Graph f(x)=(1/2)^x

Problem

ƒ(x)=(1/2)x

Solution

1. Identify the type of function. This is an exponential function of the form ƒ(x)=bx where the base b=1/2 Since 0<b<1 the graph represents exponential decay.

2. Determine the horizontal asymptote. As x increases toward infinity, (1/2)x approaches 0 Therefore, the horizontal asymptote is the line y=0

3. Calculate key points to plot the curve. Choose several values for x and find the corresponding ƒ(x) values:

• If x=−2 ƒ*(−2)=(1/2)(−2)=2=4

• If x=−1 ƒ*(−1)=(1/2)(−1)=2=2

• If x=0 ƒ(0)=(1/2)0=1

• If x=1 ƒ(1)=(1/2)1=0.5

• If x=2 ƒ(2)=(1/2)2=0.25

4. Sketch the graph by plotting the points (−2,4) (−1,2) (0,1) (1,0.5) and (2,0.25) Connect them with a smooth curve that decreases from left to right, passing through the y-intercept at (0,1) and flattening out along the x-axis.

Final Answer

The graph of ƒ(x)=(1/2)x is a decreasing exponential curve with a y-intercept at (0,1) and a horizontal asymptote at y=0

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