Question

1. $y=\frac{1}{2}\cdot4^{x}$

growth / decay
domain: ________
range: ________
y-intercept: ________
asymptote: ________

1. $y=2\cdot\left(\frac{1}{3}\

ight)^{x}$

Explanation:

Step1: Identify growth/decay type

The function is $y=\frac{1}{2}\cdot4^x$. Since the base of the exponential term $4>1$, this is exponential growth.

Step2: Find the domain

Exponential functions accept all real numbers as input, so domain is all real $x$.

Step3: Find the range

As $x\to-\infty$, $4^x\to0$, so $y\to0$; as $x\to+\infty$, $4^x\to+\infty$, so $y\to+\infty$. Since $\frac{1}{2}\cdot4^x>0$ for all real $x$, range is positive real numbers.

Step4: Calculate y-intercept

Set $x=0$: $y=\frac{1}{2}\cdot4^0=\frac{1}{2}\cdot1=\frac{1}{2}$

Step5: Find the asymptote

As $x\to-\infty$, $y\to0$, so the horizontal asymptote is $y=0$.

Answer:

Growth / Decay: Growth
Domain: $(-\infty, +\infty)$
Range: $(0, +\infty)$
y-intercept: $\frac{1}{2}$
Asymptote: $y=0$