Graph f(x)=(2/3)^x

Problem

ƒ(x)=(2/3)x

Solution

1. Identify the type of function. This is an exponential function of the form ƒ(x)=bx where the base b=2/3 Since 0<2/3<1 the function represents exponential decay.

2. Determine the y-intercept. Evaluate the function at x=0

ƒ(0)=(2/3)0=1

3. Calculate additional points to define the shape of the curve.

ƒ*(−1)=(2/3)(−1)=3/2=1.5

ƒ(1)=(2/3)1=2/3≈0.67

ƒ(2)=(2/3)2=4/9≈0.44

4. Identify the horizontal asymptote. As x increases toward infinity, the value of (2/3)x approaches 0

Horizontal Asymptote: *y=0

5. Sketch the graph. Plot the points (−1,1.5) (0,1) and (1,0.67) then draw a smooth curve that decreases from left to right, staying above the x-axis and approaching it as x becomes large.

Final Answer

ƒ(x)=(2/3)x* is a decreasing exponential curve passing through *(0,1)* with asymptote *y=0

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