Question

find the graph of the following exponential function. $y = \frac{1}{2} \cdot 2^{x}$

Explanation:

Step1: Simplify the function

First, simplify the given exponential function using exponent rules.
$y = \frac{1}{2} \cdot 2^x = 2^{-1} \cdot 2^x = 2^{x-1}$

Step2: Find key points

Calculate the value of $y$ for key $x$-values:

• When $x=0$: $y = 2^{0-1} = \frac{1}{2}$
• When $x=1$: $y = 2^{1-1} = 1$
• When $x=2$: $y = 2^{2-1} = 2$
• As $x \to -\infty$: $y = 2^{x-1} \to 0$
• As $x \to +\infty$: $y = 2^{x-1} \to +\infty$

Step3: Match to the graph

The function is a growing exponential (since the base $2>1$), approaching 0 as $x$ decreases and increasing as $x$ increases.

Answer:

The correct graph is the right-hand one (the curve that approaches the x-axis as $x \to -\infty$ and rises sharply as $x$ increases).