Basic Trigonometric Graphs

Recall that $\cos(\theta)$ and $\sin(\theta)$ are $2\pi$-periodic. This means they repeat every $2\pi$ radians. It is recommended that you know the graphs of $\cos(\theta)$ and $\sin(\theta)$ on the interval $[0,2\pi]$. The graphs of these functions, along with five important points are given below.

Recall that $\tan(\theta)$ is $\pi$-periodic. This means tangent repeats every $\pi$ radians. Note that since $\displaystyle \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, $\tan(\theta)$ is undefined when $\cos(\theta)=0$. Also, $\tan(\theta)=0$ whenever $\sin(\theta)=0$. The graph of the tangent function is given below.

Shifts of Cosine and Sine

Below is a chart summarizing transformations of the function $f(x)$.

Form	Description
$f(x) + C$	Shift up ($C>0$) or down ($C< 0$) by $C$ units
$f(x+C)$	Shift left ($C>0$) or right ($C< 0$) by $C$ units
$-f(x)$	Reflection over the $x$-axis
$f(-x)$	Reflection over the $y$-axis
$Cf(x)$	Stretch ($C>1$) or compression ($0< C< 1$) by a factor of $C$ in the $y$-axis
$f(Cx)$	Stretch ($0< C< 1$) or compression ($C >1$) by a factor of $C$ in the $x$-axis