Question

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

Answer 1

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.

Step 5. Join these points with a smooth curve.

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

And you have your graph.