**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 sin*x* is **2π** 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

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).

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).

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

Then graph and label your five key points::

Now sketch the curve:

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 *d oes 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.

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:

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

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