Problem 44 sketch the graph of the function... [FREE SOLUTION]

91影视

Calculus an Applied Approach

Ron Larson, David C, Falvo

$Math Studyset 91影视 Explanations$ Math

8 Edition

Chapter 8: Problem 44

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

Short Answer

Expert verified

The graph of $y=sec\(\pi x$\) is a series of u- shaped curves. The functions are above the x-axis between 0 and $\pi/2$, below the x-axis between $\pi/2$ and $\pi$, with vertical asymptotes at x=$\pi/2$ and x=$3\pi/2$. The pattern repeats after every $2\pi$.

Step by step solution

Identifying Key Points of Cosine Function

For graphing the secant function, we should first identify the key points of the cosine function, $y=cos(\pi x)$. We note that cosine function has a period of $2\pi$, peak value at $y=1$, and minimum value at $y=-1$. Therefore, key points that represent one period of $y=cos(\pi x)$ could be as following: (0,1), $(1/2,0)$, (1,-1), $(3/2,0)$, and (2,1). Since, we have $\pi$ in our function, this stretches horizontal axis by a factor of $\pi$. So, we will multiply all x values by $\pi$. After doing this, our modified key points become: $(0,1)$, $(\pi/2, 0)$, $(\pi, -1)$, $(3\pi/2, 0)$, and $(2\pi, 1)$.

Determine Undefined Points

Many points in the cosine function are not included in the secant function. This is because sec(x) is undefined where cos(x)=0. From our key points $(\pi/2, 0)$ and $(3\pi/2, 0)$, we see that at these x-values cosine equals zero and hence secant would be undefined. Therefore, we would add vertical asymptotes at x=$\pi/2$ and x=$3\pi/2$.

Sketching the Graph

We already have our key points and asymptotes, so we can start by plotting the asymptotes at x=$\pi/2$ and x=$3\pi/2$. Then we plot our key points, noting that sec(x) should follow the general shape of the cos(x) graph but should 'bounce' off the x-axis where cos(x) equals zero. Between 0 and $\pi/2$, the graph should be above the x-axis while between $\pi/2$ and $\pi$, it should be below. This pattern repeats after $2\pi$ which accounts for the periodicity of the function.

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Key Concepts

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

Understanding the Cosine Function

The cosine function, represented as $y = \cos(\theta)$, is a fundamental trigonometric function. It creates a wave-like curve when graphed. This function is continuous, oscillating between maximum and minimum values.

**Maximum Value:** 1
**Minimum Value:** -1
**Key Intervals:** Starts at (0,1), hits 0 at $\pi/2$, reaches -1 at $\pi$, back to 0 at $3\pi/2$, and returns to 1 at $2\pi$.

In the problem, $y = \cos(\pi x)$, the graph is horizontally stretched by a factor of $\pi$. Therefore, each key point is multiplied by $\pi$, making it crucial to recognize the new positions on the x-axis for graphing purposes.

Exploring the Secant Function

The secant function, denoted as $y = \sec(x)$, is the reciprocal of the cosine function. Where $y = \cos(x)$ equals zero, $y = \sec(x)$ is undefined.

**Formula:** $y = \sec(\theta) = \frac{1}{\cos(\theta)}$
**Undefined Points:** Occur where $\cos(x) = 0$; these points cause vertical asymptotes.

In our example, these undefined points are at $x = \pi/2$ and $x = 3\pi/2$. This necessitates drawing vertical asymptotes at these intervals, indicating where the function tends toward infinity.

Understanding Vertical Asymptotes

Vertical asymptotes represent values for which a function isn't defined, causing the graph to sharply rise or fall toward positive or negative infinity. These are crucial for sketching functions like the secant due to their undefined points.

For $y = \sec(\pi x)$, the cosine component equals zero at $x = \pi/2$ and $x = 3\pi/2$, creating asymptotes. As the function nears these x-values, it skyrockets or plunges, illustrating the secant's unique properties.

Periodicity of Trigonometric Functions

Periodicity is a key property of trigonometric functions, indicating that they repeat values in regular intervals. This characteristic forms the basis for predicting and plotting the behavior of these functions.

**Default Cosine Period:** $2\pi$
**Effect of Transformation:** When transformed as $\cos(\pi x)$, the period is reduced to 2 due to the $\pi$ multiplication.

This reduced period means the graph repeats its pattern every 2 units along the x-axis. Understanding periodicity helps to accurately predict and draw the extended graph of trigonometric functions over any specified domain.

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91影视

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

Short Answer

Step by step solution

Identifying Key Points of Cosine Function

Determine Undefined Points

Sketching the Graph

Key Concepts

Understanding the Cosine Function

Exploring the Secant Function

Understanding Vertical Asymptotes

Periodicity of Trigonometric Functions

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