Problem 91 Use a graphing utility to graph ... [FREE SOLUTION]

Chapter 4: Problem 91

Use a graphing utility to graph the function. $$ f(x)=2 \arccos (2 x) $$

Short Answer

Expert verified

The graph of the function $f(x)=2 \arccos (2 x)$ is a vertically stretched (by a factor of 2) and horizontally compressed (by a factor of 1/2) version of the base $\arccos(x)$ function. The range is [0, 2$\pi$] and the domain is -1 鈮� x 鈮� 1.

Step by step solution

Understand the Base Function

First, understand the base function. Here, the base function is the arccosine function, denoted as $\arccos(x)$. It is the inverse of the cosine function, and its graph ranges from 0 to $\pi$, inclusively. For real numbers, its domain is -1 鈮� x 鈮� 1.

Apply the Transformation

The given function has a vertical stretch and a horizontal compression, indicated by the coefficients of 2 on the arccos(x) and on x. The '2' outside the function will vertically stretch the graph by a factor of 2, and the '2' inside the function will horizontally compress it by a factor of 1/2.

Graph the Function

Next, use a graphing utility to graph $f(x)=2 \arccos (2 x)$. The shape of the graph will somewhat resemble a 'U' stretched vertically and compressed horizontally. Since the arccosine function's range is [0, $\pi$], this graph's range will be [0, 2$\pi$] after being vertically stretched. The domain remains -1 鈮� x 鈮� 1, as this does not shift or stretch horizontally or vertically.

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Key Concepts

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

Arccosine Function

The arccosine function is a fascinating element of trigonometry. It is written as $ \arccos(x) $ and serves as the inverse of the cosine function. This means it takes a value for cosine and finds the corresponding angle. For the arccosine function, angles are represented in radians.

Its range is limited to [0, $ \pi $], making it unique among trigonometric functions. This ensures it gives a single, unique angle for every valid input. Remember, for real numbers, the domain is $-1 \leq x \leq 1$. Outside of this range, the arccosine is not defined in the real number system.

The graph of $ \arccos(x) $ starts at $x = -1$ at $y = \pi$, sloping down to $y = 0$ at $x = 1$.
This descent form ensures no overlap, keeping the function simple and easy to interpret.

Vertical Stretch

A vertical stretch transforms the graph by extending it along the y-axis. In the example function $ f(x) = 2 \arccos(2x) $, the coefficient "2" outside the arccosine function causes this effect.

Imagine pulling the graph upwards, making all y-values twice as large as in the original function $ \arccos(x) $.

For example, a y-value of $\frac{\pi}{2}$ becomes $\pi$ after the stretch. This changes the graph鈥檚 range from [0, $ \pi $] to [0, 2$ \pi $].
The "U" shape of the $ \arccos(x) $ graph is accentuated as it stretches vertically.

A vertical stretch makes the arccosine function appear taller, without altering its fundamental shape or the x-values.

Horizontal Compression

Horizontal compression alters the graph by squeezing it along the x-axis. In the function $ f(x) = 2 \arccos(2x) $, the expression "2x" inside the $ \arccos $ initiates this transformation. It effectively compresses the graph horizontally by a factor of $\frac{1}{2} $.

Think of this as pushing the graph's sides inwards, making it look narrower. The domain remains unchanged at $-1 \leq x \leq 1$, but the distance between the critical points diminishes.

This alteration affects how quickly the function transitions from its start at $ x = -1 $ to its end at $ x = 1 $.
On the compressed graph, it reaches important heights faster than the original $ \arccos(x) $.

The combination of horizontal compression and vertical stretch gives the graph a distinctly modified appearance.

Graphing Utility

Graphing utilities, such as calculators or software, make working with complex functions easier. They allow you to visualize graphs of functions like $ f(x) = 2 \arccos(2x) $ without sketching them manually.

These tools handle complex calculations, apply transformations on graphs, and show results instantly. With a graphing utility, you can:

Clearly see how vertical stretching and horizontal compressions affect the graph.
Explore different transformations by altering coefficients and observing the outcomes.

Using a graphing utility helps deepen your understanding of how transformations change the behavior and appearance of functions, making it an invaluable tool for students and professionals alike.

91影视

Use a graphing utility to graph the function. $$ f(x)=2 \arccos (2 x) $$

Short Answer

Step by step solution

Understand the Base Function

Apply the Transformation

Graph the Function

Key Concepts

Arccosine Function

Vertical Stretch

Horizontal Compression

Graphing Utility

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