Learners graph the parent cotangent function, as well as the graph of functions that involve cotangent expressions.
After completing this tutorial, you will be able to complete the following:
First, recall that the domain of the function has all real numbers x except , where n is an integer and its range is all real numbers. At the values , there are vertical asymptotes. The period of is .
Transformations of the cotangent function:
Recall the basic transformations for a function f(x).
Now let`s see how these transformations affect the cotangent function.
The constants b and c affect the period and phase shift of the graph, respectively, because they both cause horizontal transformations. Note that bx + c can be written Therefore, y = a • cot (bx + c) + d has a horizontal shrink factor b according to the second transformation rule, and a horizontal shift by units to the left according to the third transformation rule. Therefore, the function is phase shifted units to the left and has period equal to when .
In the case that the function is in order to apply transformation rules 1-6:
Therefore, we have a phase shift of units to the right and a new period of . a causes a vertical stretch, and d causes an upward or downward shift of the whole graph.
a causes a vertical stretch, and d causes an upward or downward shift of the whole graph.
Example: Graph y = 4cot (2x).
Step 1: Identify the transformations of the function.
The (2x) inside the function lets us know that we have a horizontal shrink by a factor of 2, so the period of the function is . Since nothing is added or subtracted inside the function, there is no phase shift.
The 4 multiplied by the function lets us know that there is a vertical stretch by a factor of 4.
Since nothing is added or subtracted from the function, there is no vertical shift.
Identify the vertical asymptotes and x-intercepts of the function.
The function has vertical asymptotes at , where n is an integer. Because there is a horizontal shrink by a factor of 2, our function will have vertical asymptotes at , where n is an integer.
The function has x-intercepts at , where n is an odd integer. Because there is a horizontal shrink by a factor of 2, our function will have x-intercepts at .
Select a few other points between and that are known on the graph of Identify where in the Cartesian plane they would move after the transformations in Step 1 are applied.
We know that and . Due to the horizontal shrink by a factor of 2, and But let's not forget about the vertical stretch by a factor of 4. This will change the y values, so
On the coordinate plane, graph the asymptotes, the x-intercepts and the points obtained in Step 3. Then sketch the graph of the function.
If a phase shift and vertical shift are involved, do not forget to identify those transformations in Steps 2 and 3.
|Approximate Time||35 Minutes|
|Pre-requisite Concepts||Graph cosine, sine, and tangent functions; determine the asymptotes, domain, range, and period of cosine, sine, and tangent functions|
|Type of Tutorial||Procedural Development|
|Key Vocabulary||cotangent, cotangent functions, graphs of functions|