Graph of y = 2 − 3·2^x

Statement

Draw the function $y=2-3\cdot 2^x$, finding the values for large $x$ (positive and negative), for $x=0$ and the intersection with the $x$-axis.

Solution

• Horizontal asymptote: as $x\to-\infty$ we have $2^x\to 0$, so $y\to 2$. The line $y=2$ is an asymptote.
• As $x\to+\infty$: $2^x\to+\infty$, so $y\to-\infty$.
• Intersection with the $y$-axis: $x=0\Rightarrow y=2-3=-1$, point $(0;-1)$.
• Intersection with the $x$-axis: $2-3\cdot 2^x=0\Rightarrow 2^x=\tfrac23\Rightarrow x=\log_2\tfrac23\approx -0.585$.
• The function is decreasing (the coefficient $-3$ flips the growth of $2^x$) and lies below the asymptote $y=2$.

$\boxed{\text{asymptote }y=2;\ (0;-1);\ x_0=\log_2\tfrac23\approx -0.585;\ \text{decreasing}}$