Exercises - Graphing Trigonometric Functions

  1. Sketch the graphs of the following functions over the indicated domain.

    1.     $\displaystyle{y = 3\sin 2x \quad ; \quad -\pi \le x \le \pi}$

    2.     $\displaystyle{y = 3\cos(x - \frac{\pi}{3}) \quad ; \quad -2\pi \le x \le 2\pi}$

    3.     $\displaystyle{y = -\sin(2x + \frac{\pi}{2}) \quad ; \quad -2\pi \le x \le 2\pi}$

    4.     $\displaystyle{y = -3\cos(\frac{1}{2} x - \frac{\pi}{4}) \quad ; \quad -3\pi \le x \le 3\pi}$

  2. Sketch the graphs for the following.

    1.     $\displaystyle{y = 3 - 2\sin x}$

    2.     $\displaystyle{y = -1 + \cos 2x}$

    3.     $\displaystyle{y = 5\tan 3x}$

    4.     $\displaystyle{y = -3 \csc 2x}$

  3. Graph the following for $\displaystyle{-2\pi \le x \le 2\pi}$. Give the intercepts, amplitude, period, endpoints, and phase shift (when appropriate).

    1.     $\displaystyle{y = 4 \cos x}$   $\ians{}$

    2.     $\displaystyle{y = \sin \frac{2}{3} x}$   $\ians{}$

    3.     $\displaystyle{y = 4\cos(2x - \frac{3\pi}{2})}$   $\ians{}$

    4.     $\displaystyle{y = \sin(x - \frac{\pi}{6})}$   $\ians{}$

    5.     $\displaystyle{y = -\frac{1}{2} \sin x}$   $\ians{}$

  4. Graph the following over the indicated domain. Give the intercepts, amplitude, period, endpoints, and phase shift (when appropriate).

    1.     $\displaystyle{y = -\frac{8}{5} \cos (\frac{x}{5} + \frac{\pi}{3}) \quad ; \quad [-5\pi, 10\pi]}$   $\ians{}$

    2.     $\displaystyle{y = 4 \sin (2x - \frac{\pi}{6}) \quad ; \quad [-\pi, 2\pi]}$   $\ians{}$

    3.     $\displaystyle{y = \frac{5}{2} \cos (2x + \frac{\pi}{4}) \quad \textrm{from } -\pi \textrm{ to }\pi}$   $\ians{}$

    4.     $\displaystyle{y = \cos(x + \frac{\pi}{4}) \quad \textrm{from } -2\pi \textrm{ to }2\pi}$   $\ians{}$

  5. Graph the following. Label intercepts and other important points.

    1.     $\displaystyle{y = 1 + \cos x \quad \textrm{for } -2\pi \le x \le 2\pi}$   $\ians{}$

    2.     $\displaystyle{y = 2 - \sin x \quad \textrm{from } -\pi \textrm{ to } \frac{3\pi}{2}}$   $\ians{}$

    3.     $\displaystyle{y = 2 + 2\sin(\frac{x}{3} - \frac{\pi}{6}) \quad \textrm{from } -\pi \textrm{ to } 2\pi}$   $\ians{}$

    4.     $\displaystyle{y = 2 - 3\cos 2x \quad \textrm{over } [-2\pi,\pi] \quad \textrm{(omit $x$-intercepts)}}$   $\ians{}$

  6. Graph the following. Label intercepts and other important points.

    1.     $\displaystyle{y = -\tan x \quad \textrm{over } [-2\pi,2\pi]}$   $\ians{}$

    2.     $\displaystyle{y = -\sec 2x \quad \textrm{from } -\pi \textrm{ to } \pi}$   $\ians{}$

    3.     $\displaystyle{y = \frac{1}{2} \tan 2x \quad \textrm{from } -\frac{5\pi}{4} \textrm{ to } \frac{3\pi}{8}}$   $\ians{}$

    4.     $\displaystyle{y = \csc 3x \quad \textrm{for } -\frac{\pi}{2} \le x \le \frac{5\pi}{6}}$   $\ians{}$

    5.     $\displaystyle{y = 2\tan \frac{x}{2} \quad \textrm{from } -3\pi \textrm{ to } \frac{5\pi}{2}}$   $\ians{}$

    6.     $\displaystyle{y = -\csc(4x+\pi) \quad \textrm{from } -\frac{\pi}{2} \textrm{ to } \frac{\pi}{2}}$   $\ians{}$