2. what does the graph of f(x) = 2^x look like?

Answer

Explanation:

Step1: Identify the function type

The function ( f(x) = 2^x ) is an exponential function with base ( 2>1 ), so it's an increasing exponential function.

Step2: Find key points

When ( x = 0 ), ( f(0)=2^0 = 1 ), so the point ( (0,1) ) is on the graph.
When ( x = 1 ), ( f(1)=2^1 = 2 ), so the point ( (1,2) ) is on the graph.
When ( x=- 1 ), ( f(-1)=2^{-1}=\frac{1}{2}=0.5 ), so the point ( (-1,0.5) ) is on the graph.
As ( x\rightarrow-\infty ), ( 2^x\rightarrow0 ), so the graph approaches the x - axis (y = 0) as ( x ) becomes very negative.
As ( x\rightarrow+\infty ), ( 2^x\rightarrow+\infty ), so the graph increases without bound as ( x ) becomes very positive.

Step3: Sketch the graph (description)

Start from near the x - axis (approaching y = 0) when ( x ) is very negative. Pass through ( (-1,0.5) ), ( (0,1) ), ( (1,2) ), and then increase rapidly as ( x ) increases. The graph is smooth, concave up, and passes through the key points we calculated.

Answer:

The graph of ( f(x)=2^x ) is an exponential growth curve. It passes through the point ( (0,1) ), is increasing, approaches the x - axis (y = 0) as ( x\rightarrow-\infty ), and increases without bound as ( x\rightarrow+\infty ). When plotted on the given grid, it goes through ( (0,1) ), ( (1,2) ), ( (-1,0.5) ) and has the shape of an increasing exponential curve.