Summary of hand graphing techniques

 

The knowledge that you carry in the unit circle can be used to graph the sine, cosine, and tangent functions. Of course, you can graph these quickly on a graphing calculator or computer grapher, but sometimes it is very useful to quickly analyze a function by sketching it on paper. Therefore, I expect you to be able to draw the functions and label the critical points. Section 4.5 of the book has a very complete and very good step by step description of how to do this. But, here is a quick and dirty summary!

Here is a video clip that demonstrates how the unit circle can be "unwrapped" into the sine and cosine functions. Although this clip emphasizes that it is "unwrapping" sin(θ) rather than sin(x), it works just fine for you to think of the horizontal axis as x rather than θ.

 

The point that I want to stress is that you already have all the important values you need to graph these functions at your fingertips in the unit circle.

The period of sinx is because that is the distance around the unit circle.

The amplitude of sin x is 1 because that is the radius of the unit circle.

The phase shift of sin x is 0 because the unit circle starts at 0.

Your book identifies five key points for graphing y = sin x (page 518). They are the three x-intercepts, the maximum point, and the minimum point. All of these are on your unit circle.

The values of sin x correspond to the y-values, so those key points are (angle, y-value) or (0,0), (π/2, 1), (π, 0), (3π/2, -1), (2π, 0).

sine2.gif

The values of cos x correspond to the x-values, so those key points are (angle, x-value) or (0,1), (π/2, 0), (π, -1), (3π/2, 0), (2π, 1).

cos.gif

 

To sketch one period of the graph, you simple need to move these five key points; For example, to sketch the much more complicated function ,

you must first rewrite it in a form that shows the phase shift

Now you can tell the amplitude is 3

the period is

the phase shift is to the right.

To graph this you need to take each original key point and find the transformation.

1. Find the new period values

The new period is 4π so all fo the x-values will double: Instead of 0, π/2, π, 3π/2, 2π we will double the values and use 0, π, 2π, 3π, 4π

2. Apply the phase shift

Now apply the phase shift:

Doing the addition, the new x-values will be or

The original y-values for sin x are 0, 1, 0, -1, and 0.

3. Apply the amplitude

Applying the amplitude of 3, the new y-values are 0, 3, 0, -3, and 0.

The transformed points are

 

Now you can choose an appropriate scale on your graph paper

scaledaxes.png

Then graph and label your five key points::

 

 

axeswithlabeledpoints.png

Now sketch the curve:

completedsin.png

This same procedure will work for tangent, but tangent is defined by 3 points and two asymptotes. rather than five points.

The period of tangent is π, so use quadrants I and IV.

(angle, y-value/x-value) or (π/2, 1/0), (π/4, 1),(0,0/1), (-π/4, -1), (-π/2, -1/0)

Since division by 0 is undefined, this gives three points (π/4,1), (0,0) and (-π/4,-1) and two vertical asymptotes, x=π/2 and x=-π/2

.Remember that tangent does not have an amplitude (although it can have a stretch which is why we included the points at π/4.)

The period of tangent is π.

The phaseshift is 0.

tan.png

 

 

Let's look at an example of a tangent function that has a stretch, an altered period, and a phase shift,

We need to isolate the phase shift so we will factor a 2 out: and write it as a subtraction

Now we can see that the

stretch factor is 3

the period is π/2

the phase shift is -3π/2 or 3π/2 to the left

To graph this you need to take each original key point and find the transformation.

 

1. Find the new period values

The new period is π /2so all fo the x-values will be halved: Instead of π/2, π/4, 0, -π/4, -π/2 we will halve the values and use π/4, π/8, 0, -π/8, -π/4

2. Apply the phase shift

Now apply the phase shift:

Doing the addition, the new x-values will be or

The original y-values for tan x are undefined, 1, 0, -1, and undefined.

3. Apply the stretch

Applying the stretch of 3, the new y-values are undefined, 3, 0, -3, and undefined.

The transformed key values are

 

Now you can choose an appropriate scale on your graph paper:

 

tangentaxes.png

 

If you only need to graph one period, you can draw in your tangent.

oneperiodtan.png

 

 

You can also use this pattern to fill in your other periods:

transformedtan.png