Logarithmic function graph and asymptote (I)

Statement

Draw the following function after finding its vertical asymptote: $y=2-3\log_2(2x+4)$

Solution

Domain and vertical asymptote. The argument must be positive: $2x+4>0 \iff x>-2$. The vertical asymptote is the line $x=-2$.

Behaviour. The coefficient $-3$ in front of the logarithm makes the function decreasing; as $x\to-2^{+}$ we have $y\to+\infty$.

Useful points (choose $2x+4$ a power of $2$):

• $2x+4=1 \Rightarrow x=-\tfrac32,\ y=2$;
• $2x+4=2 \Rightarrow x=-1,\ y=2-3=-1$;
• $2x+4=4 \Rightarrow x=0,\ y=2-6=-4$.

A decreasing curve falling from $+\infty$ (approaching the asymptote $x=-2$) through $(-\tfrac32,2)$, $(-1,-1)$, $(0,-4)$.

$\boxed{\text{vertical asymptote }x=-2}$