Question

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

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

2021-06-18

$$\displaystyle{y}={3}{x}^{{{2}}}$$
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 $$\displaystyle{3}{x}^{{{2}}}$$