Graph of a translated exponential function

Statement

Draw the function $y=-1+2\cdot\left(\dfrac{1}{3}\right)^x$, 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 $\left(\tfrac13\right)^x\to 0$, so $y\to -1$. The line $y=-1$ is an asymptote.
• As $x\to-\infty$: $\left(\tfrac13\right)^x\to+\infty$, so $y\to+\infty$.
• Intersection with the $y$-axis: $x=0\Rightarrow y=-1+2=1$, point $(0;1)$.
• Intersection with the $x$-axis: $-1+2\left(\tfrac13\right)^x=0\Rightarrow\left(\tfrac13\right)^x=\tfrac12\Rightarrow x=\log_{1/3}\tfrac12\approx 0.63$.
• The function is decreasing (base $<1$, positive coefficient) and lies above the asymptote $y=-1$.

$\boxed{\text{asymptote }y=-1;\ (0;1);\ x_0=\log_{1/3}\tfrac12\approx 0.63;\ \text{decreasing}}$