Write two different rational functions whose graphs have the same end behavior as the graph of $$y=3x^2$$

nicekikah 2021-06-17 Answered
Write two different rational functions whose graphs have the same end behavior as the graph of $$y=3x^2$$

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unett
Answered 2021-06-18 Author has 20624 answers

\(\displaystyle{y}={3}{x}^{{{2}}}\)
Pick some factor to multiply top and bottom
\(\displaystyle{y}={3}{x}^{{{2}}}\cdot{{\lbrace}}{\frac{{{x}+{2}}}{{{x}+{2}}}}={\frac{{{3}{x}^{{{3}}}+{6}{x}^{{{2}}}}}{{{x}+{2}}}}\}\)
This adds a "hole" at \(x=-2\) simce that would be undefined, but otherwise behaves exactly like \(\displaystyle{y}={3}{x}^{{{2}}}\) because of course the factors cancel out.
2. Pick another factor to multiply top and bottom
\(\displaystyle{y}={3}{x}^{{{2}}} \cdot {{\lbrace}}{\frac{{{x}-{5}}}{{{x}-{5}}}}={\frac{{{3}{x}^{{{3}}}-{15}{x}^{{{2}}}}}{{{x}-{5}}}}\}\)
"Hole" at x=5, but otherwise the same as \(\displaystyle{y}={3}{x}^{{{2}}}\).
A simpler option could be to just add some constant to \(\displaystyle{3}{x}^{{{2}}}\)

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