Graphs of trigonometric, exponential and logarithmic functions

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Sketch the following graphs:

(a) $y=-1+\dfrac12\cos\left(\dfrac x2+\dfrac\pi2\right)$

(b) $y=1+2\sin\left(2x+\dfrac\pi3\right)$

(c) $y=2-2\tan\left(-\dfrac x2+\dfrac\pi6\right)$

(d) $y=3-\log_3(2-2x)$

(e) $y=2-3^{x}$

Solution

(a) Vertical shift $a=-1$, amplitude $\dfrac12$; from $c=\dfrac12$ the period is $T=4\pi$; the function’s values lie in $\left[-\dfrac32,\,-\dfrac12\right]$.

(b) Vertical shift $a=1$, amplitude $2$; from $c=2$ the period is $T=\pi$; the function’s values lie in $[-1,\,3]$.

(c) A tangent with $c=-\dfrac12$, so the period is $T=2\pi$; the negative sign in the argument produces a reflection of the graph about the vertical axis.

(d) The domain requires $2-2x>0$, i.e. $x<1$; the vertical asymptote is $x=1$. We have $y(0)=3-\log_3 2\approx 2.37$, and as $x\to1^-$, $y\to+\infty$.

(e) A reflected exponential, with horizontal asymptote $y=2$ approached from below; $y(0)=1$; the zero occurs at $x=\log_3 2\approx 0.63$.

$\boxed{(a)\ T=4\pi;\ \ (b)\ T=\pi;\ \ (c)\ T=2\pi;\ \ (d)\ x=1;\ \ (e)\ y=2}$