Find an appropriate graphing software viewing window for the...

Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. \begin{equation} f(x)=\frac{x^{2}-1}{x^{2}+1} \end{equation}

Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.
\begin{equation}
f(x)=\frac{x^{2}-1}{x^{2}+1}
\end{equation}

Key Concepts

Graphing Viewing Window

A graphing viewing window specifies the portion of the coordinate plane that the software displays. Selecting an appropriate window involves choosing x- and y-boundaries that reveal the key features of a function, such as intercepts, asymptotic behavior, and overall trends. This is a crucial step when using graphing software because an improperly chosen window may obscure important characteristics of the graph.

Rational Functions

Rational functions are functions expressed as the ratio of two polynomials. They often exhibit specific behaviors such as holes, vertical asymptotes, and horizontal asymptotes. Understanding the structure of rational functions aids in predicting their general shape and behavior, which is necessary when selecting a window that adequately captures both the local and global features of the graph.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity. For rational functions where the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. This concept is key in understanding the end behavior of the graph and ensuring that the viewing window reveals how the function behaves at extreme values of x.

Transcript

00:01 So, in this problem, we are given this function, it's a quotient function, and we are asked to find an appropriate viewing window for it to grab.

00:20 So what does the mean by appropriate? it means it should give a picture of the overall behavior of the function.

00:30 So with that in mind, there is more than one correct choice, but there are definitely incorrect choices.

00:37 And those are the ones with this important aspects of the function.

00:41 So if there's anything interesting about this function, if there's a curve, if there is a zero, if there is an asymptote, it should show on the viewing number.

00:56 Right, so on this kind of problem, my approach is first to graph it in this window of minus 10 to 10 by the same minus 10 to 10.

01:18 So what this means, these specifications mean is this first interval, it will correspond to the x -isputable.

01:29 And the second interval will correspond to the y -applicense.

01:34 So let's go ahead and graph this f -fx in this window.

01:39 So i did that and it got this graph, which looks pretty good.

01:46 It is a good first guess.

01:50 It's a good first guess.

01:55 So what can we say about it? so this seems to be the interesting part of it.

02:06 And on the inside end point, it seems to be going into some kind of horizontal asymptote.

02:18 And yeah, i think those are the key aspects and they're in this viewing window, so i think it's good enough...

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