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Algebra-2

Learners graph the parent cotangent function, as well as the graph of functions that involve cotangent expressions.

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After completing this tutorial, you will be able to complete the following:

- Sketch the graph of the parent cotangent function.
- Sketch the graph of a function that involves a cotangent expression.

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).

- af(x) causes a vertical stretch in the graph of f(x) by a factor of a. If a is negative, then the graph is reflected over the x-axis.
- f(bx) causes a horizontal shrink in the graph of f(x) by a factor of b.
- f(x ? c) causes a horizontal shift by c units to the right for .
- f(x + c) causes a horizontal shift by c units to the left for
- f(x) + d causes a vertical shift upward by d units for .
- f(x) ? d causes a vertical shift downward by d units for .

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.

Step 2:

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 .

Step 3:

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

Step 4:

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 |

Course | Algebra-2 |

Type of Tutorial | Procedural Development |

Key Vocabulary | cotangent, cotangent functions, graphs of functions |