We have the following functions:

So let's analyze how to transform these functions by using key features.
1. Including amplitude. To include the amplitude in a
sine or cosine functions we simply multiply the function by A, so:

The absolute value of A is the amplitude. that is:

The
tangent function have no amplitude because this function grows without a bound.
2. Including period.
Let

be a positive real number. The period of the
sine or cosine functions is obtained as follows:

The period T is given by:

Thus:

Regarding the
tangent function:

where the Period T is given by:

Thus:
3. Phase shift
The constant

in the equations:

creates a
horizontal translation (shift) of the basic functions. To the left (if positive) or to the right (if negative). The number:
is the
phase shift.4. Vertical shift
The constant

in the equations:

creates a
vertical translation (shift) of the basic functions. Upward (if positive) or downward (if negative)
5. Minimum point.
To find the
minimum point in a
sine or cosine functions let's take the functions:

The minimum point in a sine function is given when this function is minimum, that is, when

. On the other hand, the minimum point of the cosine function is given when

, then the minimum points are:

There is no a
minimum point of the
tangent function because it grows without a bound.
6. Maximum point.
To find the
Maximum point in a
sine or cosine functions let's take the functions:

So the Maximum point of the sine function is given when this function is Maximum, that is, when

. On the other hand, the maximum point of the cosine function is given when

, then the maximum points are:

There is no a
maximum point in a
tangent function because it grows negatively without a bound.
7. Example for the sine function.As shown in Figure 1 we have the graph of the following function:

So the key features are:
8. Example for the cosine function.As shown in Figure 2 we have the graph of the following function:

So the key features are:
9. Example for the tangent function.As shown in Figure 3 we have the graph of the following function:

So the key features are:
