Problem 39 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each of the given expressions. $$f(x)=\arctan x-\frac{\pi}{2}$$

Step-by-Step Solution

Answer

The graph of given function \( f(x)=\arctan x-\frac{\pi}{2} \) will be similar to the standard \( y= \arctan x \) but shifted downwards by \( \frac{\pi}{2} \), with new horizontal asymptotes at \( y = -\pi \) and \( y = 0 \).

Step 1: Sketch the base function

To begin with, the graph of \( y = \arctan x \) is needed. This function increases from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) as \( x \) goes from negative infinity to positive infinity. The graph has horizontal asymptotes at \( y = -\frac{\pi}{2} \) and \( y = \frac{\pi}{2} \). The graph is symmetrical with the origin, meaning that it mirrors along the origin.

Step 2: Adjust for the subtraction

Now that we have sketched the graph of the base function, it should be adjusted to reflect the subtraction of \( \frac{\pi}{2} \). This means the entire graph will simply be shifted down by \( \frac{\pi}{2} \) units. No changes will occur for the shape of the graph, it only translates downwards.

Step 3: Observe the new asymptotes

After shifting down, notice the new horizontal asymptotes. The older asymptotes at \( y = -\frac{\pi}{2} \) and \( y = \frac{\pi}{2} \) shifts to \( y = -\pi \) and \( y = 0 \) respectively after subtracting \( \frac{\pi}{2} \).

Key Concepts

Arctangent Function

The arctangent function, denoted as \( \arctan x \), is an essential trigonometric function. It is the inverse of the tangent function, which means it gives the angle whose tangent is a particular value. This function outputs values of angles between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \).

Unlike the sine or cosine functions, \( \arctan x \) does not repeat its values; hence, it doesn't form a periodic wave. Instead, it presents an S-shaped curve that increases continuously. The distinct characteristic of the arctangent function is that it passes through the origin, smoothly increasing as \( x \) moves from negative infinity to positive infinity. It is important to understand this behavior as it affects how transformations, such as shifts, influence the graph.

Asymptotes

Asymptotes are crucial to understanding the behavior of rational and transcendental functions like \( \arctan x \). They are lines that the graph approaches but never actually reaches. For \( \arctan x \), there are two horizontal asymptotes at \( y = -\frac{\pi}{2} \) and \( y = \frac{\pi}{2} \). These lines represent the limit of the function as \( x \) approaches positive or negative infinity.

When a function is transformed, its asymptotes might move accordingly. In the original exercise, the function \( f(x) = \arctan x - \frac{\pi}{2} \) involves moving the arctangent curve downwards by \( \frac{\pi}{2} \) units, which subsequently shifts the asymptotes to \( y = -\pi \) and \( y = 0 \). It's essential to observe these movements to accurately sketch transformed graphs.

Function Transformation

Function transformation refers to altering a graph's appearance by translating, stretching, compressing, or reflecting it. In the context of trigonometric functions, these transformations can predict changes in the graph's position or shape without altering the function's inherent properties.

The expression \( f(x) = \arctan x - \frac{\pi}{2} \) showcases a vertical shift. Every point on the graph of \( \arctan x \) moves down by \( \frac{\pi}{2} \) units. This transformation is simple compared to others like horizontal shifts or scaling, as the function's 'signature' curvature remains intact. When analyzing transformations, observe the type directly affecting whether the graph moves up or down, left or right, and whether asymptotes or intercepts change positions. Recognizing these shifts helps in anticipating the function's overall behavior post-transformation.