How do you graph #f(x)=2 sin(x/3)#?

1 Answer
Ernest Z.
Aug 11, 2015

Plot the maxima, minima, and intercepts over one period, then extend the graph in each direction.

Explanation:

Your equation is #f(x) =2sin(x/3)#

Step 1. Express your equation in the form

#f(x)=asin(bx+c)+d#

Then #a=2#, #b=1/3#, #c=0#, and #d=0#.

Step 2. Calculate the range, period, phase shift, and vertical displacement.

The amplitude is #a = 2#, so the range is [-2,2].

The period is #(2π)/b = (2π)/(1/3) = 6π#.

The phase shift is #c=0#.

The vertical shift is #d=0#.

Step 3. Divide the period #6π# into four quarters to get the key points for #f(x) = 2sin(x/3)#.

#stackrel(—————————————————————)(x=" "" "0" "(3π)/2" "color(white)(1)3π" "(9π)/2" "6π)#
#stackrel(—————————————————————)(f(x)=color(white)(1)0" "color(white)(1)2" "" "0" "-2" "0)#
#stackrel(—————————————————————)#

These points are

  • (#0,0#) = intercept
  • (#(3π)/2,2#) = maximum
  • (#3π,0#) = intercept
  • (#(9π)/2,-2#) = minimum
  • (#6π,0#) = intercept

Step 4. Plot these five key points.

Graph1

Step 5. Join these points with a smooth curve.

Graph 2

Step 6. Follow the pattern and extend your axis from #-6π# to #12π#.

Graph 3

And you have your graph.

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