Graphing sin

Question

Below sketch the graph of a full cycle of the function.

 

y=3sin[3(x+pi/3)]+2

Answer 1

First let's get a good picture of what y = sin(x) looks like. Plot it for a full phase in increments of pi/2 from [0,2pi].

 

It starts at (0,0), climbs up to 1 at pi/2, comes back down to 0 at pi, descends down to -1 at 3pi/2, and comes back to 0 at 2pi. It behaves just like the y values on the unit circle.

 

With this plot in mind, let's see how each of the extra values in your function affects the plot of sine.

 

Let's first distribute the 3 inside the sine function.

 

y = 3*sin(3x + pi) + 2

 

The "3" outside the sine function affects the amplitude. Instead of climbing up to 1 and descending down to -1, this graph will instead climb up to 3 and descend down to -3.

 

The "+ 2" on the right will affect the y-intercept just like in any other graph. Instead of sine starting off at (0,0) it will now start off at (0,2). Because of the amplitude change and the upward shift, it will now oscillate between 5 (2+3) and -1 (2-3) instead of 1 and -1.

 

The "+ pi" inside the sine function is called the phase shift. The syntax for a positive phase shift is sin(x - w), which means all of the x values will shift to the right by w units.

 

In your case you have a "sin(x + pi)", which is the same thing as sin(x - (-pi)). As such, your graph will shift negative (meaning, to the left), by pi units.

 

So now, instead of starting at (0,2) and oscillating between -1 and 5, your graph will now start at (-pi, 2) and oscillate from -1 to 5.

 

Finally we reach the "3x" inside your sine function. This will affect the period of your graph, meaning how long it takes to go through 1 cycle of oscillations. The general formula is 2*pi/B, where B is the coefficient in front of x. In your case, it's 2*pi/3.

 

This means the function goes through its cycle in 1/3 the time as sin(x), so it'll get narrower/ compress together. Instead of cycling in the interval [0,2pi] in increments of pi/2, it'll cycle from [-pi,-pi/3] in increments of 2pi/12 = pi/6. Don't forget that due to the phase shift, the first point is at -pi. And -pi+2pi/3 = -pi/3.

 

So let's summarize the points on your two plots:

 

1) sin(x)

 

(0,0), (pi/2, 1), (pi, 0), (3pi/2, -1), (2pi, 0)

 

2) 3*sin(3x + pi) + 2

 

(-pi, 2), (-5pi/6, 5), (-2pi/3, 2), (-3pi/6, -1), (-pi/3, 2)

 

Your function is higher, taller, more to the left, and narrower than a normal sine function.

 

This is the best I could do without a graphing utility handy. Hope this helps.

Answer 2

The site might not have graphing capabilities but we can visualize what it may look like. The general form for a sinusoidal function is

y=Asin(B(x-C))+D

where A is our amplitude or the displacement from the equilibrium. B is the number of cycles between 0 and 2pi and the period is 2pi/B. -C/B is our horizontal shift. And D is the equilibrium or the middle between our high and low points.