Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The graph for the function \(y = 2 \arccos x\) is a vertically stretched version of the \(\arccos x\) graph, not affecting its domain. To draw it, follow a similar shape to the original \(\arccos x\) graph, but with all y-values doubled.

Step 1: Understanding the base function

The base function in this exercise is \(\arccos x\). The function \(\arccos x\) is the inverse function of the \(\cos x\) function, restricted to \(0 \le x \le \pi\). The domain of \(\arccos x\) is \([-1, 1]\), and its range is \([0, \pi]\). Its graph is a reflection of the \(\cos x\) graph over the line \(y = x\), shifted \(\pi/2\) units up.

Step 2: Identifying the transformation involved

The function given is \(y = 2 \arccos x\). The factor 2 is a vertical stretch of the base function \(\arccos x\). It is also important to remember that this transformation does not affect the domain of the function.

Step 3: Creating the graph

Use a graphing utility to graph the function \(y = 2 \arccos x\). Make sure to set the window to include at least the domain \([-1, 1]\) and the range \([0, 2\pi]\).

Key Concepts

arccos function

The arccos function, denoted as \(\arccos x\), is the inverse of the cosine function. This means, instead of finding the cosine of an angle, you're finding the angle whose cosine is a given number. It's the opposite process of the usual trigonometric function.

• The domain of \(\arccos x\) is all values between -1 and 1, inclusive. This is because cosine values for any angle always fall between these two numbers.
• The range of \(\arccos x\) is from 0 to \(\pi\) radians. This range is due to the restriction needed for the function to remain a true inverse function.

Understanding \(\arccos x\) starts by realizing that it tells us the measure of an angle in a right triangle, based on the triangle's adjacent side length and hypotenuse length. Graphically, \(\arccos x\) can be visualized as a reflection of the cosine function across the line \(y = x\), and then adjusted to fit the specific domain and range requirements.

graph transformations

In the equation \(y = 2 \arccos x\), the 2 in front signifies a vertical stretch of the function \(\arccos x\). When you multiply the function's output by a number greater than 1, it stretches the graph vertically.

• Vertical stretches change how fast a function rises or falls. In \(2 \arccos x\), angles become twice as large as those in \(\arccos x\) at every point over the domain.
• The transformation doesn't alter the domain (which remains \([-1, 1]\)).

If we consider the graph of \(\arccos x\) and multiply its outputs by 2, you will notice the y-values are scaled upwards. Imagining this on a graph can help see how stretching impacts visual representation.

graphing utility

Using a graphing utility can significantly aid in visualizing transformations and functions like \(y = 2 \arccos x\). Graphing utilities can provide precision and insight that might otherwise be difficult to capture manually.

• Setting the proper window view is crucial. For \(y = 2 \arccos x\), ensure your graph shows at least the domain \([-1, 1]\) because that's where the function is defined.
• The range should be increased to show results in \([0, 2\pi]\), accommodating the vertical stretch of the function.
• Graphing utilities often have the capability to plot the original and modified functions simultaneously, helping you visually compare changes.

With these tools, seeing the transformations and understanding their effects on graphs becomes intuitive, helping reinforce conceptual understanding through visual evidence.

function domain and range

Every mathematical function has a domain and a range. The domain is all the possible input values (\(x\)-values) the function can accept, while the range is all possible output values (\(y\)-values) the function can produce.

• For \(\arccos x\), the domain is \([-1, 1]\). This range of inputs is due to how the cosine function behaves since it can yield results only from these values.
• The range of \(\arccos x\) is \([0, \pi]\), meaning it gives angles from 0 to \(\pi\) radians. With \(2 \arccos x\), this range is stretched to \([0, 2\pi]\) due to the vertical stretching.

Understanding domain and range is essential. They tell us the limits of what a function can do and define its behavior. When performing transformations, knowing how these concepts apply helps ensure accuracy in both algebraic and graphical depictions. Always validate these properties when analyzing or graphing functions.