Graphs of two exponential functions

Text

Sketch the graph of the following functions, giving the horizontal asymptote, two points (an $x\mid y$ table), whether they increase or decrease, and the $x$-intercept:

• (a) $y = 2\cdot 3^{x} - 1$
• (b) $y = -3\cdot \left(\dfrac{1}{2}\right)^{x} + 2$

Solution

(a) Asymptote $y=-1$; base $3>1$ with positive coefficient → increasing. Points $(0;1)$, $(1;5)$. $x$-intercept: $2\cdot3^x=1\Rightarrow x=-\log_3 2\approx-0{,}63$.

(b) Asymptote $y=2$; $\left(\tfrac12\right)^x$ decreases but the $-3$ coefficient flips it → increasing (from below toward the asymptote). Points $(0;-1)$, $(1;\tfrac12)$. $x$-intercept: $\left(\tfrac12\right)^x=\tfrac23\Rightarrow x=\log_{1/2}\tfrac23\approx0{,}585$.

$\boxed{\text{(a) } x_0\approx-0{,}63,\ y=-1;\qquad \text{(b) } x_0\approx0{,}585,\ y=2}$