Question

7.6 exponential functions
sketch the graph of each function.

1. $y = 2 \cdot 3^x$
2. $y = $

Explanation:

Step1: Find key coordinate points

Calculate $y$ for selected $x$ values:

• When $x=-1$: $y=2\cdot3^{-1}=2\cdot\frac{1}{3}=\frac{2}{3}\approx0.67$
• When $x=0$: $y=2\cdot3^{0}=2\cdot1=2$
• When $x=1$: $y=2\cdot3^{1}=2\cdot3=6$
• When $x=2$: $y=2\cdot3^{2}=2\cdot9=18$
• When $x=-2$: $y=2\cdot3^{-2}=2\cdot\frac{1}{9}=\frac{2}{9}\approx0.22$

Step2: Identify asymptote

For exponential functions $y=ab^x$ ($b>1$), the horizontal asymptote is $y=0$, since as $x\to-\infty$, $3^x\to0$, so $y\to0$.

Step3: Plot points and draw curve

Mark the points $(-2, \frac{2}{9})$, $(-1, \frac{2}{3})$, $(0,2)$, $(1,6)$, $(2,18)$ on the grid. Draw a smooth increasing curve that approaches the line $y=0$ as $x$ decreases, and rises rapidly as $x$ increases.

Answer:

The graph of $y=2\cdot3^x$ is an increasing exponential curve with a horizontal asymptote at $y=0$, passing through the points $(-2, \frac{2}{9})$, $(-1, \frac{2}{3})$, $(0,2)$, $(1,6)$, and $(2,18)$.