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

Aug 11, 2015

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

#### Explanation:

Your equation is $f \left(x\right) = 2 \sin \left(\frac{x}{3}\right)$

Step 1. Express your equation in the form

$f \left(x\right) = a \sin \left(b x + c\right) + d$

Then $a = 2$, $b = \frac{1}{3}$, $c = 0$, 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 $f \left(x\right) = 2 \sin \left(\frac{x}{3}\right)$.

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. 