Graphs of sinusoidal functions

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Sketch the graph of the following functions:

(a) $y=-\dfrac{1}{2}+\dfrac{3}{2}\cos\left(2x+\dfrac{\pi}{3}\right)$

(b) $y=2-2\sin\left(\dfrac{x}{2}-\dfrac{\pi}{6}\right)$

Solution

(a) The term $a=-\dfrac12$ is a vertical shift; the amplitude is $\dfrac32$; from $c=2$ the period is $T=\dfrac{2\pi}{2}=\pi$. Writing the argument as $2x+\dfrac\pi3=2\left(x+\dfrac\pi6\right)$ shows a horizontal shift to the left by $\dfrac\pi6$. The function’s values lie in $[-2,\,1]$, with a maximum where $2x+\dfrac\pi3=0$, i.e. $x=-\dfrac\pi6$.

(b) The term $a=2$ is a vertical shift; the amplitude is $2$ (the sine is reflected because of the negative sign); from $c=\dfrac12$ the period is $T=\dfrac{2\pi}{1/2}=4\pi$. Writing the argument as $\dfrac x2-\dfrac\pi6=\dfrac12\left(x-\dfrac\pi3\right)$ shows a horizontal shift to the right by $\dfrac\pi3$. The function’s values lie in $[0,\,4]$.

$\boxed{(a)\ T=\pi,\ [-2,1];\quad (b)\ T=4\pi,\ [0,4]}$