Question

Graph each function. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. $$ y=-3 \cot x $$

Short answer

The function \(-3 \text{cot}(x)\) has a domain \{ x | x ≠ k\text{\textpi}, k \text{is an integer} \} and a range of all real numbers.

Step-by-step solution

Understand the cotangent function

The cotangent function, \(\text{cot}(x)\), is the reciprocal of the tangent function. Its general shape consists of repeating patterns, or cycles, which are vertical asymptotes.

Determine the period and key points

The period of \(\text{cot}(x)\) is \(\text{\textpi}\) because \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\). For \(-3 \text{cot}(x)\), the period remains \(\text{\textpi}\). The key points occur where \(\text{sin}(x) = 0\), causing vertical asymptotes, and where \(\text{cot}(x) = -3, 0, 3\).

Apply the transformation

Multiplying by \(-3\) vertically stretches and reflects the graph of \(\text{cot}(x)\). The basic shape of \(\text{cot}(x)\) is then stretched by a factor of 3 and flipped.

Sketch at least two cycles

Plot key points and vertical asymptotes for two cycles. Use the interval \([0, 2\text{\textpi}]\) for simplicity, knowing a single cycle is from \((0, \text{\textpi})\). Mark points where \(\text{cot}(x) = -3,0,3\), and consequently reflect these points for \(-3 \text{cot}(x)\). The vertical asymptotes are at \(x = 0, \text{\textpi}, 2\text{\textpi}\).

Determine the domain and range

The domain of \(-3 \text{cot}(x)\) excludes points where \(\text{sin}(x) = 0\), i.e., at \{ x | x ≠ k\text{\textpi}, k \text{is an integer} \}. The range comprises all real numbers because \(\text{cot}(x)\) can take any value from \(-\text{\textinfty}\) to \(\text{\textinfty}\).

Key concepts

Cotangent Function

The cotangent function, often written as \text{cot}(x)\, is the reciprocal of the tangent function. In simpler words, if you know the tangent function, the cotangent is just \frac{1}{\text{tan}(x)}\. The cotangent graph has a pattern that repeats, known as cycles. Each cycle shows a full repeat of the curve's behavior.
Knowing the shape of the cotangent graph is important. Key points include: where the value of \text{sin}(x) = 0\ causes vertical asymptotes and where \text{cot}(x)\ touches the x-axis. When graphing \(y = -3 \text{cot}(x)\), remember that the \'-'sign reflects the graph and the \3\ stretches it vertically.

Period of Trigonometric Functions

The period of a function is the length it takes for the graph to repeat its pattern. For the cotangent function, \text{cot}(x)\, this period is \text{\textpi}\.
This means every \(\text{\textpi}\) units on the x-axis, the graph will look the same again.
When you multiply \text{cot}(x)\ by any value, like \-3\, the period doesn't change. So, for \(y = -3 \text{cot}(x)\), the period remains \text{\textpi}\. Always remember this when graphing cotangent or any trigonometric function.

Vertical Asymptotes

Vertical asymptotes are lines where the graph gets infinitely close but never touches. For \text{cot}(x)\, these asymptotes occur where \text{sin}(x) = 0\.
On your graph, these points are crucial because they help shape the way the function looks. For \(y = -3 \text{cot}(x)\), you’ll find vertical asymptotes at \(x = 0, \text{\textpi}, 2\text{\textpi},...\) and any multiples of \text{\textpi}\.
This is because \text{sin}(x) = 0\ at those points, creating gaps in the graph.

Domain and Range

The domain is all possible input values (x-values) for the function. For \(y = -3 \text{cot}(x)\), the domain includes all real numbers, except where vertical asymptotes are located. So, \(x e k\text{\textpi}\) where \k\ is an integer.
To find the range (possible output values), it’s a bit easier. The range of \text{cot}(x)\ includes all real numbers because \text{cot}(x)\ can be any value from \-\infty\ to \+\infty\. This means for \(y = -3 \text{cot}(x)\), the range also covers all real numbers. Simplifying, think of the domain as where you can\'t draw, and the range as all heights the graph can reach.