Use a graphing utility to sketch a graph of each function....

Use a graphing utility to sketch a graph of each function. Use trial and error to find a graphing window so that your graph represents the local and global behavior of the function. Include the $x$ and $y$ ranges of your window in your answer.
$$
f(x)=\frac{2 x^{2}-2}{x^{2}-3 x-5}
$$

Key Concepts

Local and Global Behavior

Local behavior refers to the characteristics of a function in a small region, such as near intercepts or local extrema, whereas global behavior describes the overall trend of the function across the entire domain. Understanding both aspects is essential for accurately capturing the function's complete graphical representation.

Graphing Window Selection

Selecting a graphing window involves choosing appropriate x and y ranges that capture both local and global behaviors of the function. A well-chosen window displays key features such as intercepts, asymptotes, and changing trends, ensuring that the overall behavior of the function is clearly visible.

Graphing Utilities

Graphing utilities are software tools used to visually represent functions on coordinate systems. They are especially useful for complex functions, allowing users to adjust the viewing window to capture key features like intercepts, asymptotes, and regions where the function changes behavior.

Rational Functions

Rational functions are functions defined by the ratio of two polynomials. They often contain points of discontinuity where the denominator is zero, and they can exhibit unique features such as vertical asymptotes and potential holes in their graphs.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually reaches. Vertical asymptotes occur where the denominator of a rational function is zero, while horizontal or oblique asymptotes describe the end behavior of the function as x approaches infinity or negative infinity.

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