$

$A$ is the amplitude of the graph
$c$ is the vertical translation of the graph and the midline is $y=c$
the period of the graph is $\frac{2\pi }{n}$ for sine and cosine, and is $\frac{\pi }{n}$ for tangent
the dilation factor from the x-axis is $\frac{1}{n}$
$b$ is the amount of horizontal translation to the right

what do we do?
read your question.
it usually restricts your function domain, for example by defining the function for a domain interval
if there are no restrictions on the domain, just draw everything that should be in the graph and

things we need

• the domain for which we sketch
• midline
• amplitude
• period
• one x-intercept. this can be used to find all x intercepts.
• y-intercept

the period is $\frac{2\pi }{n}$

to find the coordinates of the y-intercept, set $x=0$
to find the coordinates of the x-intercepts, if any, set $y=0$

amplitude is always shown as capital A¹

worked example: graph $y=2\sin (\frac{x}{2})$ for $x\in (0,4\pi )$

amplitude is 2
period is $\frac{2\pi }{0.5}=4\pi$

worked example: graph $y=3\sin (2x)$ for $x\in (-\pi ,\pi )$

amplitude is 3
period is $\frac{2\pi }{2}=\pi$

worked example: graph $y=\sin (2(x-\frac{\pi }{4}))$ for $x\in [-\frac{3\pi }{4},\frac{\pi }{2}]$
amplitude is is 1, period is $\frac{2\pi }{2}=\pi$, translated $\frac{\pi }{4}$ to the right

worked example graph $y=2\sin (3(x+\frac{\pi }{4}))$
amplitude is 2, period is $\frac{2\pi }{3}$ and translated $\frac{\pi }{4}$ to the left

calculate y-intercept: $y=2\sin (\frac{3\pi }{4})=2\sin (\frac{\pi }{4})=\sqrt{2}$

find maximum points
$\sin (3(x+\frac{\pi }{4}))=1$
$\sin (\frac{\pi }{2})=1$
$3(x+\frac{\pi }{4})=\frac{\pi }{2}$
$x+\frac{\pi }{4}=\frac{\pi }{6}$
$x=\frac{-\pi }{12}$

find minimum points
$\sin (3(x+\frac{\pi }{4}))=-1$
$\sin (\frac{3\pi }{2})=-1$
$3(x+\frac{\pi }{4})=\frac{3\pi }{2}$
$x+\frac{\pi }{4}=\frac{\pi }{2}$
$x=\frac{\pi }{4}$

find x-intercepts
$2\sin (3(x+\frac{\pi }{4}))=0$
$\sin (3(x+\frac{\pi }{4}))=0$
$\sin (0)=0$
$3(x+\frac{\pi }{4})=0$
$x=\frac{\pi }{4}$

worked example: graph $f(x)=2\cos (2x)-1$

amplitude is 2, period is $\pi$ and translated 1 down

find x-intercepts
$0=2\cos (2x)-1$
$0=\cos (2x)-\frac{1}{2}$
$\frac{1}{2}=\cos (2x)$
$\frac{1}{2}=\cos (\frac{\pi }{3})$
$2x=\frac{\pi }{3}$
$x=\frac{\pi }{6}$

worked example: graph $y=\sqrt{2}\sin (2(x+\frac{\pi }{6}))+2$

worked example: graph $y=-2\sin (2x+\pi )+1$
to graph this we need to put it in the format we recognize by factoring
$y=-2\sin (2(x+\frac{\pi }{2}))+1$
we need to flip this vertically at the end because negative at the front
amplitude is 2
period is $\frac{2\pi }{2}=\pi$
translation is $\frac{\pi }{2}$ to the left
midline is $y=1$

worked example: graph $y=3\tan (2x)$ where $x\in [-\pi ,\pi ]$
period is $\frac{\pi }{2}$

find x-intercepts
$0=3\tan (2x)$
$0=\tan (2x)$
$0=\tan (0)$
$2x=0$
$x=0$

find asymptotes
$\cos (2x)=0$
$\cos (\frac{\pi }{2})=0$
$2x=\frac{\pi }{2}$
$x=\frac{\pi }{4}$

worked example: graph $y=2\tan (\frac{x}{2})-2$ where $x\in (\frac{-\pi }{2},\pi )$

period is $2\pi$

find x-intercepts
$0=2\tan (\frac{x}{2})-2$
$y=\tan (\frac{x}{2})-1$
$1=\tan (\frac{x}{2})$
$1=\tan (\frac{\pi }{4})$
$\frac{x}{2}=\frac{\pi }{4}$
$x=\frac{\pi }{2}$

find asymptotes
$\cos (\frac{x}{2})=0$
$\cos (\frac{\pi }{2})=0$
$x=\pi$

worked example: graph $y=-\tan (x-\frac{\pi }{4})$ where $x\in (-\pi ,\pi )$

period is $\pi$

find x-intercepts

$0=-\tan (x-\frac{\pi }{4})$
$0=\tan (x-\frac{\pi }{4})$
$0=\tan (0)$
$x-\frac{\pi }{4}=0$
$x=\frac{\pi }{4}$

find asymptotes

$\cos (x-\frac{\pi }{4})=0$
$\cos (\frac{\pi }{2})=0$
$x-\frac{\pi }{4}=\frac{\pi }{2}$
$x=\frac{3\pi }{4}$

Footnotes

1. : amplitude is often shown as capital A in physics https://en.wikipedia.org/wiki/Amplitude ↩