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Chapter 8: Problem 43

sketch the graph of the function. $$ y=2 \sec 2 x $$

Short Answer

Expert verified
The graph of the function \(y = 2 \sec 2x\) consists of series of U-shapes with vertical asymptotes at \(x = \frac{(2n+1)\pi}{4}\) where \(n\) is an integer. The maximum and minimum values occur at the points \(\left(0, 2\right)\) and \(\left(\frac{3\pi}{8}, -2\sqrt{2}\right)\) respectively. The period of the graph is \(\frac{\pi}{2}\). Repeat the pattern to complete the full sketch.

Step by step solution

01

Understand Secant Function

The secant function can be defined as \(y = \sec x = \frac{1}{\cos x}\). It's undefined for \(x = \frac{(2n+1)\pi}{2}\) where \(n\) is an integer because cosine equals zero at these points, and division by zero is undefined. The graph of a secant function has a series of upright U-shapes separated by vertical asymptotes at these undefined points.
02

Determine the period and Key Points

The period of the given secant function \(y = 2 \sec 2x\) is \(\frac{\pi}{B} = \frac{\pi}{2}\). The key points of one period in the interval \(0 \leq x \leq \frac{\pi}{2}\) should be as follows: at \(x = 0, \sec 2x = 2\); at \(x = \frac{\pi}{8}, \sec 2x = \sqrt{2}\), at \(x = \frac{\pi}{4}, \sec 2x\) is undefined (asymptote), and at \(x = \frac{3\pi}{8}, \sec 2x = -\sqrt{2}\).
03

Sketch the Function

Now to graph the function: Mark the asymptotes as dashed vertical lines at \(x = \frac{\pi}{4}\), and \(x = \frac{3\pi}{4}\) for one period of \(\frac{\pi}{2}\). Then plot the key points (0, 2), \(\left(\frac{\pi}{8}, 2\sqrt{2}\right)\), \(\left(\frac{3\pi}{8}, -2\sqrt{2}\right)\). Join the points to form upright U-shapes for the graph between the asymptotes. Repeat the same for other periods.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Sketching
Graph sketching is an important skill in understanding functions, particularly trigonometric ones like the secant function. When sketching the graph of a function like \( y = 2 \sec 2x \), you need to pay attention to the function's unique characteristics. Let's break down the steps for graph sketching in this context.
  • Identify the Basic Shape: The secant function produces a series of U-shaped curves between vertical lines, which are asymptotes. These patterns help guide where you'll draw the curves.
  • Determine Key Points: By studying the equation, find significant values (key points) such as maximums, minimums, and asymptotes.
  • Consider the Period: For \( y = 2 \sec 2x \), the period is \( \frac{\pi}{2} \), meaning the U-shapes and asymptotes repeat every \( \frac{\pi}{2} \). It鈥檚 important to draw multiple periods to show the function's repetitiveness.
This structured approach will help in creating a neat and informative graph.
Trigonometric Functions
Trigonometric functions like the secant function are key players in mathematics, often used in modeling periodic phenomena. The function \( \sec x \) is defined as \( \frac{1}{\cos x} \). This definition indicates the secant function's reliance on cosine.
Some points to consider about the secant (\( \sec x \)) function include:
  • Relationship to Cosine: Because secant is the reciprocal of cosine, it inherits many properties of cosine, such as periods and undefined values where cosine is zero.
  • Amplitude and Transformation: In \( y = 2 \sec 2x \), the 2 before the secant affects the amplitude, stretching the graph vertically.
  • Understanding Transformation: The \( 2x \) inside the function tells us that we're dealing with a horizontal transformation, affecting how frequently the secant's repeating pattern occurs. A two-fold increase in frequency means a shortened period, which is \( \frac{\pi}{2} \) instead of the typical \( 2\pi \).
Understanding these attributes will give you a fuller grasp of how the secant function behaves and how it can be manipulated.
Vertical Asymptotes
Vertical asymptotes are critical features in trigonometric graphs, especially for functions like \( y = 2 \sec 2x \). Asymptotes are lines that the function approaches but never actually reaches, creating a visual guide for graphing functions. These occur where the function is undefined.
For the secant function, vertical asymptotes occur because the cosine, the denominator in \( \sec x = \frac{1}{\cos x} \), equals zero.
Key considerations for vertical asymptotes:
  • Location: In \( y = 2 \sec 2x \), the function is undefined at points where \( 2x = (2n+1)\frac{\pi}{2} \). This translates to asymptotes at \( x = \frac{(2n+1)\pi}{4} \).
  • Graphing: When sketching, draw dashed vertical lines where these asymptotes occur. They partition the graph structure, showing where each part of the secant's U-shape must stop and resume on the other side of the asymptote.
  • Effect on Periodicity: Vertical asymptotes help outline the natural period of the secant function. They appear in regular intervals, and knowing their place aids in accurately sketching multiple periods.
This understanding of vertical asymptotes aids not just in plotting secant functions, but in sketching any graph involving rational trigonometric functions.

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