Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. \(f(x)=\arctan \frac{x}{4}\)

Step-by-Step Solution

Answer

The graph of \(f(x)=\arctan \frac{x}{4}\) is a horizontal stretch of the basic \(\arctan(x)\) graph by a factor of 4. It approaches y=\(-\frac{\pi}{2}\) as x approaches \(-\infty\) and approaches y=\(\frac{\pi}{2}\) as x approaches \(\infty\). The steepest part of the graph occurs at x=4.

Step 1: Recognize the Function Type

The given function is an arctangent function which is the inverse of the tangent function. It has a general structure of \(f(x) = \arctan(x)\). But here, instead of \(x\), we have \(\frac{x}{4}\). This represents a horizontal scale of the basic \(\arctan(x)\) graph by a factor of 4.

Step 2: Understand the Scaling of the Function

The scaling of \(\frac{x}{4}\) horizontally stretches the basic \(\arctan(x)\) graph by a factor of 4. This will not change the range of the function as \(\arctan(x)\) has a fixed range between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). But the x-values at which it reaches these extremes will be quadrupled.

Step 3: Graphing the Function

With the graphing utility, input the function \(f(x)=\arctan \frac{x}{4}\) and graph it. The graph will resemble the basic \(\arctan(x)\) graph but stretched horizontally by a factor of 4. It will start from the bottom left, increasing slowly and approaching y=\(-\frac{\pi}{2}\), then rise more steeply before approaching y=\(\frac{\pi}{2}\) and increasing slowly again and continue to the top right. Because of the x/4 term, the steepest part of the graph will now occur at x=4, and not at x=1 as in the basic \(\arctan(x)\) graph.

Key Concepts

Inverse Trigonometric Functions

Understanding inverse trigonometric functions is like having a road map for a complex road system. Just as the basic trigonometric functions of sine, cosine, and tangent give us the ratio of sides in a right-angled triangle, the inverse functions guide us back from the ratio to the angle measurement. In our case, the arctangent function, denoted as \(\arctan(x)\), takes a ratio and returns an angle whose tangent is that ratio.

For example, if we know that the tangent of an angle is 1, then by using the arctangent function, we can determine that angle to be \(\frac{\pi}{4}\), since \(\tan(\frac{\pi}{4})=1\). This function is essential as it allows us to work backwards in trigonometry, finding angle measures when we have only the ratios of the sides. Interestingly, the arctangent has a range from \(\-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), meaning it can only yield angles in the first and fourth quadrants, which is particularly relevant when interpreting the output of the function in various contexts.

Horizontal Stretching of Graphs

When graphing certain functions, the effect of horizontal stretching can significantly alter the shape of the graph without changing the overall behavior. Imagine it like pulling a rubber band sideways. The middle thins out, but the ends stay in place. With functions, this stretching is achieved by multiplying the variable by a fraction, in our exercise \(\frac{x}{4}\). This transformation can 'slow down' the journey of the graph as it moves horizontally across the coordinate plane.

Why Does Stretching Occur?

Stretching a graph horizontally occurs when we divide the x-variable within the function by a factor greater than 1. For the arctangent function \(f(x)=\arctan(\frac{x}{4})\), every x-value is effectively quartered, meaning that what the function would normally do at \(x=1\), it will now do at \(x=4\). In essence, the input races four times slower towards the extremes of its range, delaying the moment it reaches its asymptotic behavior.

Graphing Utility Use in Algebra

In the digital age, graphing utilities are invaluable tools for visualizing mathematical concepts. The ability to promptly plot functions by simply entering an equation allows for immediate visual feedback on the behavior of algebraic expressions. For students particularly, this translates into a better grasp of the function's intricacies – peaks, asymptotes, and stretches.

Furthermore, graphing utilities often provide features to manipulate the graph dynamically, offering an interactive learning experience. For example, observing the effect of horizontal stretching by comparing the graphs of \(\arctan(x)\) and \(\arctan(\frac{x}{4})\) side-by-side reinforces understanding. Students can witness the change in steepness and stretching and thereby deepen their comprehension of the impact of algebraic transformations on the graph of a function.