Question

WRITING Explain how to graph a function of the form \(y=a \sec b x\).

Short answer

To graph \(y= a \sec b x\), the amplitude and period need to be determined first, which are |a| and \(2π / |b|\) respectively. The next step involves finding the vertical asymptotes, which occur where \( \cos bx = 0 \) i.e at \( x = \frac{π}{2b} + \frac{nπ}{b} \); n is an integer. Lastly, the graph is sketched by drawing 'U' or 'n' shaped curves between the asymptotes which touch the y-axis at the amplitude value, repeating the pattern every period.

Step-by-step solution

Identify the Amplitude and Period

The amplitude is given by the absolute value of 'a'. The period can be found using the equation \( 2π / |b| \). These aspects will help in understanding how stretched or compressed the graph is vertically and horizontally.

Find the Vertical Asymptotes

The vertical asymptotes of the secant function occur at points where \( \cos bx = 0 \). This occurs at \( bx = \frac{π}{2} + nπ \) where n is any integer. Therefore, the vertical asymptotes here will be at \( x = \frac{π}{2b} + \frac{nπ}{b} \). These will be used to understand the undefined regions of the secant function.

Plot the Asymptotes

Start by plotting the vertical asymptotes on the graph. These will appear at regular intervals based on the period calculated in Step 1.

Sketch the Graph

Starting at a maximum or minimum point on the y-axis, draw curves between the asymptotes, resembling the shape of a 'U' or 'n'. The graph should touch the y-axis at the maximum or minimum, which is equal to the absolute value of amplitude 'a'. As the function is periodic, this pattern will repeat every period.

Key concepts

Secant Function

The secant function, often denoted as \( \sec{x} \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, expressed as \( \sec{x} = \frac{1}{\cos{x}} \). This means that wherever the cosine is zero, the secant function will have a vertical asymptote due to division by zero. The secant function, like all trigonometric functions, is periodic, repeating its pattern of maximum and minimum curves between vertical asymptotes.

• The secant graph is characterized by alternating 'U' and 'n' shaped curves.
• It can be transformed using parameters \( a \) and \( b \) in the equation \( y = a \sec bx \).
• These transformations affect the graph's amplitude and period.

Understanding the secant function is crucial for recognizing its behavior and predicting where its vertical asymptotes will occur.

Amplitude

In the context of trigonometric functions, amplitude refers to the vertical stretching or compressing of the graph. However, it's important to note that the amplitude is not traditionally used for the secant function the same way it's used for sine or cosine because secant does not oscillate within a fixed range.
Instead, the coefficient \( a \) in the function \( y = a \sec bx \) influences the distance from the midline to the maximum or minimum point of the graph's curves.

• If \( a \) is positive, the secant graph opens upwards.
• If \( a \) is negative, the graph opens downwards.

This scaling factor ensures that the peaks and troughs of the secant graph are vertically stretched or compressed according to the absolute value of \( a \).

Period

The period of a trigonometric function refers to the length of one complete cycle of the graph. For the secant function \( y = a \sec bx \), the period is determined by the value \( b \), where the period \( T \) is calculated as \( \frac{2\pi}{|b|} \).

• The smaller the value of \( b \), the wider the period, meaning the graph's patterns stretch out more horizontally.
• A larger \( b \) compresses the cycles, closing the distance between the same points of consecutive cycles.

Understanding the period helps in predicting where each cycle of the pattern will start and end, allowing for precise placement of curves and vertical asymptotes on the graph.

Vertical Asymptotes

Vertical asymptotes are key features of the secant graph. They occur at values of \( x \) where the cosine function equals zero since the secant is the reciprocal of cosine. This results in undefined values, marked by lines where the secant function shoots towards infinity.
For the function \( y = a \sec bx \), vertical asymptotes occur when \( \cos(bx) = 0 \). Mathematically, this can be expressed as \( bx = \frac{\pi}{2} + n\pi \), simplifying to \( x = \frac{\pi}{2b} + \frac{n\pi}{b} \), where \( n \) is any integer.

• Asymptotes are plotted repeating every half-period along the x-axis.
• The presence of these asymptotes divides the graph into segments, guiding how the secant curves should be drawn.

Recognizing where vertical asymptotes occur is essential for accurately sketching the secant function, highlighting the parts of the graph that stretch towards infinity.