Graphs of exponential and logarithmic functions

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Sketch the graph of the following functions:

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

(b) $y=3\left(\dfrac{1}{2}\right)^{x}-2$;

(c) $y=2-\log_2(2-x)$.

Solution

(a) Starting from $y=2^x$ (asymptote $y=0$, passing through $(0,1)$): reflect across the $x$-axis and shift up by $3$. Horizontal asymptote $y=3$ (approached from below), $y(0)=2$, decreasing, crosses the $x$-axis where $2^x=3$, i.e. $x=\log_2 3\approx1.58$.

(b) Starting from $\left(\tfrac12\right)^x$ (decreasing, asymptote $y=0$): vertical stretch by $3$ and shift down by $2$. Asymptote $y=-2$, $y(0)=1$, decreasing, zero where $\left(\tfrac12\right)^x=\tfrac23$, i.e. $x=\log_2\tfrac32\approx0.585$.

(c) Domain $2-x>0\Rightarrow x<2$; vertical asymptote $x=2$. As $x\to2^-$, $\log_2(2-x)\to-\infty$ so $y\to+\infty$; as $x\to-\infty$, $y\to-\infty$. $y(0)=2-1=1$, $y(1)=2$. The function is increasing.

$\boxed{\text{asymptotes: } y=3;\quad y=-2;\quad x=2}$