Problem 25 - Algebra and Trigonometry

Question

Find the period and graph the function. $$ y=\frac{1}{2} \sec \left(x-\frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

The period of the function is \( 2\pi \). Graph involves vertical asymptotes and U-shaped curves.

Step 1: Understand the Form of the Function

The function given is \( y = \frac{1}{2} \sec \left(x - \frac{\pi}{6}\right) \). The standard form of a secant function is \( y = a \sec(bx - c) \). Here, \( a = \frac{1}{2} \), \( b = 1 \), and \( c = \frac{\pi}{6} \).

Step 2: Determine the Period of the Function

For the secant function, the period is calculated as \( \frac{2\pi}{b} \). Since \( b = 1 \) in this case, the period is \( \frac{2\pi}{1} = 2\pi \).

Step 3: Identify the Phase Shift and Vertical Stretch

The phase shift is calculated as \( \frac{c}{b} = \frac{\pi}{6} \). This means the graph is shifted to the right by \( \frac{\pi}{6} \). The vertical stretch factor is \( a = \frac{1}{2} \), which affects the amplitude of related cosine function, but secant doesn't have an amplitude.

Step 4: Graph Key Points

To graph \( y = \frac{1}{2} \sec \left(x - \frac{\pi}{6}\right) \), first plot the related \( y = \frac{1}{2} \cos \left(x - \frac{\pi}{6}\right) \), identifying x-intercepts, maxima, and minima based on the period \( 2\pi \). Then, plot the vertical asymptotes at points where the cosine value is zero.

Step 5: Draw the Secant Function

Using the cosine function as a reference, sketch the secant function as a series of alternating U-shaped graphs facing upwards and downards based on the periodic gaps dictated by vertical asymptotes from \( 1/2*cos(x - \frac{\pi}{6}) = 0 \).

Key Concepts

Secant Function

The secant function, represented as \( \sec(x) \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, which means \( \sec(x) = \frac{1}{\cos(x)} \). This connection makes the secant function undefined wherever the cosine function is zero.

In trigonometry, the secant function emerges frequently due to its unique properties and behavior on a graph. Unlike sine and cosine functions, secant does not have a maximum or minimum amplitude, primarily due to its vertical asymptotes, where the cosine curve intersects zero.

When working with the function \( y = \frac{1}{2} \sec (x - \frac{\pi}{6}) \):

• The coefficient \( \frac{1}{2} \) represents a vertical stretch factor.


• The expression \( x - \frac{\pi}{6} \) indicates a horizontal translation or phase shift of the graph to the right by \( \frac{\pi}{6} \).

Understanding these transformations is key in successfully graphing and interpreting the secant function.

Graphing Trigonometric Functions

Graphing trigonometric functions involves understanding their behavior over specific intervals and how they transform. For functions like secant, this means first looking at the related cosine function.

Here are the main steps for graphing a secant function such as \( y = \frac{1}{2} \sec(x - \frac{\pi}{6}) \):

• Plot the cosine curve: Start by sketching \( y = \frac{1}{2} \cos(x - \frac{\pi}{6}) \). This helps in locating the x-intercepts, maxima, and minima.


• Identify vertical asymptotes: These occur at points where the cosine function equals zero. The secant graph will never touch these lines.


• Sketch the secant function: Using the cosine curve as a guide, draw a series of U-shaped branches. These branches will mirror the position of where the cosine function is positive above the x-axis and negative below it.

Secant functions usually appear as repeating patterns of U-shapes on a coordinate grid, alternating direction (upwards or downwards) between each vertical asymptote.

Period of Trigonometric Functions

The period of a trigonometric function is the distance over which the function's shape repeats. For a standard secant function, the period corresponds to the period of its counterpart, the cosine function.

In the function \( y = \frac{1}{2} \sec(x - \frac{\pi}{6}) \), calculating the period involves identifying the coefficient \( b \) in the formula \( a \sec(bx - c) \). The typical formula used is \( \frac{2\pi}{b} \).

• In this equation, \( b = 1 \). Therefore, the period is \( \frac{2\pi}{1} = 2\pi \).


• This means every \( 2\pi \) interval, the secant graph will repeat its pattern of vertical asymptotes and U-shaped branches.

Recognizing the period of a trigonometric function helps in efficiently graphing and analyzing its behavior across different phases and shifts. For transformations like phase shifts and vertical stretches, understanding the period is crucial to adjust the graph accordingly.