Problem 11 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=-\cot x$$

Step-by-Step Solution

Answer

The period of \( y = -\cot x \) is \( \pi \). Reflect and sketch the graph following the standard cotangent shape, inverted over the x-axis with asymptotes at \( x = 0, \pm \pi, \ldots \).

Step 1: Identify the standard form of the cotangent function

The standard form of the cotangent function is given by \( y = a \cdot \cot(bx - c) + d \). In your exercise, the function is \( y = -\cot x \), which takes the form of the standard cotangent equation with values: \( a = -1 \), \( b = 1 \), \( c = 0 \), and \( d = 0 \).

Step 2: Determine the period of the cotangent function

The period of the cotangent function \( \cot(bx) \) is given by the formula \( \frac{\pi}{b} \). In this function, since \( b = 1 \), the period is \( \frac{\pi}{1} = \pi \).

Step 3: Graph key points of the cotangent function

For \( y = -\cot x \), graphing key points will help. The function \( \cot x \) has vertical asymptotes at \( x=0, \pm \pi, \pm 2\pi, \ldots \) and zeros at \( x=\pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots \). The graph is reflected over the x-axis due to the negative sign in front of \( \cot x \).

Step 4: Sketch the graph

Using the period and key points, plot the asymptotes and curve between them. For \( y = -\cot x \), start with an interval from \( x = 0 \) to \( x = \pi \), placing vertical asymptotes at these points. The curve will start from positive infinity at \( x = 0 \), cross through zero at \( x = \frac{\pi}{2} \), and approach negative infinity as \( x \) approaches \( \pi \). Repeat the pattern across other intervals.

Key Concepts

Period of Trigonometric Functions

The period of a trigonometric function refers to the length of the interval over which the function repeats. For the cotangent function, the period is relatively straightforward to determine. The standard form of a trigonometric function like cotangent is given by \( y = a \cdot \cot(bx - c) + d \).
For the function \( y = \cot(bx) \), the period is calculated using the formula \( \frac{\pi}{b} \). This means the function repeats every \( \pi \) units along the x-axis. In the exercise with \( y = -\cot x \), since \( b = 1 \), the period is \( \frac{\pi}{1} = \pi \).
This implies that every \( \pi \) units, the graph repeats its pattern of ascending and descending. Understanding this concept is crucial as it tells us how often the full wave or pattern of the function occurs, which is essential when sketching graphs or analyzing trigonometric patterns.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their key features, such as asymptotes, zeros, and the points where they hit maximum and minimum values. For cotangent, these features help determine the shape of the graph.

• Vertical asymptotes: Lines where the function approaches infinity or negative infinity. For cotangent, vertical asymptotes occur where the sine component is zero, such as at \( x = 0, \pm \pi, \pm 2\pi, \ldots \).
• Zeros: Points where the cotangent equals zero. These occur where the cosine is zero, hence at \( x = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots \).

For \( y = -\cot x \), the graph begins at positive infinity, crosses zero at \( x = \frac{\pi}{2} \), and goes to negative infinity as it approaches \( x = \pi \). The negative sign implies a reflection over the x-axis, altering the typical cotangent hills and valleys.
Understanding these key features and how the function repeats over its period helps create an accurate and informative graph.

Properties of the Cotangent Function

The cotangent function, represented by \( \cot x \), has unique properties that distinguish it from other trigonometric functions. Here, focusing on its behavior and characteristics can enhance our understanding.

• Domain: Cotangent is undefined where sine is zero, meaning at multiples of \( \pi \): \( x = 0, \pm \pi, \pm 2\pi, \ldots \).
• Range: The outputs of \( \cot x \) include all real numbers from negative infinity to positive infinity.
• Asymptotes: The function has vertical asymptotes at its undefined points, creating a pattern of repeated slopes between these lines.
• Symmetry: Cotangent is an odd function, meaning it is symmetric about the origin. This symmetry remains even when multiplied by a negative value, like in \( y = -\cot x \).

Remember, the cotangent function decays towards its vertical asymptotes and crosses zero at its midpoint within each period. Understanding these properties provides insights into analyzing the behavior of the cotangent function under transformations and reflections.